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Maths is a critical scientific discipline that is all pervading and highly cross functional. Mathematics finds its applications in almost all facets of human civilisations. Mathematics very closely and strongly resonates with the success achieved through the pages of human history. Maths, with it's latent wisdom, hidden knowledge, implementations, palpable outcomes, and the related ensuing visible manifestations, has left an indelible mark upon all the human evolutionary milestones, viz., pre - historic, ancient, medieval and modern educational systems. Apropos, various global educational systems had been developed, being globally pursued and are still being evolved aggressively at a global level.

The pedagogical approaches, time relevant governance, valid curricular content, compliance guidelines, regulatory frameworks, principle policies, governance frameworks, educational monitoring bodies, review authorities, watchdogs, citizens' feedbacks, time tested populist policies, sociology studies, societal necessities, ease of learning, social studies SST, conducive learning environment, ease of comprehensibility, rating bodies, real life learning applicability, learning oriented frameworks and learning support procedures employed in any given educational environment are primarily dictated by the objectives and goals of the applicable and particular educational systems.

Mathematics teaching methods, approaches and procedures include the following, viz., Classical Maths Pedagogy, Conventional Math approach, Historical Mathematics approach, Discovery oriented Mathematics, Recreational Mathematics, Problems' Outcomes oriented Maths, Standards based Mathematics, Relational Mathematics approach, Rote Learning oriented Maths and much more. In conventional education, rote learning is employed to teach multiplication tables, definitions, formulae, and other areas of mathematics. Rote Mathematical learning is the teaching of mathematical findings, definitions and concepts, via repetition and memorization, usually without meaning or backed by mathematical logic.

Relational Mathematics is another approach towards Math teaching near me. It uses class subjects to comprehend everyday problems and connects the topic to current events. This method emphasises on the various applications of mathematics. It assists students in understanding why they need to study it, as well as in applying it to real - world problems outside of the classroom. Third approach is the Standards based Mathematics. It is a vision for pre - college oriented mathematics education.

Standards based Mathematics is centred on improving students' grasp of mathematical concepts and methods. Fourth approach is the Recreational oriented Mathematics. Here, Mathematical problems that are enjoyable can stimulate students to learn maths. This approach can boost the appreciation of the applications of mathematics. Fourth approach is Mathematical Problem solving. It is used to develop new mathematical knowledge, often by building on pupils' past expertise.

The tasks might range from simple word problems to problems from major mathematical challenges. Fifth approach is New Maths approach. It is a technique of teaching mathematics that emphasise upon various abstract ideas such as set theory, functions and more. Sixth approach is about the Discovery based Maths. It emphasises upon a constructivist style of teaching wherein discovery learning based mathematics is focused upon. It is problem - based or inquiry - based learning and employs open - ended questions, as well as various manipulative materials.

Seventh approach is the Conventional approach to Mathematics. It encompasses the progressive and systematic progression through the hierarchy of various mathematical conceptions, ideas, and processes. The Mathematics instructors must be well-versed in elementary mathematics to deliver the tenets of Conventional Maths. Eighth approach is Computer Based Maths education. It involves the use of computers to teach mathematics through Web based applications or WebApp. Latter have also been developed to assist pupils in learning Maths. Ninth approach is the Computer software based that leverages computing systems, tools and platforms.

Ninth approach uses the capabilities and efficiencies of various computing models, as well as various computing possibilities. With the advancement in technology and the advent of information technology IT revolution, the adoption of computing tools, advanced softwares and Cloud based platforms ( AWS, Azure, GCP, OCI, etc., ) have become the de facto standards. There is no possibility of continuing with traditional approaches, methodologies and related pedagogical approaches.

Maths' applications, cross functional dependencies and the ensuing mathematical treatments are being fully influenced and deeply embedded into such information technology IT frameworks. Various coding governance frameworks, programming initiatives, algorithmic structures, data structures, low level and high level computing languages have mathematics at their foundations.

Basis the challenges and the problems at hand, as well as the quantum of impact that could be quantified or on basis of the potential to be gauged, the symbiosis among math and other cross functional disciplines, including that of computer sciences and engineering, information technology IT, informatics practices IP, Artificial Intelligence AI , Web Applications Web App, Data Science DS , Typography Computer Applications TCA, Python Programming PP , GeoSpatial Technology GeoSpatial, Multi Media Applications MM , etc., are arrived at.

Thus, this approach and the adoption of cutting edge technological landscape are inalienable aspects of mathematics. The symbiosis is now mandatory and inseparable, leading to value addition for the ever evolving human societies, evolving sociological civilisations, microeconomics frameworks, Gender Studies Knowledge Systems HRGS, psychological constructs, Human relationship Studies, macroeconomics frameworks, Historical Knowledge systems, Environmental Studies, Geographical Studies, as well as the political systems, Civics structures existing and anticipated in the near future on our Planet Earth.

All this is achieved through leveraging various computing algorithms, Programming Softwares, tools, Platforms, cloud initiatives, Data Structures Frameworks, Computer Coding languages, data sciences, informatics practices, information and communication technologies ICT, web applications Web App, information technology IT, Computers and Communication Technology CCT, Artificial Intelligence AI Frameworks, Data Science DS Courses, Informatics Practices IP Courses, Information Technology IT Frameworks, Geospatial Technology Information System GIS Frameworks, Multi Media Information MM Frameworks, Web Application Development studies, Coding courses, Python Programming Language, Library Information Systems LIS and other cross - functional disciplines.

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Following is the general outline of Central Board of Secondary Education CBSE / NCERT prescribed syllabus for Standard Mathematics ( Conventional Maths ) Syllabi for Classes 1 - I, 2 - II, 3 - III, 4 - IV, 5 - V ( Classes 1st, 2nd, 3rd, 4th, 5th ) :

What is Long, What is Round, Counting in Groups, How Much Can You Carry, Counting in Tens, Patterns, Footprints, Jugs and Mugs, Tens and Ones, My Funday, Add our Points, Lines and Lines, Give and Take, The Longest Step, Birds Come, Birds Go, How Many Ponytails

Where to Look From, Fun with Numbers, Give and Take, Long and Short, Shapes and Designs, Fun with Give and Take, Time Goes On, Who is Heavier, How Many Times, Play with Patterns, Jugs and Mugs, Can We Share, Smart Charts, Rupees and Paise

Building with Bricks, Long and Short, A Trip to Bhopal, Tick-Tick-Tick, The Way The World Looks, The Junk Seller, Jugs and Mugs, Carts and Wheels, Halves and Quarters, Play with Patterns, Tables and Shares, How Heavy? How Light?, Fields and Fences, Smart Charts, Building with Bricks, Long and Short, A Trip to Bhopal, Tick-Tick-Tick, The Way The World Looks, The Junk Seller, Jugs and Mugs, Carts and Wheels, Halves and Quarters, Play with Patterns, Tables and Shares, How Heavy? How Light?, Fields and Fences, Smart Charts

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Following is the outline of Central Board of Secondary Education CBSE / NCERT prescribed syllabus for conventional Mathematics. This syllabi covers standard Maths for different Classes, viz., 6 - VI, 7 - VII, 8 - VIII, 9 - IX, 10 - X, 11 - XI, 12 - XII ( Classes 6th, 7th, 8th, 9th, 10th, 11th, 12th ) :

Class 6 - VI ( Class 6th )

A Note For The Teachers, Knowing Our Numbers, Whole Numbers, Playing With Numbers, Laying With Numbers, Basic Geometrical Ideas, Understanding Elementary Shapes, Integers, Fractions, Decimals, Data Handling, Mensuration, Algebra, Ratio And Proportion, Symmetry, Practical Geometry, Brain-Teasers

Class 7 - VII ( Class 7th )

Integers, Fractions and Decimals, Data Handling, Simple Equations, Lines and Angles, The Triangle and its Properties, Congruence of Triangles, Comparing Quantities, Rational Numbers, Practical Geometry, Perimeter and Area, Algebraic Expressions, Exponents and Powers, Symmetry, Visualising Solid Shapes

Class 8 - VIII ( Class 8th )

Rational Numbers, Linear Equations in One Variable, Understanding Quadrilaterals, Practical Geometry, Data Handling, Squares and Square Roots, Cubes and Cube Roots, Comparing Quantities, Algebraic Expressions and Identities, Visualising Solid Shapes, Mensuration, Exponents and Powers, Direct and Inverse Proportions, Factorisation, Introduction to Graphs, Playing with Numbers

Class 9 - IX ( Class 9th )

Number Systems :

Irrational Numbers, Real Numbers and their Decimal Expansions, Representing Real Numbers on the Number Line, Operations on Real Numbers, Laws of Exponents for Real Numbers,

Polynomials :

Polynomials in One Variable, Zeroes of a Polynomial, Remainder Theorem, Factorisation of Polynomials, Algebraic Identities,

Coordinate Geometry :

Cartesian System, Plotting a Point in the Plane if its Coordinates are given,

Linear Equations In Two Variables :

Linear Equations, Solution of a Linear Equation, Graph of a Linear Equation in Two Variables, Equations of Lines Parallel to x-axis and y-axis,

Introduction To Euclid’s Geometry :

Introduction, Euclid’s Definitions, Axioms and Postulates, Equivalent Versions of Euclid’s Fifth Postulate,

Lines And Angles :

Basic Terms and Definitions, Intersecting Lines and Non-intersecting Lines, Pairs of Angles, Parallel Lines and a Transversal, Lines Parallel to the same Line, Angle Sum Property of a Triangle,

Triangles :

Congruence of Triangles, Criteria for Congruence of Triangles, Some Properties of a Triangle, Some More Criteria for Congruence of Triangles, Inequalities in a Triangle,

Quadrilaterals :

Angle Sum Property of a Quadrilateral, Types of Quadrilaterals, Properties of a Parallelogram, Another Condition for a Quadrilateral to be a Parallelogram, The Mid-point Theorem,

Areas Of Parallelograms And Triangles :

Figures on the same Base and Between the same Parallels, Parallelograms on the same Base and between the same Parallels, Triangles on the same Base and between the same Parallels,

Circles :

Circles and its Related Terms : A Review, Angle Subtended by a Chord at a Point, Perpendicular from the Centre to a Chord, Circle through Three Points, Equal Chords and their Distances from the Centre, Angle Subtended by an Arc of a Circle, Cyclic Quadrilaterals,

Constructions :

Basic Constructions, Some Constructions of Triangles,

Heron’s Formula :

Area of a Triangle – by Heron’s Formula, Application of Heron’s Formula in finding Areas of Quadrilaterals,

Surface Areas And Volumes :

Surface Area of a Cuboid and a Cube, Surface Area of a Right Circular Cylinder, Surface Area of a Right Circular Cone, Surface Area of a Sphere, Volume of a Cuboid, Volume of a Cylinder, Volume of a Right Circular Cone, Volume of a Sphere,

Statistics :

Collection of Data, Presentation of Data, Graphical Representation of Data, Measures of Central Tendency,

Probability :

Probability – an Experimental Approach,

Proofs In Mathematics :

Mathematically Acceptable Statements, Deductive Reasoning, Theorems, Conjectures and Axioms, What is a Mathematical Proof?,

Introduction To Mathematical Modelling :

Review of Word Problems, Some Mathematical Models, The Process of Modelling, its Advantages and Limitations

Class 10 - X ( Class 10th )

Real Numbers :

Introduction, Euclid’s Division Lemma, The Fundamental Theorem of Arithmetic, Revisiting Irrational Numbers, Revisiting Rational Numbers and Their Decimal Expansions, Summary

Polynomials :

Introduction, Geometrical Meaning of the Zeroes of a Polynomial, Relationship between Zeroes and Coefficients of a Polynomial, Division Algorithm for Polynomials, Summary

Pair of Linear Equations in Two Variables :

Introduction, Pair of Linear Equations in Two Variables, Graphical Method of Solution of a Pair of Linear Equations, Algebraic Methods of Solving a Pair of Linear Equations, Substitution Method, Elimination Method, Cross-Multiplication Method, Equations Reducible to a Pair of Linear Equations in Two Variables

Quadratic Equations :

Introduction, Quadratic Equations, Solution of a Quadratic Equation by Factorisation, Solution of a Quadratic Equation by Completing the Square, Nature of Roots

Arithmetic Progressions :

Introduction, Arithmetic Progressions, nth Term of an AP, Sum of First n Terms of an AP

Triangles :

Introduction, Similar Figures, Similarity of Triangles, Criteria for Similarity of Triangles, Areas of Similar Triangles, Pythagoras Theorem

Coordinate Geometry :

Introduction, Distance Formula, Section Formula, Area of a Triangle

Introduction to Trigonometry :

Introduction, Trigonometric Ratios, Trigonometric Ratios of Some Specific Angles, Trigonometric Ratios of Complementary Angles, Trigonometric Identities

Some Applications of Trigonometry :

Introduction, Heights and Distances

Circles :

Introduction, Tangent to a Circle, Number of Tangents from a Point on a Circle

Constructions :

Introduction, Division of a Line Segment, Construction of Tangents to a Circle

Areas Related to Circles :

Introduction, Perimeter and Area of a Circle — A Review, Areas of Sector and Segment of a Circle, Areas of Combinations of Plane Figures

Surface Areas and Volumes :

Introduction, Surface Area of a Combination of Solids, Volume of a Combination of Solids, Conversion of Solid from One Shape to Another, Frustum of a Cone Statistics :

Introduction, Mean of Grouped Data, Mode of Grouped Data, Median of Grouped Data, Graphical Representation of Cumulative Frequency Distribution Probability :

Introduction, Probability — A Theoretical Approach

Appendix A1 : Proofs in Mathematics

A1.1 Introduction,

A1.2 Mathematical Statements Revisited,

A1.3 Deductive Reasoning,

A1.4 Conjectures, Theorems, Proofs and Mathematical Reasoning,

A1.5 Negation of a Statement,

A1.6 Converse of a Statement,

A1.7 Proof by Contradiction,

A1.8

Appendix A2 : Mathematical Modelling

A2.1 Introduction,

A2.2 Stages in Mathematical Modelling,

A2.3 Some Illustrations,

A2.4 Why is Mathematical Modelling Important?

Class 11 - XI ( Class 11th )

Sets :

Introduction, Sets and their Representations, The Empty Set, Finite and Infinite Sets, Equal Sets, Subsets, Power Set, Universal Set, Venn Diagrams, Operations on Sets, Complement of a Set, Practical Problems on Union and Intersection of Two Sets

Relations and Functions :

Introduction, Cartesian Product of Sets, Relations, Functions

Trigonometric Functions :

Introduction, Angles, Trigonometric Functions, Trigonometric Functions of Sum and Difference of Two Angles, Trigonometric Equations

Principle of Mathematical Induction :

Introduction, Motivation, The Principle of Mathematical Induction

Complex Numbers and Quadratic Equations :

Introduction, Complex Numbers, Algebra of Complex Numbers, The Modulus and the Conjugate of a Complex Number, Argand Plane and Polar Representation, Quadratic Equations

Linear Inequalities :

Introduction, Inequalities, Algebraic Solutions of Linear Inequalities in One Variable and their Graphical Representation, Graphical Solution of Linear Inequalities in Two Variables, Solution of System of Linear Inequalities in Two Variables

Permutations and Combinations :

Introduction, Fundamental Principle of Counting, Permutations, Combinations

Binomial Theorem :

Introduction, Binomial Theorem for Positive Integral Indices, General and Middle Terms

Sequences and Series :

Introduction, Sequences, Series, Arithmetic Progression ( A.P. ), Geometric Progression ( G.P. ), Relationship Between A.M. and G.M., Sum to n terms of Special Series

Straight Lines :

Introduction, Slope of a Line, Various Forms of the Equation of a Line, General Equation of a Line, Distance of a Point From a Line

Conic Sections :

Introduction, Sections of a Cone, Circle, Parabola, Ellipse, Hyperbola

Introduction to Three Dimensional Geometry :

Introduction, Coordinate Axes and Coordinate Planes in Three Dimensional Space, Coordinates of a Point in Space, Distance between Two Points, Section Formula

Limits and Derivatives :

Introduction, Intuitive Idea of Derivatives, Limits, Limits of Trigonometric Functions, Derivatives

Mathematical Reasoning :

Introduction, Statements, New Statements from Old, Special Words/Phrases, Implications, Validating Statements

Statistics :

Introduction, Measures of Dispersion, Range, Mean Deviation, Variance and Standard Deviation, Analysis of Frequency Distributions

Probability :

Introduction, Random Experiments, Event, Axiomatic Approach to Probability

Appendix 1 : Infinite Series

A.1.1 Introduction,

A.1.2 Binomial Theorem for any Index,

A.1.3 Infinite Geometric Series,

A.1.4 Exponential Series,

A.1.5 Logarithmic Series

Appendix 2 : Mathematical Modelling

A.2.1 Introduction,

A.2.2 Preliminaries,

A.2.3 What is Mathematical Modelling

Class 12 - XII ( Class 12th ) - Part - 1 ( I )

Relations and Functions :

Introduction, Types of Relations, Types of Functions, Composition of Functions and Invertible Function, Binary Operations

Inverse Trigonometric Functions :

Introduction, Basic Concepts, Properties of Inverse Trigonometric Functions

Matrices :

Introduction, Matrix, Types of Matrices, Operations on Matrices, Transpose of a Matrix, Symmetric and Skew Symmetric Matrices, Elementary Operation ( Transformation ) of a Matrix, Invertible Matrices

Determinants :

Introduction, Determinant, Properties of Determinants, Area of a Triangle, Minors and Cofactors, Adjoint and Inverse of a Matrix, Applications of Determinants and Matrices

Continuity and Differentiability :

Introduction, Continuity, Differentiability, Exponential and Logarithmic Functions, Logarithmic Differentiation, Derivatives of Functions in Parametric Forms, Second Order Derivative, Mean Value Theorem

Application of Derivatives :

Introduction, Rate of Change of Quantities, Increasing and Decreasing Functions, Tangents and Normals, Approximations, Maxima and Minima

Appendix 1: Proofs in Mathematics

A.1.1 Introduction,

A.1.2 What is a Proof?

Appendix 2: Mathematical Modelling

A.2.1 Introduction,

A.2.2 Why Mathematical Modelling?,

A.2.3 Principles of Mathematical Modelling

Class 12 - XII ( Class 12th ) - Part - 2 ( II )

Integrals :

Introduction, Integration as an Inverse Process of Differentiation, Methods of Integration, Integrals of some Particular Functions, Integration by Partial Fractions, Integration by Parts, Definite Integral, Fundamental Theorem of Calculus, Evaluation of Definite Integrals by Substitution, Some Properties of Definite Integrals

Application of Integrals :

Introduction, Area under Simple Curves, Area between Two Curves

Differential Equations :

Introduction, Basic Concepts, General and Particular Solutions of a Differential Equation, Formation of a Differential Equation whose General Solution is given, Methods of Solving First order, First Degree Differential Equations

Vector Algebra :

Introduction, Some Basic Concepts, Types of Vectors, Addition of Vectors, Multiplication of a Vector by a Scalar, Product of Two Vectors

Three Dimensional Geometry :

Introduction, Direction Cosines and Direction Ratios of a Line, Equation of a Line in Space, Angle between Two Lines, Shortest Distance between Two Lines, Plane, Coplanarity of Two Lines, Angle between Two Planes, Distance of a Point from a Plane, Angle between a Line and a Plane

Linear Programming :

Introduction, Linear Programming Problem and its Mathematical Formulation, Different Types of Linear Programming Problems

Probability :

Introduction, Conditional Probability, Multiplication Theorem on Probability, Independent Events, Bayes' Theorem, Random Variables and its Probability Distributions, Bernoulli Trials and Binomial Distribution

Applied Mathematics & Applied Maths - Significance & Scope :

Secondary school education prepares pupils to pursue future employment opportunities once they graduate from high school. Mathematics is an important subject that assists students in making career decisions. Mathematics is commonly employed in higher education as a supplementary topic in fields such as Economics, Commerce, Business Studies, Accounting Researches, Social Sciences SST, Sociology , ICT, CCT, Botany, Zoology, Civics Researches, Languages ( Sanskrit, Hindi, Urdu, English ) Informatics IP, Entrepreneurship researches, Legal Practices, Fashion Researches and more.

Even many other engineering disciplines covering Computer Sciences CS, Computer Coding, Information Technology IT, Web Application Development, Artificial Intelligence AI, Machine Learning ML, Multimedia MM Development, Mass Media MM Development, Library Information Systems LIS, Python & Java Programming, Electrical Engineering, Electronics Engineering, Automotive Engineering, BioTechnology Researches, Physics, Chemistry, Biology, Typography Computer Applications, Shorthand Computing, Data Sciences DS, Informatic Practices IP are deeply looked into.

It has been found that the Mathematics syllabus in senior secondary grades designed for Science topics ( like, Environmental Sciences EVS, Geospatial Computing ) may not be suitable for pupils wishing to study Commerce or Social Science - based subjects in university education. Keeping this in mind, one more elective course in the Mathematics curriculum is being designed for Senior Secondary courses with the goal of providing students with meaningful mathematics expertise that may be applied in subjects other than Physical Science.

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Following is the outline of Central Board of Secondary Education CBSE / NCERT prescribed syllabus for Applied Mathematics ( Applied Maths ) syllabi for Class 11 - XI ( Class 11th ) :

Class 11 - XI ( 11th ) - Applied Maths & Applied Mathematics

1. Number Theory :

a. Prime Numbers: Intersecting properties of prime number without proof, Ramanujan’s work on Prime number, Encryption and prime number

b. Ratio, Proportion and Logarithms: Business Application related to Ratio and Proportion. Practical Applications of Logarithms and Anti Logarithms

2. Interpretation of Data :

Interpretation of Data represented in the form of charts, graphs, Frequency distribution, Histogram, Pie-chart etc.

3. Analysis of Data :

Arithmetic Mean, Median, Mode, Geometric and Harmonic Mean, Range, Mean deviation, Standard Deviation, Variance, coefficient of variation, skewness.

4. Commercial Mathematics :

Profit and Loss, Simple interest, compound interest, depreciation, Effective rate of interest, present value, net present value, future value, annuities.

5. Set Theory :

Set and their representations, Empty set, Finite and Infinite sets, Equal sets, subsets, power set, universal set, Venn diagrams, union and intersections of sets, complement of set.

6. Relation and Function :

Pictorial representation of a function, domain, co-domain and range of function, Function as special type of Relation, it’s Domain and range.

7. Algebra :

a. Complex Number: Concept of iota, imaginary numbers, arithmetic operation on complex number.

b. Sequence and Series: Introduction of sequences, series, Arithmetic and Geometric Progression. Relationship between AM and GM, sum of n terms etc.

c. Permutations and Combinations: Basic concepts of Permutations and Combinations, Factorial, permutations, results, combinations with standard results, Binomial Theorem ( statement only ).

8. Trigonometry :

Trigonometric identities, calculation of Height and distance involving angles of all degrees till 90.

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Following is the outline of Central Board of Secondary Education CBSE / NCERT prescribed syllabus for Applied Mathematics ( Applied Maths ) syllabi for Class 12 - XII ( Class 12th ) :

Class 12 - XII ( 12th ) - Applied Maths & Applied Mathematics

1. Fundamentals of Calculus :

Basics of Limits & continuity, differentiation of non-trigonometric functions, Basic applications of derivatives in finding Marginal cost, Marginal Revenues etc. Increasing and Decreasing Functions, Maxima / Minima. Integration as reverse process of differentiation, integration of simple algebraic functions.

2. Algebra :

Introduction of Matrices, Algebra of Matrices, Determinants of Square matrices (Application only).

3. Logical Reasoning :

Number series, Coding, decoding and odd man out, direction tests, blood relations, syllogism, Binary numbers, logical operations and truth table.

4. Commercial Mathematics :

Calculating EMI, calculations of Returns, Compound annual growth rate (CAGR), Stocks, Shares, Debenture, valuation of Bonds, GST, Concept of Banking.

5. Probability :

Introduction to probability of an event, Mutually exclusive events, conditional probability, Law of Total probability. Basic application of Probability Distribution (Binomial Distribution, Poisson Distribution and Normal Distribution).

6. Two dimensional Geometry :

Slope of a line, equation of a line in point slope form, slope intercept form and two point form.

7. Linear Programming :

Introduction, related terminology such as constraints, objective function, optimization, different types of LP, mathematical formulation of LP problem, graphical method of solution for problems in two variables.

8. Analysis of time based Data :

a. Index numbers: meaning and uses of index number, construction of index numbers, construction of consumer price indices.

b. Time series & trend analysis: Component of time series, additive models, Finding trend by moving average method.

Apart from the above outline of recently proposed syllabus of Applied Mathematics, there are several projects that are real-life based and could be taken up to meet the requirements of the governing Boards, viz., CBSE and other Boards.

For example :

Algorithmic approach of Sieve of Erastosthene’s, Ramanujan’s theory of prime numbers: Use of prime numbers in coding and decoding of messages, Bertrnad’s postulate, etc..

Engineering Mathematics & Engineering Maths :

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Following is the general outline of College Level Engineering Maths for various Technical courses. ( This Engineering Mathematics syllabus has wide scope, encompassing various Engineering Courses, Polytechnic Courses, Diploma Courses & other related College Level Courses ) :

Linear Algebra:

Algebra of matrices, inverse, rank, system of linear equations, symmetric, skew symmetric and orthogonal matrices. Hermitian, skew-Hermitian and unitary matrices. Eigenvalues and eigenvectors, diagonalisation of matrices, Cayley-Hamilton Theorem.

Calculus:

Functions of single variable, limit, continuity and differentiability, Mean value theorems, Indeterminate forms and L'Hospital rule, Maxima and minima, Taylor's series, Fundamental and mean value-theorems of integral calculus. Evaluation of definite and improper integrals, Beta and Gamma functions, functions of two variables, limit, continuity, partial derivatives, Euler's theorem for homogeneous functions, total derivatives, maxima and minima, Lagrange method of multipliers, double and triple integrals and their applications, sequence and series, tests for convergence, power series, Fourier Series, Half range sine and cosine series.

Complex variables:

Analytic functions, Cauchy-Riemann equations, Application in solving potential problems, Line integral, Cauchy's integral theorem and integral formula (without proof), Taylor's and Laurent' series, Residue theorem (without proof) and its applications.

Vector Calculus:

Gradient, divergence and curl, vector identities, directional derivatives, line, surface and volume integrals, Stokes, Gauss and Green's theorems (without proofs) applications.

Ordinary Differential Equations:

First order equation (linear and nonlinear), Second order linear differential equations with variable coefficients, Variation of parameters method, higher order linear differential equations with constant coefficients, Cauchy- Euler's equations, power series solutions, Legendre polynomials and Bessel's functions of the first kind and their properties.

Partial Differential Equations:

Separation of variables method, Laplace equation, solutions of one dimensional heat and wave equations.

Probability and Statistics:

Definitions of probability and simple theorems, conditional probability, Bayes Theorem, random variables, discrete and continuous distributions, Binomial, Poisson, and normal distributions, correlation and linear regression.

Numerical Methods:

Solution of a system of linear equations by L-U decomposition, Gauss-Jordan and Gauss-Seidel Methods, Newton's interpolation formulae, Solution of a polynomial and a transcendental equation by Newton-Raphson method, numerical integration by trapezoidal rule, Simpson's rule and Gaussian quadrature, numerical solutions of first order differential equation by Euler's method and 4th order Runge-Kutta method.

Differential Calculus - I:

Leibnitz’s theorem, Partial derivatives, Euler’s theorem for homogeneous functions, Total derivatives, Change of variables, Curve tracing: Cartesian and Polar coordinates.

Differential Calculus - II:

Taylor’s and Maclaurin’s Theorems, Expansion of function of several variables, Jacobian, Approximation of errors, Extrema of functions of several variables, Lagrange’s method of multipliers ( Simple applications)

Linear Algebra:

Inverse of a matrix by elementary transformations, Rank of a matrix ( Echelon & Normal form), Linear dependence, Consistency of linear system of equations and their solution,. Characteristic equation, Eigen values and eigen vectors, Cayley-Hamilton Theorem,A brief introduction to Vector Spaces,Subspaces. Rank & Nullity. Linear transformations.

Multiple Integrals:

Double and triple integrals, Change of order of integration, Change of variables, Application of integration to lengths, Volumes and Surface areas – Cartesian and Polar coordinates. Beta and Gamma functions, Dirichlet’s integral and applications.

Vector Calculus:

Point function, Gradient, Divergence and Curl and their physical interpretations, Vector identities, Directional derivatives. Line,Surface and Volume integrals, Applications of Green’s, Stoke’s and Gauss divergence theorems (without proofs)

Differential Equations:

Linear differential equations of nth order with constant coefficients, Complementary function and Particular integral, Simultaneous linear differential equations, Solution of second order differential equations by changing dependent & independent variables, Normal form, Method of variation of parameters, Applications to engineering problems (without derivation).

Series Solution and Special Functions:

Series solution of second order ordinary differential equations with variable coefficient (Frobenius method), Bessel and Legendre equations and their series solutions, Properties of Bessel function and Legendre polynomials.

Laplace Transform:

Laplace transform, Existence theorem, Laplace transforms of derivatives and integrals, Initial and final value theorems, Unit step function, Dirac- delta function, Laplace transform of periodic function, Inverse Laplace transform, Convolution theorem, Application to solve simple linear and simultaneous differential equations.

Fourier Series and Partial Differential Equations:

Periodic functions, Fourier series of period 2, Euler’s Formulae, Functions having arbitrary periods, Change of interval, Even and odd functions, Half range sine and cosine series, Harmonic analysis. Solution of first order partial differential equations by Lagrange’s method, Solution of second order linear partial differential equations with constant coefficients.

Applications of Partial Differential Equations:

Classification of second order partial differential equations, Method of separation of variables for solving partial differential equations, Solution of one and two dimensional wave and heat conduction equations, Laplace equation in two dimension, Equation of transmission lines.

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From the perspective of Maths Courses, Mathematics Syllabi, Math Curriculi and aligned mathematical, as well as analytical treatment, we can reflect upon the following related and critical Mathematical sub - disciplines :

Trigonometry,

Arithmetic,

Algebra,

Probability,

Statistics,

Geometry,

Mensuration,

Constructions,

Functions,

Sets,

Matrices,

Determinants,

Inverse Trigonometric Functions,

Differentiation,

Number Systems,

Polynomials,

Integration,

Integral Calculus,

Differential Equations,

Continuity & Differentiability,

Linear equations,

Linear Programming,

Operations Research,

Marketing Research

Applied Mathematics,

Statistics,

Probability,

Engineering Mathematics

Let's relook each of the above mentioned sub disciplines in the following sections. It is for a detailed, holistic and comprehensive Mathematical learning, as well as deeper understanding :

Trigonometry:

Trigonometry is quite old, proven and established Mathematical discipline. Trigonometry in Maths revolves around the study of the dependencies among the ratios of the sides of a right-angled triangle and their various angles. There are several trigonometric ratios that are used to study this relationship. Few of them are sine, cosine, tangent, cotangent, secant, and cosecant. The term trigonometry originated in Greece in the 16th century. Trigonometry is a Latin derivative of the Greek mathematician Hipparchus' concept.

Trigonometry is well-known for its numerous identities. These trigonometric identities are frequently used to rewrite trigonometrical expressions with the goal of simplifying an expression, finding a more useful form of an expression, or solving an equation.

Sumerian astronomers investigated angle measurement by dividing circles into 360 degrees. They, and later the Babylonians, investigated the ratios of the sides of similar triangles and discovered some properties of these ratios, but did not turn this into a systematic method for determining triangle sides and angles. A similar investigation method was used by the ancient Nubians.

The term was coined by Bartholomaeus Pitiscus, who published his Trigonometria in 1595. Trigonometry grew into a major branch of mathematics as a result of the demands of navigation and the growing need for accurate maps of large geographic areas. Trigonometric series were influenced by the works of Scottish mathematicians James Gregory in the 17th century and Colin Maclaurin in the 18th century. Brook Taylor defined the general Taylor series in the 18th century as well. Gemma Frisius described the method of triangulation that is still used in surveying today for the first time. Leonhard Euler was the first to fully incorporate complex numbers into trigonometry.

Trigonometry is a very important branch of mathematics. Trigonometry is derived from the words 'Trigonon' and 'Metron,' which mean triangle and measure, respectively. It is the study of the relationship between a right-angled triangle's sides and angles. It thus aids in determining the unknown dimensions of a right-angled triangle by employing formulas and identities based on this relationship.

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Trigonometry fundamentals are concerned with angle measurement and angle-related problems. In trigonometry, there are three basic functions: sine, cosine, and tangent. Other important trigonometric functions can be derived from these three basic ratios or functions: cotangent, secant, and cosecant. These functions underpin all of the important concepts covered in trigonometry. As a result, in order to understand trigonometry, we must first learn these functions and their respective formulas.

The three sides of a right-angled triangle are as follows :

Perpendicular - The opposite side of the angle.

Base - This is the angle's adjacent side.

Hypotenuse - The hypotenuse is the side opposite the right angle.

In trigonometry, there are six fundamental ratios that aid in establishing a relationship between the ratio of sides of a right triangle and the angle, say x. If x is the angle formed between the base and hypotenuse of a right-angled triangle, then

sin x = Perpendicular / Hypotenuse

cos x = Hypotenuse / Base

tan x = Base / Perpendicular

The values of the other three functions, cot x, sec x, and cosec x, are proportional to tan x, cos x, and sin x, as shown below.

cot x = 1 / tan x = 1 / ( Base / Perpendicular )

sec x = 1 / cos x = 1 / ( Hypotenuse / Base )

cosec x = 1 / sin x = Hypotenuse / Perpendicular

The trigonometric table is composed of trigonometric ratios that are related to one another - sine, cosine, tangent, cosecant, secant, cotangent. These ratios are abbreviated as sin, cos, tan, cosec, sec, cot, and are used to calculate standard angle values. To learn more about these ratios, consult the trigonometric table chart.

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Trigonometric angles are the angles in a right-angled triangle that can be used to represent various trigonometric functions. Standard angles in trigonometry include 0, 30, 45, 60, and 90. These angles' trigonometric values can be found directly in a trigonometric table. Other important angles in trigonometry include 180, 270, and 360.

In trigonometry, various formulas depict the relationships between trigonometric ratios and angles for different quadrants. The following are the basic trigonometry formulas :

Trigonometry Ratio Formulas :

sin θ = Opposite Side / Hypotenuse

cos θ = Adjacent Side / Hypotenuse

tan θ = Opposite Side / Adjacent Side

cot θ = 1 / tan θ = Adjacent Side / Opposite Side

sec θ = 1 / cos θ = Hypotenuse / Adjacent Side

cosec θ = 1 / sin θ = Hypotenuse / Opposite Side

Trigonometry Formulas Involving Pythagorean Identities :

sin²θ + cos²θ = 1

tan²θ + 1 = sec²θ

cot²θ + 1 = cosec²θ

Trigonometry has been used throughout history in fields such as mechanical engineering, astronomical general sciences, Geographical Information Systems, Library Information Systems, Physics, and so on. Among its varied applications are Geospatial Information Systems, Astronomy, Deep Oceanography, Earth Seismology, Meteorology, Physical Sciences, Astronomical Measurements, Machine Learning, Acoustics Technology, navigational sciences, Artificial Intelligence, Electronics Technology, Electrical Technology , Informatics Practices , Computers & Communication Technology , Information & Communication Technology and many other scientific fields.

Trigonometry Sine and Cosine Law :

a / sinA = b / sinB = c / sinC

Here, a, b, and c are the lengths of the triangle's sides, and A, B, and C are the angles of the triangle.

The unit circle can be used to calculate the sine, cosine, and tangent values of basic trigonometric functions.

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Trigonometric function graphs can be used to investigate various properties of a trigonometric function such as domain, range, and so on. Thus, the domain and range of sin and cosine functions can be written as,

sin x : Range [ -1, +1 ] ; Domain ( - infinity, + infinity )

cos x : Domain ( - infinity, + infinity ) ; Range [ -1, +1 ]

An equation is called an identity in Trigonometric when it is true for all values of the variables involved. Similarly, a trigonometric identity is an equation involving trigonometric ratios of an angle that is true for all values of the angles involved. You will learn more about the Sum and Difference Identities in trigonometry :

For example, sin θ / cos θ = [ Opposite / Hypotenuse ] ÷ [ Adjacent / Hypotenuse ] = Opposite / Adjacent = tan θ

Therefore, tan θ = sin θ / cos θ is a trigonometric identity.

Trigonometric functions were one of the first applications for mathematical tables. Such tables were incorporated into mathematics textbooks, and students were taught how to look up values and interpolate between the values listed in order to achieve greater accuracy. Trigonometric functions were scaled differently in slide rules.

The main trigonometric functions are calculated using buttons on scientific calculators ( sin, cos, tan, and sometimes cis and their inverses ). Most allow for a variety of angle measurement methods, including degrees, radians, and, in some cases, gradians. The trigonometric functions are available in most computer programming languages' function libraries. The floating point unit hardware built into the microprocessor chips found in the majority of personal computers includes instructions for calculating trigonometric functions.

From the perspective of Trigonometric applications to real life challenges and problems, they are several and vast. Trigonometry is useful for determining the length of various geographical masses, water bodies like rivers, seas, oceans, ponds, measuring the height of hills & mountains, and so on. Spherical trigonometry has been used in astronomical sciences to determine the positions of the sun, moon, and stars. Our Maths Trigonometry teachers, trainers, Trigonometry offline maths home tution near me, online maths tuition for class 10 near me, home ib maths tutor near me offline, home maths tuition teacher near me offline, home tuition for maths near me offline, Trigonometry home tuition maths offline near me, online maths teacher online near me, online academy for mathematics & english cost near me offline, Trigonometry home maths tutor at home near me, online best maths and science tuition near me offline, online best tutors for maths near me offline, Trigonometry online best maths tuition near me at home, online private maths tutors near me offline, home tutor near me math at home offline, home diploma maths tuition near me offline, home maths tutors for class 1 near me offline, Trigonometry home maths tutor for class 10 near me offline, home maths teacher for class 12 near me offline, home maths tutor for class 11 near me offline, home maths teacher for class 11 near me offline, Trigonometry home math classes near me offline, offline mathematic coaching in greater noida west offline near me are quite proficient in the Mathematics sub discipline of Trigonometry. Our Trigonometry Maths teachers are quite well versed with the contemporary Mathematical Trigonometric foundations and their synergistic applications to varied complex problems at hand.

In addition to the six ratios mentioned previously, there are several trigonometric functions that were historically important but are now rarely used. These include the chord ( crd(θ) ), the versine ( versin(θ) ) , the coversine ( coversin(θ) ) , the haversine ( haversin(θ) ), the exsecant ( exsec(θ) ), and the excosecant ( excsc(θ) ). More relationships between these functions can be found in the List of trigonometric identities.

Algebra :

Algebra ( from Arabic ( al - jabr ) reunion of broken parts ) is the study of variables and the rules for manipulating these variables in formulas; it is a thread that runs through almost all of mathematics.

Elementary algebra deals with manipulating variables ( commonly represented by Roman letters ) as if they were numbers and is thus required in all mathematical applications. Abstract algebra is the name given to the study of algebraic structures such as groups, rings, and fields, which is mostly used in education. Linear algebra, which deals with linear equations and linear mappings, is used in modern geometry presentations and has numerous practical applications ( in artificial intelligence, computers & communication technology, information & communication technology, computer science & engineering, astronomical sciences, information technology , metereology, weather forecasting, geospatial information systems, library information systems, electrical technology, physics, computer coding, computer programming, informatics practices, electronics technology, for example ). Many areas of mathematics belong to algebra, some of which have the word "algebra" in their name, and some may not have.

The term algebra is used to name not only an area of mathematics and some subareas, but also some types of algebraic structures, such as an algebra over a field, which is commonly referred to as an algebra. A subarea and its main algebraic structures are sometimes referred to by the same phrase, such as Boolean algebra and a Boolean algebra. An algebraist is a mathematician who specialises in algebra.

Algebra began with computations similar to arithmetic, with letters representing numbers. This enabled proofs of properties that are true regardless of the numbers involved. At Wise Turtle Academy, we research and deliver Algebra learning support services through our competent tutors, teachers, home science and maths tutor near me offline, home tuition maths and science near me, Algebra online maths tution online, home engineering maths classes near me offline, home mathematics coaching centre near me offline, Algebra home tuitions for maths near me, home vedic maths tuition near me offline, home math tutoring classes near me offline, offline science and maths home tuition near me, online maths tuition 12th near me, home grade 11 math tutor near me offline, Algebra home class 11 maths online teaching classes near me, online igcse maths tuition near me offline, home nat 5 maths tutor near me offline, offline maths and english home tutors near me, Algebra home 8th class maths tuition near me offline, home maths tutors for class 8 near me offline, home maths tutors for class 7 near me offline, Algebra home maths tutors for class 6 near me offline, home maths tutors for class 5 near me offline, home maths tutors for class 4 near me offline, Algebra online math classes near me online, home maths tutor for class 12 near me offline, home maths tutors for class 3 near me offline, Algebra home maths tutors for class 2 near me offline, home maths tutors for class 1 near me offline and others. We emphasize upon the gist and crux of Algebra. We simplify the Algebraic learning curve.

Historically, and still today, the study of algebra begins with the solution of equations like the quadratic equation above. Then there are more general questions like "does an equation have a solution?" and "how many solutions are there?".

Mathematics was divided into only two subfields prior to the 16th century: arithmetic and geometry. Even though some methods developed much earlier are now considered algebra, the emergence of algebra and, soon after, infinitesimal calculus as subfields of mathematics dates only from the 16th or 17th century. Many new fields of mathematics emerged in the second half of the nineteenth century, the majority of which used both arithmetic and geometry, and almost all of which used algebra.

Today, algebra encompasses many branches of mathematics, as evidenced by the Mathematics Subject Classification, where none of the first level areas ( two digit entries ) are referred to as algebra. Sections 08 - General algebraic systems, 12 - Field theory and polynomials, 13 - Commutative algebra, 15 - Linear and multilinear algebra; matrix theory, 16 - Associative rings and algebras, 17 - Non associative rings and algebras, 18 - Category theory; homological algebra, 19 - K - theory, and 20 - Group theory are now included in algebra. 11 - Number theory and 14 - Algebraic geometry both make extensive use of algebra.

Linear algebra is one example of a subfield of algebra with the word algebra in its name. Others, such as group theory, ring theory, and field theory, do not. This section contains a list of mathematical areas with the word "algebra" in their names.

Elementary algebra is the portion of algebra that is typically taught in elementary mathematics courses.

Abstract algebra is the study and axiomatic definition of algebraic structures such as groups, rings, and fields.

Linear algebra is the study of the specific properties of linear equations, vector spaces, and matrices.

Boolean algebra is a branch of algebra that abstracts computation by using the truth values false and true.

The study of commutative rings is known as commutative algebra.

The implementation of algebraic methods as algorithms and computer programmes is known as computer algebra.

The study of algebraic structures that are fundamental to the study of topological spaces is known as homological algebra.

Universal algebra is the study of properties that are shared by all algebraic structures.

Algebraic number theory is the study of number properties from an algebraic standpoint.

Algebraic geometry is a branch of geometry that specifies curves and surfaces as polynomial equation solutions.

Algebraic combinatorics is the study of combinatorial problems using algebraic methods.

Relational algebra is a collection of financial relations that are closed under certain operators.

Elementary algebra is the most fundamental type of algebra. It is taught to students who are assumed to have no knowledge of mathematics beyond basic arithmetic principles. Only numbers and their arithmetical operations ( such as +,-,*,% ) occur in arithmetic. In algebra, numbers are frequently represented by symbols known as variables ( such as a, n, x, y or z ).

A polynomial is an expression that is the product of a constant and a finite number of variables raised to whole number powers. A polynomial expression is an expression that can be rewritten as a polynomial using addition and multiplication commutativity, associativity, and distributivity. For example, ( x - 1 ) ( x + 3 ) is a polynomial expression that is not, in fact, a polynomial. A polynomial function is a function defined by a polynomial or, more precisely, a polynomial expression.

The factorization of polynomials, that is, expressing a given polynomial as a product of other polynomials that cannot be factored any further, and the computation of polynomial greatest common divisors, are two important and related problems in algebra. Finding algebraic expressions for the roots of a polynomial in a single variable is a related class of problems.

It has been proposed that elementary algebra be taught to students as young as eleven years old, though in recent years, public lessons have begun at the eighth grade level ( 13 years old ). However, algebra instruction in some US schools begins as early as ninth grade. In line with it, we at Wise Turtle Academy, strive to deliver Algebra learning support through various formats and approaches, including but not limited to algebra home maths teachers for class 8 near me offline, advanced engineering maths home tuition Greater Noida West offline, home maths class 10 tuition near me offline, home maths coaching near me for class 10 offline, best maths home tutors near me offline, home maths and physics tutor near me offline, home maths tuition near me for class 10 offline, modern engineering maths home tuition Noida Extension offline, home english and math tutor near me offline, home finite math tutors near me offline, home english and maths classes near me offline, home physics chemistry maths tuition near me offline, home 12th maths tuition near me offline, advanced engineering maths home tuition classes Noida Extension offline, home btech maths tuition near me offline, home maths tuition for engineering near me offline, home maths & english tuition near me offline, online engineering maths home tuitions Greater Noida West offline, applied maths home tuition class Noida Extension offline, offline maths and science home tutors near me, home maths teachers for class 7 near me offline, algebra home maths teachers for class 6 near me offline, online engineering maths home tuitions Noida Extension offline, home maths teachers for class 5 near me offline, applied maths home tuition in Noida Extension offline, home maths teachers for class 4 near me offline, algebra home maths teachers for class 3 near me offline, online engineering maths home tuition classes Greater Noida West offline, home bio maths tuition near me offline, online engineering maths home tuition class Greater Noida West offline, home maths teachers for class 2 near me offline, algebra offline mathematics lessons in greater noida west offline, home maths teachers for class 1 near me offline, algebra online maths tutor in noida extension offline, home bio maths tuitions near me offline, mathematics home tutor in greater noida west offline and others.

Abstract algebra generalises the familiar concepts found in elementary algebra and number arithmetic. The following are the fundamental concepts in abstract algebra.

Sets: Rather than focusing solely on the various types of numbers, abstract algebra addresses the broader concept of sets: collections of objects known as elements. Sets are all collections of familiar types of numbers. The set of all two-by-two matrices, the set of all second-degree polynomials (ax2 + bx + c), the set of all two-dimensional vectors of a plane, and the various finite groups, such as the cyclic groups, which are the groups of integers modulo n, are also examples of sets. Set theory is a branch of logic rather than an algebraic branch.

Binary operations: The concept of addition (+) is generalised to the concept of binary operation. Without the set on which the operation is defined, the concept of binary operation has no meaning. For two elements a and b in a set S, a * b is another element in the set; this condition is known as closure. When defined on different sets, addition ( + ), subtraction ( - ), multiplication ( * ), and division ( / ) can be binary operations, as can addition and multiplication of matrices, vectors, and polynomials.

Identity elements: The numbers zero and one are generalised to represent an operation's identity element. The identity element for addition is zero, and the identity element for multiplication is one.

Inverse elements: The concept of inverse elements is derived from negative numbers. The inverse of a is written -a for addition, and the inverse of a^-1 for multiplication.

Associativity: Integer addition has the property of associativity. That is, grouping the numbers to be added has no effect on the sum.

Commutativity: Real-number addition and multiplication are both commutative. That is, the order of the numbers has no effect on the outcome.

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Arithmetic :

Arithmetic is a branch of mathematics that studies the properties of the traditional operations on numbers — addition, subtraction, multiplication, division, exponentiation, and root extraction. Giuseppe Peano, an Italian mathematician, formalised arithmetic in the nineteenth century with his Peano axioms, which are still very important in the field of mathematical logic today.

The prehistory of arithmetic is limited to a few artefacts that may indicate the concept of addition and subtraction, the best-known of which is the Ishango bone from central Africa, dating from between 20,000 and 18,000 BC, though its interpretation is disputed.

According to the earliest written records, the Egyptians and Babylonians used all four basic arithmetic operations as early as 2000 BC: addition, subtraction, multiplication, and division. These artefacts do not always reveal the exact process used to solve problems, but the characteristics of the specific numeral system have a strong influence on the complexity of the methods. The hieroglyphic system for Egyptian numerals, like the later Roman numeral system, descended from counting tally marks.

The place - value concept and positional notation were independently devised by the gradual development of the Hindu - Arabic numeral system, which combined the simpler methods for computations with a decimal base, and the use of a digit representing 0. This enabled the system to consistently represent both large and small integers, an approach that eventually supplanted all others. The Indian mathematician Aryabhata incorporated an existing version of this system into his work in the early sixth century AD, and experimented with different notations. Brahmagupta established the use of 0 as a separate number in the 7th century, and determined the results for multiplication, division, addition, and subtraction of zero and all other numbers — except division by zero.

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Arithmetic was one of the seven liberal arts taught in universities during the Middle Ages. The growth of algebra in the mediaeval Islamic world, as well as in Renaissance Europe, was a result of the enormous simplification of computation enabled by decimal notation.

To aid in numerical calculations, various tools have been invented and widely used. They were various types of abaci prior to the Renaissance. Slide rules, nomograms, and mechanical calculators, such as Pascal's calculator, are more recent examples. Electronic calculators and computers have largely replaced them.

Although addition, subtraction, multiplication, and division are the most basic arithmetic operations, arithmetic also includes more advanced operations such as percentage manipulations, square roots, exponentiation, logarithmic functions, and even trigonometric functions, in the same vein as logarithms. Arithmetic expressions must be evaluated in the order in which they are intended to be evaluated. There are several ways to specify this, the most common of which is to use parentheses and rely on precedence rules, or to use a prefix or postfix notation, which uniquely fix the order of execution by themselves. A field is any collection of objects on which all four arithmetic operations ( except division by zero ) can be performed and where these four operations obey the usual laws ( including distributivity ).

Any integer greater than 1 has a unique prime factorization ( a representation of a number as the product of prime factors ), excluding the order of the factors, according to the fundamental theorem of arithmetic. 252, for example, has only one prime factorization:

252 = 22 × 32 × 71

This theorem was first introduced in Euclid's Elements, along with a partial proof (known as Euclid's lemma). Carl Friedrich Gauss proved the fundamental theorem of arithmetic first.

One of the reasons why 1 is not considered a prime number is because of the fundamental theorem of arithmetic. Other reasons include Eratosthenes' sieve and the definition of a prime number itself ( natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers ).

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However, any numeral system based on powers of 10, such as Greek, Cyrillic, Roman, or Chinese numerals, may conceptually be described as "decimal notation" or "decimal representation." In common usage, the term "decimal representation" only refers to the written numeral system employing arabic numerals as the digits for a radix 10 ("decimal") positional notation.

The four basic operations of addition, subtraction, multiplication, and division were initially introduced by the Indian mathematician Brahmagupta. In mediaeval Europe, this was referred to as the "Modus Indorum" or "Way of the Indians." The representation or encoding of numbers using the same symbol for each order of magnitude (e.g., the "ones place," "tens place," and "hundreds place") and utilising the same symbols to express fractions with a radix point is known as positional notation, sometimes known as "place-value notation" (e.g., the "tenths place", "hundredths place").

Number theory and "arithmetic" were synonymous terms before the 19th century. The addressed issues included primality, divisibility, and the resolution of integer equations like Fermat's Last Theorem. These issues were all closely related to the fundamental operations. Although fairly simple to explain, it seemed that the most of these issues were highly challenging and might require extremely complex mathematics involving ideas and techniques from numerous other fields of mathematics in order to be solved. As a result, analytical number theory, algebraic number theory, Diophantine geometry, and arithmetic algebraic geometry emerged as new disciplines of number theory. In order to solve issues that can be stated, advanced techniques that go beyond the traditional methods of arithmetic are required, as shown by Wiles' demonstration of Fermat's Last Theorem.

We can thus admit the significance of the sub discipline of Arithmetic and the contribution to the development of the Mathematics in toto. Other specialisations have been tremendously impacted and influenced by Arithmetic. Some of the impacted and influenced specialisations are artificial intelligence, computers & communication technology, information & communication technology, computer science & engineering, astronomical sciences, information technology , metereology, weather forecasting, geospatial information systems, library information systems, electrical technology, physics, computer coding, computer programming, informatics practices, electronics technology, deep space exploration initiatives and others.

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Algorithms for the arithmetic of natural numbers, integers, fractions, and decimals are frequently given a lot of attention in primary mathematics instruction ( using the decimal place - value system ). Algorithm is another name for this field of research.

Education professionals have long questioned this curriculum and argued for the early teaching of more fundamental and intuitive mathematical concepts due to the complexity and unjustified look of these algorithms. The New Math of the 1960s and 1970s, which aimed to teach arithmetic in the spirit of axiomatic progression from set theory, an echo of the dominant approach in higher mathematics, was one major movement in this direction. Islamic scholars also employed mathematics to teach how to apply the laws governing zakat and irth.

Probability :

The probability of an occurrence is a number used in science to describe how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place. A certain occurrence has a chance of 1, while an impossible event has a probability of 0. The odds of two complementing events A and B happening, either A happens or B happens, sum up to 1. A straightforward illustration is tossing a fair ( impartial ) coin. The likelihood of both the possible outcomes ( heads and tails ) is equal if a coin is fair.

In probability theory, a subfield of mathematics used in fields like statistics, mathematics, science, finance, gambling, artificial intelligence, machine learning, computer science, and game theory to, among other things, draw conclusions about the expected frequency of events, these ideas have been given an axiomatic mathematical formalisation. Moreover, the mechanics and regularities that underlie complex systems are described using probability theory.

The word probability is derived from the Latin word probabilitas, which also means "probity," a measure of a witness's credibility in a court case and sometimes associated with aristocracy in Europe. This is quite different from the present definition of probability, which measures the strength of the available empirical evidence and is derived through inductive reasoning and statistical inference.

A contemporary advancement in mathematics is the study of probability. Gambling demonstrates that there has long been a need to quantify the concepts of probability, but precise mathematical representations only developed much later. The slow progress of probability mathematics has its justifications. Although the mathematical study of probability was inspired by games of chance, basic questions are nevertheless obfuscated by gamblers' superstitions.

The theory of probability represents its notions in formal words, or in terms that may be considered independently of their meaning, like other theories do. Rules of mathematics and logic are used to manipulate these formal words, and any outcomes are then understood or applied to the original problem domain.

At least two formalisations of probability, namely the Cox formulation and the Kolmogorov formulation, have proved effective. Sets are understood as events in Kolmogorov's formulation ( also see probability space ), and probability is a measure on a class of sets. The focus of Cox's theorem is on creating a consistent way to assign probability values to propositions because probability is treated as a primitive (i.e., not further examined).

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The everyday use of probability theory includes risk modelling and evaluation. Actuarial science is used by the insurance sector and markets to establish pricing and make trading choices. In the analysis of entitlements, financial regulation, and environmental regulation, governments use probabilistic approaches.

The impact of the perceived likelihood of any protracted conflict in the Middle East on oil prices, which have repercussions on the economy as a whole, is an illustration of how probability theory is used in equity trading. The price of a commodity can go up or down based on a trader's judgement on the likelihood of war, which also informs other dealers of that view. As a result, neither an independent nor a necessary rational assessment of the probability is made.

Probability can be used to examine patterns in biology and ecology, as well as financial assessment ( e.g., the spread of diseases ) ( e.g., biological Punnett squares ). Similar to finance, risk assessment may be used as a statistical technique to determine the probability of unfavourable occurrences happening and can help with the implementation of protocols to avoid running into such situations. Games of chance are created using probability so that casinos may always turn a profit while yet paying out winnings to players frequently enough to keep them coming back.

Reliability is a vital aspect of probability theory's practical application. Reliability theory is used in product design for many consumer goods, including cars and consumer electronics, to lower the likelihood of failure.

The intersection or joint probability of two events A and B occurring on a single performance of an experiment is represented by the symbol 'inverted U' or P(A V B). Mutually exclusive occurrences are those where either event A or event B can happen, but never both at once. The likelihood of an event A given the occurrence of another event B is known as conditional probability. The expression "the probability of A, given B" is conditional probability.

Bayes' rule, which has applications in probability theory, connects the probabilities of event A to event B, both before ( before to ) and after ( posterior to ) conditioning on another event B. The ratio of the probability of the two events is what determines the chances on event A to event B. The rule can be rephrased as posterior is proportional to previous times likelihood when arbitrarily many occurrences of A are of interest, not just two.

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Statistics :

Mathematical statistics, as opposed to methods for gathering statistical data, is the application of probability theory, a subfield of mathematics, to statistics. In particular, mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure theory are employed in this.

The planning of studies, particularly the design of randomised experiments and the preparation of surveys utilising random sampling, are under the purview of statistical data collecting.

The initial analysis of the data frequently adheres to the predetermined protocol for the study. The outcomes of a study's data can also be examined to assess auxiliary hypotheses motivated by the preliminary findings or to propose further investigations. Mathematical statistics is used in the secondary analysis of the data from a planned study using methods from data analysis.

The categories of data analysis are :

Descriptive statistics are the area of statistics that summarises and describes the characteristics of the data.

Inferential statistics, a branch of statistics that uses a model of the data to draw conclusions. For instance, choosing a model for the data, determining whether the data meet the requirements of the chosen model, and estimating the associated uncertainty are all part of inferential statistics ( e.g. using confidence intervals ).

Although other types of data are often used, randomised study data is where data analysis tools perform at their best. For instance, from observational studies and natural experiments, where the inference is subject to the model that the statistician selects.

Statistics is undoubtedly an inalienable discipline of mathematics and finds it's applications into various domains of scientific nature and scope.

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In a random experiment, survey, or statistical inference technique, each measurable subset of the potential outcomes is given a probability via a function known as a probability distribution. Examples include experiments with non-numerical sample spaces, where the distribution would take the form of a categorical distribution; experiments with discrete random variable-encoded sample spaces, where the distribution can be described by a probability mass function; and experiments with continuous random variable-encoded sample spaces, where the distribution can be described by a probability density function. The use of more generic probability measures may be necessary in more complicated experiments, such as those involving stochastic processes specified in continuous time.

Either a probability distribution is multivariate or univariate. A multivariate distribution (a joint probability distribution) gives the probabilities of a random vector, which is a set of two or more random variables, taking on various combinations of values. A univariate distribution gives the probabilities of a single random variable taking on various alternative values. The binomial distribution, the hypergeometric distribution, and the normal distribution are all significant and frequently encountered univariate probability distributions. One type of multivariate distribution that is frequently used is the multivariate normal distribution.

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Various special distributions abound under Statistics. The most prevalent continuous distribution is the normal distribution. Bernoulli distribution for a single Bernoulli trial's result, such as success or failure or a yes or no answer. Given a certain total number of independent occurrences, the number of "positive occurrences" ( such as accomplishments, yes votes, etc. ) will follow a binomial distribution. For binomial-type data, a negative binomial distribution is used, but the quantity of interest is the number of failures that must occur before a certain number of successes do.

Geometric distribution for observations of the negative binomial type, where the number of successes is one and the quantity of interest is the number of failures prior to the first success. For a limited range of values, discrete uniform distribution ( e.g. the outcome of a fair die ) exists. With continuously dispersed values, a continuous uniform distribution occurs.

Poisson distribution, for the frequency of a Poisson-type event over a specific time period. For the duration preceding the following Poisson-type event, exponential distribution. Gamma distribution during the interval prior to the subsequent k Poisson-type occasions occur. Chi-squared distribution, which is important for drawing conclusions about the sample variance of normally distributed samples, is the distribution of a sum of squared standard normal variables ( chi-squared test ).

The Student's t distribution, which is useful for estimating the mean of normally distributed samples with unknown variance, is the distribution of the square root of a scaled chi squared variable and the ratio of a standard normal variable. Beta distribution, corresponding to the Bernoulli distribution and the binomial distribution, with a single probability ( real number between 0 and 1 ) exists.

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Drawing conclusions from data that are prone to random variation, such as observational mistakes or sample variance, is a process known as statistical inference. When applied to clearly defined scenarios, such a system of inference and induction techniques must provide appropriate results. It must also be sufficiently broad to be used in a variety of contexts. Using sample data, inferential statistics are used to test hypotheses and make estimates. Since descriptive statistics describe a sample, inferential statistics infer predictions about a wider population that the sample represents.

Geometry :

Geometry, which derives from the Ancient Greek words "gemetra" ( meaning "land measurement" ); "ge" ( meaning "earth, land," and "v" ( metron ) "a measure"). Together with arithmetic, it is one of the earliest subfields of mathematics. It is concerned with spatial characteristics like the separation, shape, size, and relative placement of objects. A geometer is a mathematician who specialises in geometry.

Euclidean geometry, which incorporates the concepts of point, line, plane, distance, angle, surface, and curve as fundamental concepts, was nearly entirely the focus of geometry up until the 19th century. The field of geometry has been divided into numerous subfields based on the underlying techniques ( differential geometry, algebraic geometry, computational geometry, algebraic topology, discrete geometry, also known as combinatorial geometry, etc. ) or the Euclidean space properties that are ignored ( affine geometry, which ignores the consideration of distance and parallelism, and projective geometry, which only takes into consideration point alignment but not distance and parallelism ).

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Geometry, which was first used to represent the physical universe, is now used in nearly every branch of science as well as the visual arts, architecture, and other related fields. There are many uses for geometry in seemingly unrelated branches of mathematics. Some of the critical concepts in geometry are as follows :

One of the most significant books ever written, Euclid's Elements, adopted an abstract approach to geometry. Euclid developed a number of axioms, or postulates, reflecting the fundamental or obvious characteristics of points, lines, and planes. He then rigorously inferred additional qualities using mathematical reasoning. Euclid's rigorous method of approaching geometry became known as axiomatic or synthetic geometry because of its distinguishing quality.

In general, points are regarded as the basic units of geometry construction. They can also be defined in synthetic geometry by the characteristics that they must possess, as in Euclid's definition of "that which has no part". They are typically defined in modern mathematics as components of the axiomatically defined set known as space. This is not the case with synthetic geometry, where a line is another fundamental object that is not understood as the set of the points through which it travels. With these modern definitions, every geometric shape is defined as a set of points.

A line is a "breadthless length" that "lies equally with respect to the points on itself," according to Euclid. Given the variety of geometries in modern mathematics, the idea of a line is directly related to how the geometry is described. In analytical geometry, for example, a line in the plane is frequently defined as the collection of points whose coordinates satisfy a particular linear equation, but in a more abstract context, such as incidence geometry, a line may be an independent object, distinct from the collection of points that lie on it.

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The dimensions of an object in one, two, and three dimensions are described by length, area, and volume, accordingly. The Pythagorean theorem can be used to determine a line segment's length in both analytic and Euclidean geometry. Area and volume can be described and computed in terms of lengths in a plane or three-dimensional space, or they can be defined as fundamental quantities distinct from length.

Many precise formulas for area and volume of various geometric objects have been discovered by mathematicians. In calculus, integrals like the Riemann integral or the Lebesgue integral can be used to define area and volume.

Mensuration :

Mensuration is an area of mathematics that deals with measuring various geometric figure properties and other things. An object or event's attributes are quantified through measurement so that they can be compared to those of other things or occurrences. Measurement, then, is the process of establishing how big or little a physical quantity is in relation to a fundamental reference quantity of the same kind. Measurement's breadth and applications depend on the setting and field. As stated in the International Vocabulary of Metrology published by the International Bureau of Weights and Measures, measurements do not apply to nominal qualities of things or occurrences in the natural sciences and engineering. However, measures can have numerous levels in other disciplines, such as statistics and the social and behavioural sciences. These levels include nominal, ordinal, radio and interval scales.

Trade, science, technology, and quantitative research across many areas all depend on measurement. To enable comparisons in the various spheres of human existence, numerous measurement systems have existed historically. These were frequently accomplished through regional agreements between business partners or collaborators. The contemporary International System of Units was created as a result of advancements made since the 18th century in the direction of unifying, universally acknowledged International Systems of Units standards ( SI ). All physical measures are condensed in this system to a mathematical combination of seven base units. In the discipline of metrology, measuring science is pursued.

Comparing an unknown quantity to a known or standard quantity is the definition of measurement.

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Kind, magnitude, unit, and uncertainty are four categories that can be used to group measurements of properties. They make it possible to compare measurements in an unambiguous manner. The degree of measurement is a taxonomy for a comparison's methodological nature. For instance, it is possible to compare two states of a property using ratio, difference, or ordinal preference. The type is frequently implied rather than expressly stated in the definition of a measuring process.

The magnitude is the characterization's numerical value, which is often measured using an appropriate measuring device. A unit gives the magnitude that is obtained as a ratio to a feature of an artefact used as a standard or a natural physical quantity a mathematical weighting factor. Errors that are random and systemic are represented by uncertainty.

The International System of Units ( SI ) is most frequently used in measurements as a basis for comparison. The system specifies seven basic units: the kilogram, the metre, the candela, the second, the ampere, the kelvin, and the mole. These units are all defined without using a specific physical object to act as a standard. In contrast to standard artefacts, which are susceptible to deterioration or destruction, artifact - free definitions fix measurements at a precise value associated to a physical constant or other invariable phenomena in nature. Instead, the measuring unit can only ever be altered by improving the accuracy with which the value of the constant to which it is connected is determined.

Units of measurement are obtained from previous agreements, with the exception of a few essential quantum constants. There is no requirement in nature for an inch to be a specific size or for a mile to be a more accurate unit of measurement than a kilometre. But over the course of human history, standards of measurement changed to provide communities with a set of universal norms, initially out of convenience and subsequently out of necessity. Rules governing measuring were initially created to stop commercial fraud.

Considering the complex architectural patterns and building blocks prevalent in the mathematical space, measurement is just an inseparable element of any scientific field, including the contemporary, classical and modern advanced Mensuration. Our competent tutors, teachers, analysts, professors, lecturers and the like strive to earnestly deliver an unmatched experience through various formats encompassing Mensuration maths private online tutions near me, offline maths classes in greater noida offline, Mensuration mathematics private home tuitions in greater noida west, maths private online tutors near me, online engineering maths teachers near me offline, Mensuration maths private online tution near me, mathematics private home teacher in noida extension, online engineering math teacher near me offline, Mensuration maths private online teachers in greater noida west, math private online lessons near me, Mensuration mathematics private online tuitions in noida extension, mathematics private home tuitions in noida extension, online engineering math teachers near me offline, Mensuration math private online lesson near me, maths private home teacher in noida extension, Mensuration math private online teachers near me and others.

There must be a considerable deal of work put into making measurements as accurate as possible since good measurement is crucial in many professions and because all measurements are inherently approximations. Consider the challenge of quantifying the amount of time it takes an object to fall one metre ( about 39 in ). Physics may be used to demonstrate that any object should descend one metre in around 0.45 seconds in the gravitational field of the Earth. Yet, a few of the sources of inaccuracy that occur are as follows :

This calculation uses 32 feet per second ( or 9.8 metres per second squared ) as the acceleration of gravity. However, this measurement is only accurate to two significant digits, therefore it is not exact. The gravitational field of the Earth changes with varying distance from Earth's centre. The physical sciences often use the classical definition of measurement, which states that it is the determination or estimation of ratios of quantities. Quantity and measurement are synonymous terms; qualities that can, in theory, be measured are considered to be quantitative. Euclid's Elements served as a precursor to the classical concept of quantity, which may be traced back to John Wallis and Isaac Newton.

Measurement is described in the representational theory as "the association of numbers with phenomena that are not numbers." Additive conjoint measurement is another name for the representational theory that is technically complex. Numbers are assigned in this type of representational theory based on correspondences or resemblances between the structures of numerical and qualitative systems. If these structural similarities can be proven, a property can be quantified. Numbers need simply to be assigned according to a rule in weaker kinds of representational theory, such as that which is inherent in the work of Stanley Smith Stevens.

Constructions :

The production of lengths, angles, and other geometric figures using just an idealised ruler and a pair of compasses is known as straightedge - and - compass construction in geometry. It is also referred to as ruler - and - compass construction, Euclidean construction, or classical construction.

The idealised ruler, or "straightedge", is thought to be infinitely long, have just one edge, and be devoid of any markings. The compass may not be used to directly transfer distances because it is presumed to have no maximum or minimum radius and to "collapse" when lifted from the page. ( This is a moot constraint because a distance can be communicated even with a collapsing compass using a multi-step process; see compass equivalence theorem )

In straightedge-and-compass constructions, the "straightedge" and "compass" are idealised representations of actual rulers and compasses.

The straightedge is a line that is endlessly long and has no markings. Only a line segment between two locations or an existing line segment may be drawn with it. The compass has no markings and can have any size radius ( unlike certain real - world compasses ). The centre and a point on the circle can be used as starting points to create circles and circular arcs. The compass could either fold after being removed off the page, destroying its "stored" radius, or it could not. Constructions of lines and circles have zero breadth and unlimited precision.

All straightedge - and - compass constructions are made up of five fundamental constructions that are applied repeatedly utilising previously created points, lines, and circles. These are drawing a line between two points, making a circle with a centre at a different location and a point in the middle, establishing the point where two ( non - parallel ) lines converge, establishing one or two spots where a line and a circle converge ( if they intersect ), making a single point or two places when two circles converge ( if they intersect ).

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The following are the most popular straightedge-and-compass constructions, viz, Making a segment into the perpendicular bisector, locating a segment's halfway, creating a line that is perpendicular to a point and a line, cutting a corner, mirroring a line's point, building a line across a circle's tangent point, making a circle out of three noncollinear points and tracing a line from a specific location along a specific line.

Depending on the intricacy of the tools needed for their solution, the ancient Greeks divided buildings into three main groups. A construction was classified as planar if it just required a straightedge and compass, solid if it also required one or more conic sections ( other than the circle ), and all other constructions were placed in the third category. This classification fits in well with the perspective of contemporary algebra. Complex numbers that can only be stated using square roots and field operations ( as previously mentioned ) have a planar architecture. A complicated number with a sound structure also contains cube root extraction.

If a point can be drawn with a straightedge, compass, and ( perhaps imaginary ) conic drawing tool that can draw any conic with already formed focus, directrix, and eccentricity, the point has a sound construction. Frequently, a smaller toolkit can be used to construct the same set of points. Every complex number with a sound construction can be built, for instance, by using a compass, straightedge, and a sheet of paper with the parabola y=x2 and the points ( 0,0 ) and ( 1,0 ) on it. Similarly, a device that can draw any ellipse with pre-built foci and main axes ( imagine two pins and a piece of rope ) is equally effective.

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The mathematical theory of origami is more effective than building with a straightedge and compass. With a compass and conic sketching tool, folds meeting the Huzita - Hatori axioms can create the exact same collection of points as the extended structures. As a result, two of the classic difficulties can be resolved using origami by using it to solve cubic equations ( and subsequently, quartic equations ).

With a markable ruler, Archimedes, Nicomedes, and Apollonius provided constructs. This would enable them, for instance, to take a line segment, two lines ( or circles ), and a point; they could then draw a line that goes through the point and crosses the two lines, such that the separation between the points of intersection is equal to the line segment. Due to the fact that the new line tends to the point, the Greeks named this neusis ( Latin for "inclination," "tendency," or "verging" ).

Functions :

A function in mathematics from a set X to a set Y allocates exactly one element of Y to each element of X. The set X is referred to as the function's domain, while the set Y is referred to as the function's codomain. Initially, functions represented the idealised relationship between two changing quantities. A planet's position, for instance, depends on time. In the past, the idea was developed with the infinitesimal calculus at the end of the 17th century, and the functions that were taken into consideration until the 19th century were differentiable ( that is, they had a high degree of regularity ). By the close of the 19th century, the idea of a function was codified in terms of the set theory.

An assignment of an element from set Y to each element of set X constitutes a function from set X to set Y. The sets X and Y are collectively referred to as the function's domain and codomain, respectively. The notation f: X—>Y denotes a function, its domain, and its codomain. The value of a function f at an element x of X, denoted by the symbol f( x ), is referred to as the image of x under f or the value of f applied to the input x. Although some authors establish a distinction between "maps" and "functions," functions are also sometimes referred to as maps or mappings.

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When a function is defined, the domain and codomain are not always explicitly stated, and without performing some ( possibly challenging ) computation, one may only be aware that the domain is a subset of a larger set. Usually, "a function from X to Y" refers to a function that may have a proper subset of X as its domain in mathematical analysis. A real - valued function of a real variable, for instance, may be referred to as a "function from the reals to the reals." Nevertheless, a "function from the reals to the reals" merely designates a set of real numbers that has a non - empty open interval and not the entire set of real numbers as the domain of the function.

In functional notation, a function is given a name right away, such as f, and is defined by what it does to the explicit input x using a formula in terms of x. Leonhard Euler invented functional notation in 1734. There are some frequently used functions that are denoted by a symbol made up of many letters ( usually two or three, generally an abbreviation of their name ). Roman type is typically used in this situation as opposed to italic font for single-letter symbols, such as "sin" for the sine function.

Arrow notation describes the rule of a function inline, without requiring a name to be given to the function. Functional notation is frequently substituted with index notation. In other words, one writes rather fx than writing f (x). For functions whose domain is the set of natural numbers, this is frequently the case. In this example, the element fn is referred to as the nth element of the series. Such a function is known as a sequence.

In some specialised areas of mathematics, there are additional notations for functions. For instance, to demonstrate the underlying duality, linear forms and the vectors they act upon are expressed using a dual pair in functional analysis and linear algebra. This is comparable to how quantum physics uses bra - ket notation. The lambda calculus function notation is used to convey the fundamental ideas of function abstraction and application in logic and computing theory. Commutative diagrams, which extend and generalise the arrow notation for functions mentioned above, are used in category theory and homological algebra to express networks of functions in terms of how they and their components commute with one another.

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Although a function is sometimes referred to as a map or a mapping, some authors distinguish between the terms "map" and "function." For instance, the term "map" is frequently used to refer to a "function" having a unique structure ( e.g. maps of manifolds ). For the sake of succinctness, map is frequently substituted for homomorphism in this case ( e.g., linear map or map from G to H instead of group homomorphism from G to H ). Because the structure of the codomain explicitly relates to the definition of the function, some authors reserve the term "mapping" for that situation.

Some authors, like Serge Lang, only use the term "function" to describe mappings whose codomain is a subset of the real or complex numbers, while using the term "mapping" to describe functions that are more broadly defined. A map is a type of evolution function in the theory of dynamical systems that is used to build discrete dynamical systems. also see Poincaré's map. Whatever the definition of a map, words like domain, codomain, injective, and continuous have the same meaning as those used to describe functions.

A polynomial function is, more broadly speaking, a function that can be expressed in terms of a formula that only uses additions, subtractions, multiplications, and exponentiations of nonnegative integers. A rational function is the same, and divisions are permitted. Within algebraic functions both nth roots and roots of polynomials are allowed. A bijective function from the positive real numbers to the real numbers is the natural logarithm. The exponential function, its opposite, transforms the real numbers into positive numbers.

The antiderivative of one function can be referred to as several different functions. This is how the natural logarithm works. In a broader sense, it is possible to define many functions, including the majority of special functions, as solutions of differential equations. The exponential function, which is the only function that is equal to its derivative and takes the value 1 for x = 0, is likely the most straightforward illustration. Functions on the domain in which power series converge can be defined.

A graph is frequently used to present an understandable illustration of a function. It is simple to determine from a function's graph whether it is increasing or decreasing, which is an illustration of how a graph aids in understanding a function. Bar charts can also be used to represent some functions. In the Cartesian plane there is a 2-dimensional coordinate system. This might be used in part to produce a plot that illustrates the function. Plots are used so frequently that they too are referred to as the function's graph. In various coordinate systems, functions can also be graphically represented.

Functions are quite varied and diversified. The range of diversity is vast, manifold and the ways of interpretation too vary across multiple channels. Other branches of maths like sets also play a critical role in the comprehension of functions. Seeing this challenge we at Wise Turtle Academy attempt to deliver quality oriented and value added learning support through varied formats including Functions maths lesson in greater noida west, offline mathematics lecture in greater noida west offline, Functions math private home teacher in noida extension, mathematics private home tution in noida extension, Functions mathematics private online teachers in greater noida west, offline math lectures in greater noida west offline, Functions maths private online teacher in greater noida west, mathematics private online tuitions in greater noida west, Functions offline maths class in greater noida offline, offline math lessons in greater noida west offline, Functions maths private home tutors in noida extension, Functions math private home tutors in greater noida west and others.

A table of values can be used to represent a function. This allows a function to be fully described if the function's domain is finite. On the other hand, if the domain of a function is continuous, a table can present the values of the function at particular domain values. Interpolation can be used to calculate the function's value if an intermediate value is required.

The natural numbers, integers, and functions with a finite set as their domain are frequently represented using bar charts. In this example, an x-axis interval represents an element of the domain, and a rectangle with a f ( x ) value at its base represents the value of the function that corresponds to that element ( possibly negative, in which case the bar extends below the x - axis ).

The empty function, often known as the empty map, is a special function that goes from the empty set to each set X. The empty set is the graph of an empty function. For the theory to be coherent and to prevent exceptions regarding the empty set in many sentences, empty functions must exist. There is exactly one empty function for each set according to the standard set-theoretic definition of a function as an ordered triplet ( or equivalent ones ).

Older English - language literature used the phrases "one - to - one" and "onto" more frequently; "injective," "surjective," and "bijective" were originally French terminology created in the second half of the 20th century by the Bourbaki group and imported into English. Just to be clear, a "one - to - one function" is an injective function, but a "one - to - one correspondence" refers to a bijective function.

A function g that makes f a restriction of g is said to be an extension of that function. The method of analytical continuation, which enables extending functions whose domain is a small portion of the complex plane to functions whose domain is virtually the entire complex plane, is a typical use of this notion.

A function that depends on multiple parameters is referred to as a multivariate function or function of several variables. These operations are often used. For instance, a car's location on a road depends on the distance travelled and the average speed. A function of n variables is, more precisely, a function whose domain is a set of n-tuples. An example of a bivariate function is the multiplication of integers, which has the set of all pairs ( 2-tuples ) of integers as its domain and the set of integers as its codomain. Every binary operation is same. Any mathematical operation is, in general, defined as a multivariate function.

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The new infinitesimal calculus was founded on the concept of function, which emerged in the 17th century. At that time, all functions were presumed to be smooth, and only real-valued functions of a real variable were taken into account. However, the concept was rapidly expanded to include functions for many variables as well as complex variables. A function's mathematically precise definition and the definition of functions with unrestricted domains and codomains were both presented in the second half of the 19th century.

Today, functions are employed in all branches of mathematics. When the word "function" is employed in the context of basic calculus, it refers to a real-valued function of a single real variable. In the second or third year, the more comprehensive definition of a function is typically introduced.

A real function is a real-valued function of a real variable, meaning it has an interval as part of its domain and has the field of real numbers as its codomain. These operations are referred to as functions in this section. The most frequent functions in mathematics and its applications are continuous, differentiable, and even analytical, which indicates some regularity. These functions can be seen via their graphs because of this regularity. All functions in this section are differentiable in some range.

The domain of polynomial functions, which are defined by polynomials, is the entire set of real numbers. Constant functions, linear functions, and quadratic functions are among them. Real numbers are the domain of rational functions, which are quotients of two polynomial functions, with a certain number of them eliminated to prevent division by zero.

If the sign of the derivative is consistent across the interval and the function is differentiable, it is monotonic. A real function f has an inverse function, which is a real function with the domain f(I) and image I, if it is monotonic in the interval I. Inverse trigonometric functions, where the trigonometric functions are monotonic, are defined in terms of trigonometric functions in this way. Another illustration: The natural logarithm has an inverse function that is a bijection between the real numbers and the positive real numbers since it is monotonic on the positive real numbers and has the full real line as its image. The exponential function is this inverse function. The implicit function theorem defines a large number of additional real functions.

A function is referred to as a vector-valued function if the elements of the codomain are vectors. Particularly helpful applications for these abilities include modelling physical attributes. As an illustration, a vector-valued function is one that assigns a velocity vector to each fluid point.

A function space is a collection of scalar-valued or vector-valued functions that together form a topological vector space in mathematical analysis, more specifically in functional analysis. A function space that forms the foundation of the theory of distributions, for instance, is formed by real smooth functions with a compact support ( i.e., they are zero outside some compact set ).

By enabling the study of function qualities using their algebraic and topological features, function spaces play a crucial role in advanced mathematical analysis. For instance, the study of function spaces leads to all theorems proving the existence and uniqueness of solutions to ordinary or partial differential equations.

Functions exhibit multi dimensional relationships among various parameters. Functions are good for various machine learning initiatives, as well as basic to advanced mapping and modelling techniques. We have aced these complexities and are quite adept at transferring the knowledge base across to others, through various ways including maths private home tutors in greater noida west, maths private home tutor in noida extension, offline math lecture in greater noida west offline, advanced engineering maths home lessons near me offline, maths private home lessons in greater noida west, math private home tutor in greater noida west, mathematics private online teacher in greater noida west, mathematics tuitions in noida extension, offline mathematic class in greater noida west offline, offline math lesson in greater noida west offline, maths private home lesson in greater noida west, advanced engineering maths home lesson near me offline, maths private online tuitions in greater noida west, maths private home tutor in greater noida west and others.

A number of techniques for defining functions of real or complex variables begin with a local definition at a point or in its vicinity and then extend the function through continuity to a much broader domain. When examining complex functions, which are often analytical functions, the usefulness of the concept of multi-valued functions becomes more apparent. In most cases, the complex plane constitutes the domain to which a complex function may be extended via analytic continuation. Yet, one frequently obtains different values while extending the domain along two distinct paths.

Given that a function's domain and codomain must both be sets, the definition of a function in this article necessitates the understanding of sets. As it is typically not difficult to examine only functions whose domain and codomain are sets that are well defined, even if the domain is not explicitly stated, this is not a problem in ordinary mathematics. While it can occasionally be helpful to focus on more universal functions.

In computer programming, a function is typically a component of a computer programme that actualizes the conceptually abstract idea of function. It is a programme unit that generates an output for each input, in other words. Nonetheless, every subroutine is referred to as a function in many programming languages, even when there is no output and the functionality consists just of changing some data in the computer's memory.

The paradigm of programming known as "functional programming" entails solely employing subroutines that behave like mathematical functions when creating programmes. For instance, the function if then else accepts three functions as arguments and, depending on whether the first function's result is true or false, either the second or third function's result is returned.

With the exception of terminology specific to computers, "function" in computer science refers to mathematical operations. The computability of a function is a crucial quality in this field. Many models of computing have been created to give this idea — and the related concept of algorithm — exact meanings. The traditional ones include general recursive functions, lambda calculus, and Turing machines. The central proposition of computability theory states that all alternative models of computation that have ever been suggested either define the same set of computable functions or a smaller set than what is defined by these three models of computation. The Church-Turing thesis asserts that every definition of a computable function that is philosophically acceptable also defines the same functions.

The partial functions from integers to integers known as general recursive functions can be defined through the use of the operators, constant functions, successors, projection functions, minimization, composition and primitive recursion.

They are only defined for functions from integers to integers, but because of the following characteristics, they can model any computable function :

Every series of symbols may be represented as a sequence of bits, and a bit sequence can be read as the binary representation of an integer. A computation is the manipulation of finite sequences of symbols ( number digits, formulas, etc. ).

Mathematical functions are thus quite difficult concepts and theories to comprehend. Any new beginner will naturally face serious challenges digesting the problems associated with the specialisation of Functions. To assist and overcome these problems Wise Turtle Academy stands prepared and ready to dole out various programmes in various formats, including but not limited to maths lessons in greater noida west, mathematics private online lessons in greater noida west, maths private online tutions in greater noida west, mathematics private home lesson in greater noida west, offline mathematics class in greater noida west offline, offline mathematics tutorial in greater noida west offline, online applied maths class 12 coaching near me, offline mathematics class in greater noida offline, math private online teachers in greater noida west, maths tutorials in greater noida west, maths tution in noida extension, mathematics private home tutors in greater noida west and others.

Functional programming's theoretical foundation is lambda calculus, a theory that specifies computable functions without relying on set theory. It is made up of terms that are either variables, function definitions ( often known as "terms" ), or terms to which functions are applied. The axioms of the theory, the equivalence, the reduction, and the conversion, can be seen as principles of computing and are used to manipulate terms.

Domain and codomain of a function are not part of lambda calculus in its original form. In typed lambda calculus, they have essentially been introduced as type in the theory. Compared to untyped lambda calculi, most types of typed lambda calculi can define fewer functions.

According to this definition, a "graph" is a collection of item pairs. The functions from the real numbers to themselves are the most appropriate to graphs in the sense of diagrams. All functions can be defined by sets of pairs, but creating a graphic for functions between sets may not be feasible ( such as sets of matrices ).

Functions have varied applications that span across Astronomy, Deep Oceanography, Geospatial Information Systems, Earth Seismology, Physical Sciences, Astronomical Measurements, Machine Learning, Acoustics Technology, navigational sciences, Artificial Intelligence, Electronics Technology, Electrical Technology , Air Sensors & Detection Systems, Informatics Practices , Radars, Computers & Communication Technology , Meteorology, Information & Communication Technology , Under Water Technologies and many other scientific fields.

This is a result of the extensionality axiom, which states that two sets are identical if and only if they include the same members. Some authors exclude codomain from a function's definition, in which case the idea of equality must be treated carefully.

Although most of the mathematical operations that are encountered in elementary courses are considered elementary in this context, some elementary operations, such as those involving the roots of complex polynomials, are not considered elementary in the common sense.

Sets :

The mathematical representation of a collection of unique items is called a set ; A set contains elements or members, which can be any type of mathematical object, including variables, other sets, numbers, symbols, points in space, lines, and other geometric shapes. A set with a single element is a singleton, while a set with no elements is an empty set. A set can either be infinite or have a finite number of elements. If all of the elements in two sets are identical, then the sets are equal.

Modern mathematics is rife with sets. In fact, since the first part of the 20th century, Zermelo-Fraenkel set theory has been the de facto method for constructing solid foundations for all areas of mathematics. The concept of a set first appeared in mathematics at the end of the nineteenth century. Bernard Bolzano coined the German word for set, Menge, in his book Paradoxes of the Infinite.

A set is a collection of definite, distinct objects of our perception or thought that are referred to as set elements. The most important property of a set is that it can have elements, also known as members. When two sets have the same elements, they are equal. Sets A and B are equal if every element of A is an element of B and every element of B is an element of A; this property is known as set extensionality.

The simple concept of a set has proven extremely useful in mathematics, but paradoxes arise when no restrictions on how sets can be constructed are imposed. Russell's paradox demonstrates that the "set of all sets that do not contain themselves" does not exist. Cantor's paradox demonstrates that "the set of all sets" does not exist. A set, according to naive set theory, is any well-defined collection of distinct elements, but problems arise due to the ambiguity of the term well-defined.

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The qualities of sets have been defined by axioms in further attempts to address these dilemmas since the original formulation of naive set theory. The idea of a set is treated as a fundamental concept in axiomatic set theory. The axioms serve as a foundation from which first-order logic can be used to infer the validity or falsity of specific mathematical propositions ( statements ) regarding sets. Yet, it is impossible to utilise first-order logic to demonstrate that any such specific axiomatic set theory is paradox-free, in accordance with Gödel's incompleteness theorems.

Sets are frequently identified in mathematical publications by capital letters in italics, such as A, B, and C. A set may also be referred to as a collection or family, particularly if its components are also sets. A set is defined by listing each member of it within curly brackets and commas in roster or enumeration notation. The ordering of the elements in roster notation is meaningless because the only thing that matters in a set is whether each element is present or not ( in contrast, in a sequence, a tuple, or a permutation of a set, the ordering of the terms matters ).

The list of members can be condensed for sets with several items, especially those that adhere to an implicit pattern, by using an ellipsis. A set with an unlimited number of elements is known as an infinite set. An ellipsis, or two ellipses, are used to denote an infinite set in roster notation, indicating that the list goes on forever.

The set - builder notation describes a set as a pick from a bigger set that is based on an elemental condition. The description can be read as "F is the set of all integers n such that n is an integer in the range from 0 to 19 inclusive." The vertical bar "|" in this notation denotes "such that." Some authors choose to omit the vertical bar in favour of a colon, ":".

A rule is used in an intensional definition to determine membership. Examples include set-builder notation-based definitions and semantic definitions.

A set's elements are listed in an extensional definition's description of the set. Enumerative definitions are another name for them. A roster with an ellipsis is an example of an ostensive definition, which uses examples of elements to explain a collection.

The only set that is empty or null contains no members. It is represented as an emptyset. A is referred to as being a subset of B or contained in B, if every element of set A is also in set B. A collection of sets is graphically represented by an Euler diagram, where each set is represented by a planar region with its members inside, enclosed by a loop.

The region representing A is entirely contained within the region representing B if A is a subset of B. When two sets do not share any elements, the regions do not overlap. In comparison, a Venn diagram is a graphical representation of n sets where the n loops split the plane into n zones. Mathematicians commonly refer to certain sets of mathematical significance, and as a result, they have unique names and notational norms to help them be recognised.

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The number of elements in each of the aforementioned sets of integers is limitless. Each of the sets below it is a subset of the others. Superscript plus and minus signs are occasionally used to indicate sets of positive or negative numbers. In more technical terms, a function is a special kind of relation, one that connects each element of A to precisely one element of B. It is a rule that assigns each "input" element of A, a "output" that is an element of B.

In sets, we deal with many types of functions, namely, injective ( or one - to - one ) if it maps any two different elements of A to different elements of B; surjective ( or onto ) if there is at least one element of A that maps to each element of B; and bijective ( or a one - to - one correspondence ) if the function is both injective and surjective. In this case, there are no unpaired elements because each element of A is paired with a unique element of B.

A bijective function is also known as a one-to-one correspondence, a surjective function is known as a surjection, and an injective function is known as an injection. The number of members in a set S is its cardinality, indicated by the symbol | S |. For instance, | B | Equals 3 if B = "blue, white, red." Roster notation does not count repeated members, therefore | blue, white, red, blue, white | also equals three. Formally, two sets are said to have the same cardinality if their relationships are one - to - one. The empty set has zero cardinality.

Some sets have an infinite or limitless list of elements. For instance, the set of natural numbers in math is infinite. In actuality, all of the unique sets of numbers listed in the previous section are limitless. The cardinality of infinite sets is infinite.

Several infinite cardinalities have different sizes. The fact that the set of real numbers has more cardinality than the set of natural numbers is perhaps one of the most important conclusions from set theory. The term "countable sets" refers to sets whose cardinality is less than or equal to that of "math N"; these sets can either be finite sets or countably infinite sets ( sets with the same cardinality as "math N") ; some authors use the term "countable" to refer to "countably infinite" sets.

There is no set with cardinality precisely between the cardinality of the natural numbers and the cardinality of a straight line, according to Georg Cantor's continuum theory, which was first forth in 1878. Paul Cohen demonstrated in 1963 that the continuum hypothesis is independent of the Zermelo - Fraenkel set theory and the axiom of choice that make up the ZFC axiom system.

Modern mathematics is rife with sets. In abstract algebra, for instance, structures like groups, fields, and rings are sets that are closed under one or more operations. The creation of relations is one of the primary uses of naive set theory. The Cartesian product A x B is a subset of a relation between domains A and B.

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The set of all S-subsets is known as the power set of a set. As they are both subsets of S, the empty set and S are both parts of the power set of S. A set of nonempty subsets of a set S is referred to as a partition if each member x in S is contained in exactly one of these subsets. Since no two sets of the partition have any elements in common, the subsets are pairwise disjoint, and S is the union of all the subsets of the partition.

Assume a universal set U has been fixed, which contains all of the elements under discussion, and that A is a subset of U. The set of all items (from U) that do not belong to A is known as the complement of A. It could be written as Ac or A′. Given any two sets A and B, respectively, their union is the collection of all objects that belong to A or B, or both is referred to as A U B. Their intersection is the collection of all objects that belong to both A and B and is known as A intersection B.

According to one of De Morgan's laws, (A ∪ B)′ = A′ ∩ B′. (that is, the elements outside the union of A and B are the elements that are outside A and outside B).

The product of the cardinalities of A and B is the cardinality of A x B. When A and B are both finite, this is a basic fact. Multiplication of cardinal numbers is defined to make this true when one or both are infinite.)

Any set's power set is transformed into a Boolean ring by adding its symmetric difference and multiplying its intersection. A method for counting the elements in a union of two finite sets in terms of the dimensions of the two sets and their intersection is the inclusion-exclusion principle.

Matrices :

A matrix, sometimes known as matrices, is a rectangular array or table of letters, numbers, or other symbols organised in rows and columns that is used to represent a mathematical object or a characteristic of one.

Matrix representations of linear mappings without additional details enable explicit computations in linear algebra. As a result, a significant portion of linear algebra involves the study of matrices, and the majority of the characteristics and operations of abstract linear algebra may be described in terms of matrices. The composition of linear maps, for instance, is represented by matrix multiplication.

Not every matrix has a connection to linear algebra. This is especially true for incidence matrices and adjacency matrices in graph theory. Unless otherwise stated, all matrices in this article that are connected to linear algebra represent linear maps or can be interpreted as such.

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In matrix theory, square matrices — those with the same number of rows and columns — play a significant role. One of the most typical examples of a noncommutative ring is formed by square matrices in a particular dimension. For example, a square matrix is invertible if and only if it has a nonzero determinant, and the eigenvalues of a square matrix are the roots of a polynomial determinant. The determinant of a square matrix is a number connected to the matrix, which is crucial for the study of a square matrix.

Matrix representations of coordinate changes and geometric transformations ( such as rotations ) are frequently utilised in geometry. Many computing issues in numerical analysis are resolved by matrix reduction.

The entries of a matrix are a rectangular array of numbers ( or other mathematical objects ). Standard operations like addition and multiplication can be performed on matrices. A matrix over a field F is often just a rectangular array of F's constituent elements. Real and complex matrices are those whose entries are real or complex numbers, respectively.

This frequently entails computation with matrices with enormous dimensions. The majority of mathematical and scientific disciplines make use of matrices, either directly or indirectly through the use of geometry and numerical analysis.

The entries or elements of the matrix are its numbers, symbols, or phrases. In a matrix, the terms "rows" and "columns" refer to the horizontal and vertical rows of entries, respectively.

The quantity of rows and columns in a matrix determines its size. As long as they are positive integers, there is no restriction on how many rows and columns a matrix ( in the conventional sense ) can have. An m n matrix, also known as an m-by-n matrix, is a matrix having m rows and n columns. M and n are referred to as the matrix's dimensions. As an illustration, matrix A above is a 3 by 2 matrix.

Row vectors are matrices with a single row, and column vectors are matrices with a single column.

A square matrix is one that has the same number of rows and columns. An infinite matrix is one that has an infinite number of rows, columns, or both. Consider a matrix with no rows or columns, often known as an empty matrix, in some situations, such as computer algebra applications.

A single-row matrix that is occasionally used to represent a vector. A single - column matrix that is occasionally used to represent a vector. A matrix with the same number of rows and columns that is occasionally used to describe a linear transformation from one vector space to another, such as reflection, rotation, or shearing.

Matrix addition, scalar multiplication, transposition, matrix multiplication, row operations, and submatrix are some of the fundamental operations that can be used to alter matrices.

Some matrix operations share common characteristics with operations on numbers, such as addition being commutative, which means that the sum of the matrix is independent of the order of the summands. The transpose is compatible with scalar multiplication and addition. If and only if the number of columns in the left matrix equals the number of rows in the right matrix, we can say that two matrices have been multiplied.

In stark contrast to ( rational, real, or complex ) numbers, whose product is independent of the order of the parts, matrix multiplication is not commutative.

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Several less common operations on matrices that can be regarded as forms of multiplication exist in addition to the standard matrix multiplication just mentioned, such as the Hadamard product and the Kronecker product. These appear when attempting to solve matrix equations like the Sylvester equation.

Row operations come in three different flavours :

Adding a row to another row is known as row addition. Row multiplication, which is the process of multiplying each row's entries by a non-zero constant; Row switching, which involves a matrix's two rows being switched.

Solving linear equations and locating matrix inverses are two examples of the many applications for these procedures.

A matrix's submatrix can be created by removing any group of rows and / or columns. By calculating the determinant of specific submatrices, one can determine the minors and cofactors of a matrix.

A square submatrix called a principal submatrix is created by deleting particular rows and columns. From author to author, the term changes. Some writers define a principal submatrix as a submatrix in which the set of remaining row indices and column indices are the same. According to some authors, a principal submatrix is one in which the top k rows and columns, for any value of k, are the only ones left. This kind of submatrix is also referred to as a principal principal submatrix.

Matrix multiplication relates to map composition according to the one-to-one connection between matrices and linear maps. The greatest number of the matrix's linearly independent column vectors as well as its maximum number of linearly independent row vectors make up the rank of a matrix A. It is, in essence, the size of the linear map image that is represented by A. According to the rank-nullity theorem, the number of columns in a matrix is equal to the dimension of its kernel times its rank.

A is referred to as an upper triangular matrix if all of its entries below the main diagonal are zero. Similar to this, A is referred to as a lower triangular matrix if all of its elements above the primary diagonal are zero. A is referred to as a diagonal matrix if all entries other than the main diagonal are zero.

It is both a unique form of diagonal matrix and a square matrix of order n. It is known as an identity matrix because it multiplies a matrix without changing it. A scalar matrix is a nonzero scalar multiple of an identity matrix. The scalar matrices form a group under matrix multiplication that is isomorphic to the multiplicative group of nonzero elements of the field if the matrix entries are taken from a field.

Hermitian matrices, which satisfy A = A, are frequently used in place of symmetry in complex matrices. The star or asterisk indicates the matrix's conjugate transpose, which is the transposition of the complex conjugate of A.

Real symmetric matrices and complex Hermitian matrices both have an eigenbasis according to the spectral theorem, meaning that any vector can be expressed as a linear combination of eigenvectors. All eigenvalues are real in both scenarios. This theorem can be extended to situations in infinite dimensions involving matrices with an unlimited number of rows and columns.

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The symmetric matrix is said to be positive-semidefinite if the quadratic form f only produces non-negative values ( positive or zero ) ( or negative -semidefinite if only non - positive values are produced ). The matrix is indefinite precisely when it is neither positive - semidefinite nor negative - semidefinite.

If and only if all of a symmetric matrix's eigenvalues are positive, the matrix is positive-definite, which also means that it is invertible. Two 2 - by - 2 matrix options are shown in the table to the right.

The terminology and outcome are the same for complex matrices, with symmetric matrix, hermitian form, sesquilinear form, and conjugate transpose replacing, respectively, quadratic form, bilinear form, and transpose.

A square matrix with real entries whose columns and rows are orthogonal unit vectors is known as an orthogonal matrix. A matrix A is equivalently said to be orthogonal if its transpose equals its inverse.

An orthogonal matrix with determinant + 1 is a specific orthogonal matrix. Every orthogonal matrix with determinant +1 is a pure rotation without reflection as a linear transformation, which keeps the orientation of the transformed structure intact. Conversely, every orthogonal matrix with determinant -1 reverses the orientation by combining a pure reflection and a ( possibly null ) rotation. The identity matrices are simple rotations with angle zero and have determinant 1.

A unitary matrix is an orthogonal matrix's complex analogue.

The total of the diagonal entries makes up a square matrix A's trace, or tr(A). Despite the fact that, as was already mentioned, matrix multiplication is not commutative, the trace of the product of two matrices is unaffected by the factors' order.

Thus, the trace of the product of more than two matrices is independent of cyclic permutations of the matrices. Nevertheless, this does not generally hold true for arbitrary permutations ( such as, for instance, tr(ABC) ≠ tr(BAC) ). Moreover, a matrix's trace and its transpose are equivalent.

A integer that encodes specific matrix characteristics is known as the determinant of a square matrix A, represented by det(A) or |A|. If and only if a matrix's determinant is nonzero, the matrix is invertible.

Three by three matrix determinants have six terms ( rule of Sarrus ). The Leibniz formula, which is longer, applies these two principles to all dimensions. A square matrix's product's determinant is equal to the sum of its determinants.

The determinant is unaffected by adding a multiple of any row to another row or a multiple of any column to another column. The determinant is affected by changing two rows or two columns by multiplying it by 1. Any matrix may be converted into a lower ( or upper ) triangular matrix using these operations, and for such matrices, the determinant is equal to the product of the entries on the major diagonal, offering a way to figure out the determinant of any matrix. The determinant is finally expressed using the Laplace expansion in terms of minors, or determinants of smaller matrices.

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The determinant of a 1 - by - 1 matrix, which is its only entry, or even the determinant of a 0 - by - 0 matrix, which is 1, can be used to start a recursive definition of determinants that can be seen to be equivalent to the Leibniz formula. Cramer's rule, which states that the division of the determinants of two related square matrices equals the value of each of the system's variables, can be utilised to use determinants to solve linear equations.

Many methods can be used to do matrix calculations. Both direct algorithms and iterative methods can be used to tackle a wide variety of issues. By identifying a series of vectors xn that converge to an eigenvector as n approaches towards infinity, for instance, it is possible to determine the eigenvectors of a square matrix.

The effectiveness and precision of each algorithm should be evaluated in order to select the one that is best suited for solving each unique challenge. Numerical linear algebra is the field that studies these topics. The complexity of algorithms and their numerical stability are the two key factors, as they are in other numerical circumstances.

Finding upper limits or estimations of how many basic operations, like additions and scalar multiplications, are required to execute a particular algorithm, such the multiplication of matrices, is how complexity of an algorithm is determined. With the definition provided above, calculating the matrix product of two n-by-n matrices requires n3 multiplications because n multiplications are required for each of the n2 items in the product.

This "naive" algorithm is outperformed by the Strassen algorithm, which only requires n2.807 multiplications. A more sophisticated method incorporates particular computer device features.

In many real-world scenarios, more details about the matrices at play are known. Sparse matrices, or matrices with the majority of their entries being zero, are an important instance. For example, the conjugate gradient method is a specially tailored methodology for solving linear systems Ax = b for sparse matrices A.

A numerically stable algorithm is one in which small variations in the input values do not result in significant variations in the output. The conditioning of linear algebraic issues, like calculating a matrix's inverse, can be captured by a matrix's norm.

While most programming languages for computers enable arrays, they lack built - in commands for matrices. Alternatively, almost all widely used programming languages offer matrix operations on arrays through external libraries that are readily available. One of the first numerical applications of computers was matrix manipulation. Since its second edition implementation in 1964, the original Dartmouth BASIC contained built-in commands for matrix arithmetic on arrays. Certain engineering desktop computers, such the HP 9830, included ROM cartridges with matrices-specific BASIC instructions as early as the 1970s. Many mathematical programmes can be used to help manipulate matrices, and certain computer languages like APL were created for the purpose.

Matrix rendering can be done in a number of different ways to make it more user-friendly. Common names for them include matrix decomposition and matrix factorization techniques. The appeal of all these methods is that they maintain certain matrix properties, such as determinant, rank, or inverse, making it possible to calculate these values after the transformation, or that for specific types of matrices, certain matrix operations are algorithmically simpler to perform.

Lower ( L ) and upper triangular matrices combine to form the LU decomposition factor matrices ( U ). Once this decomposition has been calculated, the straightforward forward and reverse substitution method can be used to solve linear systems more effectively. Similarly, it is simpler to calculate triangular matrices' inverses algorithmically. Similar algorithms include the Gaussian elimination, which converts any matrix to row echelon form. In both cases, the matrix is multiplied by appropriate elementary matrices, which entails permuting the rows or columns or adding multiples of one row to another row. Any matrix A can be expressed using singular value decomposition as the product UDV*, where U and V are unitary matrices and D is a diagonal matrix.

It is significantly simpler to determine the power of a diagonal matrix by obtaining the corresponding powers of the diagonal elements than it is to exponentiate for A. In order to solve linear differential equations, matrix logarithms, and matrix square roots, it is usually necessary to compute the matrix exponential eA. Other techniques, like the Schur decomposition, can be used to prevent numerically ill-conditioned situations.

Several generalisations can be made for matrices. While linear algebra codifies the characteristics of matrices in the concept of linear maps, abstract algebra uses matrices with entries in more generic fields or even rings. It is feasible to think about matrices with an endless number of rows and columns. Tensors are another expansion, which can be thought of as higher-dimensional arrays of numbers. Tensors differ from vectors and matrices, which are both rectangular or two-dimensional arrays of numbers, in that vectors are frequently realised as sequences of numbers. Matrix groups are formed by matrices when specific conditions are met. Matrix rings are rings that matrices can generate under specific circumstances. Despite the fact that the product of matrices is not always commutative, some matrices can nonetheless create fields known as matrix fields.

The matrices with real or complex number elements are the subject of this article. In contrast to real or complex numbers, matrices can be thought of with far more diverse types of entries. Any field, i.e., a set where addition, subtraction, multiplication, and division operations are specified and behave appropriately, may be used in place of R or C as a first step towards generalisation. Examples include rational numbers or finite fields. Matrix over finite fields are used, for instance, in coding theory. Wherever eigenvalues are taken into account, since they are polynomial roots, they may only exist in a field that is greater than the matrix's elements; for example, they may be complex in the case of a matrix with real entries.

It is then possible to consider each square matrix to have a complete set of eigenvalues by reinterpreting the entries of a matrix as components of a bigger field (for instance, by viewing a real matrix as a complex matrix whose entries happen to be all real). As an alternative, only matrices having entries in an algebraically closed field, like C, can be taken into account right away.

In mathematics more generally, matrices containing entries in a ring R are frequently utilised. In that a division operation is not required, rings are a more broad concept than fields. In this situation as well, matrices can be added to and multiplied in the same ways. All square n-by-n matrices over R are represented by the set M(n, R) (sometimes written Mn(R) ), which is a ring known as the matrix ring and is isomorphic to the endomorphism ring of the left R-module Rn. The ring M(n, R) is also an associative algebra over R if the ring R is commutative, or if its multiplication is commutative. The Leibniz formula can still be used to define the determinant of square matrices over a commutative ring R.

Such a matrix generalises the case over a field F, where every nonzero element is invertible, and is invertible if and only if its determinant is invertible in R. [60] Supermatrices are matrices over superrings.

Not all entries in matrices are placed in the same ring, or even in any ring at all. Block matrices, which can be thought of as matrices whose entries are matrices themselves, are one unique but typical example. The entries do not have to be square matrices or rings, but their widths must meet certain compatibility requirements.

The category of all matrices with entries in a field k and multiplication as composition is identical to the category of finite-dimensional vector spaces and linear mappings over this field, which restates these facts more intuitively.

More generally, the R-linear maps between the free modules Rm and Rn for any ring R with unity can be represented as the set of m x n matrices. These mappings can be composed when n = m, which results in the matrix ring of n x n matrices, which represents the endomorphism ring of Rn.

A group is a mathematical structure made up of a collection of items and a binary operation, which, under specific conditions, combines any two objects into a third. A matrix group is one in which the objects are matrices and the group operation is multiplication. The most general matrix groups—known as general linear groups—are those composed of all invertible matrices of a given size because a group's elements must all be invertible.

More matrix groups can be defined using any attribute of matrices that is maintained by matrix products and inverses. For instance, a subgroup of matrices with a certain dimension and a determinant of 1 is (that is, a smaller group contained in) is called a special linear group.

Determinant 1 or 1 exists in every orthogonal matrix. Special orthogonal group is the name of the subgroup formed by orthogonal matrices with determinant 1. Consider the regular representation of the symmetric group to see that any finite group is isomorphic to a matrix group. Matrix groups, which are rather well understood, can be used to study general groups using representation theory.

It is also possible to take into account matrices that have an endless number of rows and/or columns despite the fact that because they are infinite objects, they cannot be explicitly written down. All that requires is that there is a clearly defined entry for each element in the set indexing rows and each element in the set indexing columns ( these index sets need not even be subsets of the natural numbers). While it is still possible to describe the fundamental operations of addition, subtraction, scalar multiplication, and transposition without any issues, matrix multiplication may require endless summations to define the resulting entries, and these are not often defined.

For the following reason, only matrices with all of their columns having a limited number of nonzero entries can be utilised to represent linear maps when using infinite matrices. It is necessary to choose bases for both spaces in order for a matrix A to describe a linear map f: VW. By definition, this means that every vector in the space can be written as a (finite) linear combination of basis vectors, and that only a finite number of entries vi are nonzero when written as a (column) vector v of coefficients.

Now, the columns of A describe the images by f of distinct basis vectors of V in the basis of W. But, this description is only relevant if the number of nonzero entries in these columns is finite. Yet, there is no restriction on the rows of A because every one of its entries, even if it is given as an infinite sum of products, contains only a finite number of nonzero terms and is thus properly defined. In the product Av, there are only a finite number of nonzero coefficients of v involved. Also, this amounts to creating a linear combination of A's columns that only involves a finite number of them in reality.

Products of two matrices of the specified type are well defined, of the same type, and correspond to the composition of linear maps (assuming that the column-index and row-index sets match).

The requirement of row or column finiteness can be removed if R is a normed ring. Finite sums can be replaced by totally convergent series once the norm has been established. For instance, a ring is formed by the matrices whose column sums are absolutely convergent sequences. Similar to this, a ring is formed by matrices whose row sums are absolutely convergent series.

Moreover, infinite matrices can be used to define operators on Hilbert spaces, where issues with convergence and continuity recur, necessitating the imposition of further restrictions. The more powerful and abstract tools of functional analysis can be utilised in place of matrices, whose explicit point of view tends to obscure the issue.

A matrix with zero rows, columns, or both is referred to as an empty matrix. When dealing with maps involving the zero vector space, empty matrices are helpful. A 3-by-3 zero matrix, for instance, corresponds to the null map from a 3-dimensional space V to itself, while BA is a 0-by-0 matrix if A is a 3-by-0 matrix and B is a 0-by-3 matrix. Although empty matrices can be created and used for computation, there is no standard notation for them. Regarding the empty product appearing in the Leibniz formula for the determinant as 1, the determinant of the 0-by-0 matrix is 1.

Matrix-based structures have many uses in both mathematics and other areas. Some of them only benefit from a set of integers' compact matrix representation. For instance, in game theory and economics, the reward matrix encodes the payout for two players based on which option they select from a predetermined (limited) set of options. Document-term matrices like the tf-idf are used in text mining and automatic thesaurus creation to keep track of the frequency of particular terms across several documents.

Complex number addition and multiplication, as well as matrix multiplication, are equivalent. For instance, 2-by-2 rotation matrices, as mentioned before, represent multiplication with a complex number having absolute value 1. Quaternions and Clifford algebras in general are susceptible to a similar interpretation.

Matrix-based encryption methods from the past, such the Hill cypher, were very popular. These codes are, nevertheless, rather simple to crack because of the linear nature of matrices. A three-dimensional object can be projected onto a two-dimensional screen to simulate a theoretical camera observation. Computer graphics uses matrices to describe objects and to calculate transformations of objects using affine rotation matrices. The study of control theory requires a thorough understanding of matrices over polynomial rings.

Since the application of quantum theory to analyse molecular bonding and spectroscopy, chemistry has used matrices in a variety of ways. Examples include the Fock matrix and the overlap matrix, which are utilised in the Hartree-Fock method to solve the Roothaan equations and produce the molecular orbitals.

Matrix-based structures have many uses in both mathematics and other areas. Some of them only benefit from a set of integers' compact matrix representation. For instance, in game theory and economics, the reward matrix encodes the payout for two players based on which option they select from a predetermined (limited) set of options. Document-term matrices like the tf-idf are used in text mining and automatic thesaurus creation to keep track of the frequency of particular terms across several documents.

A fundamental idea in graph theory is the adjacency matrix of a finite graph. It keeps track of which edges connect the graph's vertices. Logical matrices are matrices with only two distinct values (1 and 0 representing, for instance, "yes" and "no," respectively). Information concerning edge distances is contained in the distance (or cost) matrix. Similar ideas can be extended to websites connected by links, cities connected by roads, etc., in which case the matrices tend to be sparse, that is, to have few nonzero entries, unless the connection network is exceptionally dense. Hence, network theory can make use of specially designed matrix methods.

The second derivatives of a differentiable function : R n R with respect to the various coordinate directions make up the Hessian matrix for that function. It encodes data pertaining to the function's local growth behaviour.

If the Hessian matrix is positive definite, then the function has a local minimum. Finding the global minima or maxima of quadratic functions that are closely connected to the ones attached to matrices can be done using quadratic programming .

The Jacobi matrix of a differentiable map f is another matrix that is widely utilised in geometrical circumstances:

According to the implicit function theorem, f is locally invertible at that point if n > m and the rank of the Jacobi matrix reaches its maximum value, m.

The matrix of coefficients for the highest-order differential operators in the equation can be used to categorise partial differential equations. This matrix is positive definite for elliptic partial differential equations.

It significantly affects the range of potential solutions to the enumerated equation.

The finite element method is a crucial computational approach for resolving partial differential equations and is frequently used to model intricate physical systems. It makes an effort to approximate a particular equation's solution using piecewise linear functions, where the pieces are selected in accordance with a small enough grid, which may then be transformed into a matrix equation.

Stochastic matrices are square matrices with entries that are non-negative and add to one and whose rows are probability vectors. To define Markov chains with a finite number of states, stochastic matrices are utilised. The probability distribution for the next position of a particle that is currently in the state that corresponds to the row is given by a row of the stochastic matrix. The eigenvectors of the transition matrices can be used to read the properties of the Markov chain-like absorbing states, or the states that every particle finally reaches.

Matrix calculations are frequently used in statistics in a variety of ways. The goal of descriptive statistics is to describe data sets, which are frequently represented as data matrices and can then be reduced in size using dimensionality. The mutual variance of numerous random variables is encoded in the covariance matrix.

It relates to the singular value decomposition of matrices and can be expressed in terms of matrices. Matrix entries in random matrices are chosen at random from appropriate probability distributions, such as the matrix normal distribution. They are used outside of probability theory in fields like number theory and physics.

Modern physics heavily relies on linear transformations and the symmetries that go along with them. For instance, the behaviour of elementary particles under the spin group is used to categorise them as representations of the Lorentz group of special relativity in quantum field theory. The physical description of fermions, which act as spinors, includes concrete representations involving the Pauli matrices and more generic gamma matrices. There is a group -theoretical representation for the three lightest quarks involving the special unitary group SU; for their calculations, physicists use a practical matrix representation known as the Gell-Mann matrices, which is also used for the SU gauge group that serves as the foundation of the contemporary description of strong nuclear interactions, quantum chromodynamics. In turn, the Cabibbo-Kobayashi-Maskawa matrix.

The Cabibbo-Kobayashi-Maskawa matrix, on the other hand, expresses the fact that the fundamental quark states that are crucial for weak interactions are different from but linearly connected to the fundamental quark states that describe particles with particular and distinct masses.

The operators of the theory were represented as infinite - dimensional matrices acting on quantum states in Heisenberg's 1925 original quantum mechanics model. This also goes by the name of matrix mechanics. The density matrix, which describes the "mixed" state of a quantum system as a linear mixture of elementary, "pure" eigenstates, is one example in particular.

For explaining the scattering experiments that are the foundation of experimental particle physics, another matrix is a crucial tool: The scalar product of outgoing particle states and a linear combination of ingoing particle states can be used to describe collision reactions, such as those that take place in particle accelerators when non - interacting particles approach one another and collide in a small interaction zone to create a new set of non-interacting particles.

The S-matrix, a matrix that stores all information about potential interactions between particles, provides the linear combination. The description of linearly connected harmonic systems is a general use of matrices in physics. The equations of motion for such systems can be expressed as matrices, with the kinetic term being represented by the mass matrix multiplied by the generalised velocity and the interactions being represented by the force matrix multiplied by the displacement vector. The optimum method for finding solutions is to diagonalize the matrix equation and find the system's eigenvectors, or normal modes. These kinds of methods are essential for understanding the internal vibrations of systems made up of mutually bound component atoms, which make up the internal dynamics of molecules. They are also required to describe oscillations in electrical circuits and mechanical vibrations. They are also required to describe oscillations in electrical circuits and mechanical vibrations.

Further matrix applications are provided by geometrical optics. The wave aspect of light is not taken into consideration in this approximate theory. As a result, a model is created in which light rays are actually geometrical rays. The action of a lens or reflective element on a given light ray can be described as the multiplication of a two-by-two matrix called a ray transfer matrix analysis when the deflection of light rays by optical elements is small. The matrix encodes the characteristics of the optical element, and the vector's components are the slope and distance from the optical axis of the light ray. There are actually two different types of matrices: a translation matrix that describes how the plane is translated, and a refraction matrix that describes how light is refracted at a lens surface.

The matrix that results from the product of the matrices of the components accurately describes the optical system, which consists of a combination of lenses and / or reflective elements.

In electronics, conventional mesh analysis and nodal analysis result in a set of linear equations that can be expressed as a matrix. Matrix-based descriptions of the behaviour of numerous electrical components are possible. Let A be a two-dimensional vector with elements representing the input voltage v1 and the input current i1 of the component, and let B be a two-dimensional vector with elements representing the output voltage v2 and the output current i2 of the component.

Matrices have been used to solve linear equations for a very long time, but until the 1800s, they were known as arrays. The Nine Chapters on the Mathematical Art, a Chinese work from the 10th to 2nd century BCE that includes the idea of determinants, is the first instance of the use of array methods to solve simultaneous equations. Gerolamo Cardano, an Italian mathematician, published Ars Magna in 1545, introducing the approach to Europe. The same array techniques were employed in 1683 by the Japanese mathematician Seki to resolve simultaneous equations. In his work Elements of Curves from 1659, Dutch mathematician Jan de Witt illustrated transformations using arrays (1659). Gottfried Wilhelm Leibniz conducted over 50 experiments and advertised the use of arrays for storing data or solutions between 1700 and 1710.

The word "matrix" ( Latin for "womb," "dam" ( non - human female animal kept for breeding ), "source," "origin," "list," and "register," derived from mater — mother ) was first used by James Joseph Sylvester in 1850. Sylvester understood a matrix as an object giving rise to several determinants now known as minors, that is, determinants of smaller matrices that derive from the original one.

In a dissertation on geometric transformations, Arthur Cayley used matrices rather than the previously used rotated versions of the coefficients under consideration. Instead, he demonstrated that the associative and distributive laws held true by defining operations like addition, subtraction, multiplication and division as transformations of those matrices. Cayley looked into and showed both the commutative and non - commutative natures of matrix addition and multiplication. The use of arrays in early matrix theory was almost entirely restricted to determinants, hence Arthur Cayley's abstract matrix operations were groundbreaking. He played a key role in the proposal of an equation system - free matrix paradigm. Cayley conceived and proved the Cayley-Hamilton theorem in his 1858 book A memoir on the theory of matrices.

The contemporary bracket notation for matrices was invented by the English mathematician Cuthbert Edmund Cullis in 1913. These ideas were taken further by Eisenstein, who also made the observation that matrix products are non-commutative in today's terms. Cauchy was the first to establish broad generalisations regarding determinants.

There are many influences on how determinants are studied today. Gauss connected coefficients of quadratic forms and linear maps in three dimensions to matrices as a solution to number-theoretic issues. These ideas were taken further by Eisenstein, who also made the observation that matrix products are non-commutative in today's terms. Cauchy was the first to establish broad generalisations regarding determinants.

Real numbers make up symmetric matrices' eigenvalues. Jacobi studied "functional determinants," later referred to as Jacobi determinants by Sylvester, which can be used to explain geometric transformations at the local ( or infinitesimal ) level. Kronecker's Vorlesungen über die Theorie der Determinanten and Weierstrass' Zur Determinantentheorie, both published in 1903, were the first works to treat determinants axiomatically, as opposed to earlier more established approaches. Determinants had by this time been thoroughly established.

In the aforementioned memoir, Cayley proved the Cayley - Hamilton theorem for 2 x 2 matrices, while Hamilton proved it for 4 x 4 matrices. Several theorems were first proven for smaller matrices alone. Frobenius expanded the theorem to encompass all dimensions when working with bilinear forms ( 1898 ). The Gauss - Jordan elimination was also developed at the end of the 19th century.

Wilhelm Jordan invented the Gauss-Jordan elimination at the end of the 19th century, generalising a specific example that is today known as the Gauss elimination. Due in part to their employment in classifying the extremely complex number systems of the previous century, matrices came to play a crucial role in linear algebra around the beginning of the 20th century.

The study of matrices with infinitely many rows and columns resulted from the development of matrix mechanics by Heisenberg, Born, and Jordan.

Subsequently, by further expanding functional analytical concepts like linear operators on Hilbert spaces, which, generally speaking, correspond to Euclidean space but with an infinite number of independent directions, von Neumann completed the mathematical formulation of quantum mechanics.

The term "matrix" is used by Bertrand Russell and Alfred North Whitehead in their Principia Mathematica (1910–1913) in reference to their axiom of reducibility. This axiom was put forth as a way to sequentially reduce any function to a lower type.

The term for any function, regardless of how many variables it has, that has no discernible variables. Then, by using generalisation, any possible function other than a matrix derives from a matrix by taking into account the premise that the function in question is true for all feasible values or for some value of one of the arguments, with the other argument( s ) remaining indeterminate.

In his 1946 Introduction to Logic, Alfred Tarski coined the word "matrix" to refer to the idea of truth.

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Following are the named matrices list :

Multiplicity of an eigenvalue as a root of the characteristic polynomial is an example of algebraic multiplicity.

Dimension of the eigenspace connected to an eigenvalue hints at the geometric multiplicity.

Using the Gram-Schmidt method, a set of vectors is orthonormalized.

Uneven matrix.

Multivariable calculus' specialised notation is called matrix calculus.

A matrix function is a formula that converts one matrix into another.

Matrix multiplication algorithm.

A generalisation of matrices that can have any number of indices is a tensor.

Determinants :

The determinant in mathematics is a scalar quantity that is a function of the rows and columns of a square matrix. It describes some aspects of the matrix and the linear map that the matrix represents. In particular, the matrix must be invertible and the linear map it represents must be an isomorphism for the determinant to be nonzero. Matrices products' determinant is the product of its constituent determinants. det(A), det A, or |A| are used to indicate the determinant of a matrix A.

A n * n matrix's determinant can be defined in a number of equivalent ways. The determinant is represented by the Leibniz formula as a sum of signed products of matrix elements, where each summand represents the product of n distinct entries, and the total number of these summands is " n! " and " n's factorial " ( first n positive integers' product ).

The determinant of a n x n matrix is expressed via the Laplace expansion as a linear combination of determinants of ( n - 1 ) x ( n - 1 ) submatrices. By using a series of simple row operations to create a diagonal matrix, Gaussian elimination may express the determinant as the product of the diagonal elements.

Determinants can also be described by some of their characteristics, such as the four following ones. The determinant is the sole function defined on the n x n matrices. The identity matrix's determinant is 1, and adding a row ( or a column ) that is a multiple of another row ( or column ) has no effect on the determinant. Instead, exchanging two rows ( or two columns ) multiplies the determinant by 1; multiplying a row ( or a column ) by a number multiplies the determinant by this number; and adding a row (or a column) that is a multiple of another row ( or column ) has no changes over the determinant.

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Mathematical determinants can be found everywhere. In a system of linear equations, for instance, a matrix is frequently used to represent the coefficients, and determinants can be used to solve these equations ( Cramer's formula ), although other approaches to solution are computationally far more efficient.

The characteristic polynomial of a matrix, whose roots are the eigenvalues, is defined using determinants. A determinant in geometry is used to express the signed n-dimensional volume of an n-dimensional parallelepiped. This is utilised in calculus, specifically for variable changes in multiple integrals, together with exterior differential forms and the Jacobian determinant.

By directly evaluating the definition for determinants of 2 x 2 matrices, it is possible to demonstrate several important aspects of the determinant, and these qualities hold for determinants of bigger matrices as well. These are what they are: First, the identity matrix's determinant is 1. Second, if two rows are identical, the determinant is zero.

If the two columns match, this still applies. The determinant is additionally multiplied by that amount if any column is multiplied by some integer r ( i.e., all entries in that column are multiplied by that number ).

The matrix A can be used to represent two linear maps, one that maps the standard basis vectors to the rows of A and the other that maps them to the columns of A, if the matrix elements are real integers. In either scenario, the parallelogram formed by the images of the basis vectors serves as a representation of the unit square under the mapping. The parallelogram in the accompanying diagram has vertices at ( 0, 0 ), ( a, b ), ( a + c, b + d ), and ( c, d) and is defined by the rows of the aforementioned matrix.

The area of the parallelogram is the absolute value of ad - bc, which indicates the scale factor by which A transforms areas. The parallelogram created by the columns of A is typically a different parallelogram, but the area will be the same because the determinant is symmetric with regard to rows and columns. The orientated region of the parallelogram is formed by the absolute value of the determinant and its sign. When the angle between the first and second vectors defining the parallelogram rotates in a clockwise direction, the oriented area is identical to the conventional area except that it is negative, opposite to the direction of the identity matrix.

As a result, the mapping symbolised by A's determinant provides the scaling factor and the orientation. The matrix's defined linear mapping is equi-areal and orientation-preserving when the determinant is equal to one.

These concepts are connected by the concept of the bivector. It can be thought of as an oriented plane segment in two dimensions when two vectors are imagined.

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The determinant provides the parallelotope's signed n-dimensional volume. It explains the n-dimensional volume scaling factor of the linear transformation created by A in a more generic way. The sign indicates whether the transformation reverses orientation or retains it. Particularly, if the determinant is zero, the parallelotope has a volume of zero and is not fully n - dimensional, indicating that the image of A has a dimension that is less than n. Hence, A does not yield an invertible linear transformation because it is neither onto nor one - to - one.

The Leibniz formula, an explicit formula utilising sums of products of certain elements of the matrix, can be used to define the determinant of a square matrix A, that is, one with the same number of rows and columns. The determinant can also be defined as the singular function that depends on the matrix elements meeting specific criteria. By simplifying the matrices in question, determinants can also be computed using this method.

When the copies of the first two columns of the matrix are written beside it, as in the illustration, the rule of Sarrus is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements. The determinant of a 3 x 3 matrix cannot be determined using this method in greater dimensions.

The determinant of an n-by n-matrix in higher dimensions is expressed by the Leibniz formula as an expression involving permutations and their signatures. Sigma is a function that reorders this set of numbers by permuting the set {1, 2,...n}. The value in the i - th position following the reordering is shown below by the symbol "sigma subscript". The symmetric group, also known as the set of all such permutations, is frequently abbreviated S[n].

These three features can be directly demonstrated by looking at the Leibniz formula if the determinant is defined using it as described above. These three features are also used by some writers to directly approach the determinant. It can be demonstrated that there is only one function that can assign each n x n - matrix A a value that meets these three criteria. This demonstrates that the definition produced by the Leibniz formula and the more abstract method of approaching the determinant are equivalent.

Expanding the determinant by multi-linearity in the columns into a ( large ) linear combination of determinants of matrices, where each column is a standard basis vector, is all that is necessary to show this.

Although it appears less technical, this characterization cannot fully replace the Leibniz formula in defining the determinant since without it, it is unclear whether an adequate function even exists.

In general, the determinant is multiplied by the sign of any permutation of the columns. The determinant is zero if any column can be represented as a linear combination of the other columns ( matrix's columns form a linearly dependent set ). This covers a particular scenario where a column has all zero entries, in which case the determinant of that matrix is 0.

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The value of the determinant is unaffected by adding a scalar multiple of one column to another. This is a result of multilinearity and the fact that the determinant alternates, which causes it to change by a factor of the determinant of a matrix with two equal columns, which determinant is 0.

In fact, a diagonal matrix can be used to reduce such a matrix by properly multiplying the columns with fewer nonzero values by the columns with more entries without changing the determinant. Using the linearity in each column of such a matrix, one can reduce it to the identity matrix, in which case the formula is valid due to the very first defining characteristic of determinants. The identity permutation is the only permutation sigma that contributes something other than zero, hence this formula can also be inferred from the Leibniz formula.

Theoretically significant, the characterising features and their ramifications can also be employed to calculate determinants for actual matrices. In reality, Gaussian elimination may be used to transform any matrix into an upper triangular form, and this algorithm's phases have a controlled impact on the determinant.

The determinant of A is the same as the determinant of A's transposition. Examining the Leibniz formula will demonstrate this. This suggests that the word "column" can be completely swapped out with the word "row" in each of the characteristics.

The determinant is a multiplicative map, meaning that for square matrices A and B of equal size, the product of their determinants is the determinant of the matrix. This crucial point can be demonstrated by noting that both sides of the equation are alternating and multilinear as functions of the columns of A for a fixed matrix B. Furthermore, when A is the identity matrix, they both take the value det B. This assertion is supported by the alternating multilinear maps' special characterisation.

If a matrix A's determinant is not zero, the matrix is perfectly invertible. This is inferred from the multiplicative nature of det and the inverse formula using the adjugate matrix. This characteristic still applies to, respectively, the products and inverses of matrices with non-zero determinants (determinant one). Hence, a group called the generic linear group is formed by the collection of such matrices ( of fixed size n ).

In a broader sense, the term "special" refers to the subgroup of another matrix group of determinant one matrices. Examples are the special unitary group and the special orthogonal group, which contain all rotation matrices depending on whether n is 2 or 3.

The product formula for rectangular matrices is a generalisation of the Cauchy - Binet formula. For compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix, this expression can alternatively be rewritten as a multiplicative formula.

The determinant of a matrix A is expressed through Laplace expansion in terms of the minor determinants of smaller matrices. Iterative Laplace expansion can be used to compute determinants, although this method is slow for large matrices. The determinants of extremely symmetric matrices like the Vandermonde matrix can be computed with its help, though.

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The cofactors matrix is transposed to form the adjugate matrix adj ( A ). The inverse of a nonsingular matrix can be expressed using the adjugate matrix. The formula for the determinant of a 2 × 2 continues to hold, assuming acceptable extra assumptions, for a block matrix, i.e., a matrix formed of four submatrices.

According to Sylvester's Determinant Theorem, given A and B, a m x n matrix and a n x m matrix, respectively, so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix.

The eigenvalues and the characteristic polynomial of a matrix, two additional fundamental ideas in linear algebra, are intimately related to the determinant. Pseudo-determinant refers to the sum of all non-zero eigenvalues. If all of a Hermitian matrix's eigenvalues are positive, the matrix is said to be positive definite. According to Sylvester's criterion, this is comparable to the submatrices' determinants. By definition, the sum of A's diagonal entries and eigenvalues are both equal to the trace tr(A).

By translating the traces and determinant into terms of the eigenvalues, it is possible to demonstrate these inequality. As a result, they illustrate the widely known fact that the harmonic mean is smaller than the geometric mean, which is smaller than the arithmetic mean, which is smaller still than the root mean square.

The Leibniz formula demonstrates that a polynomial function is the determinant of real or equivalently for complex square matrices. It is differentiable everywhere, in particular. Jacobi's formula can be utilised to express its derivative.

When describing Lie algebras connected to specific matrix Lie groups, this identity is employed. For instance, the equation det A = 1 defines the special linear group SL. The aforementioned formula demonstrates that its Lie algebra is the unique linear Lie algebra made up of matrices with trace zero.

A determinant was initially described as a characteristic of a system of linear equations. Determinants were utilised historically long before matrices. Whether or not the system has a special solution is determined by the determinant which occurs precisely if the determinant is non - zero. In this sense, determinants were first employed in the 3rd century BCE Chinese math treatise The Nine Chapters on the Mathematical Art written by Chinese scholars. In Europe, Cardano first used a determinant-like object in 1545 to describe solutions to linear systems of two equations.

Determinants proper were first developed by Leibniz in 1693 and Seki Takakazu in Japan in 1683. Without providing any evidence, Cramer ( 1750 ) asserted his rule. The issue of plane curves travelling through a specific set of points led Cramer and Bezout ( 1779 ) to determinants. Determinants were originally acknowledged as distinct functions by Vandermonde in 1771.

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Vandermonde had already provided a specific instance, but Laplace ( 1772 ) provided the general approach of expanding a determinant in terms of its complementary minors. Lagrange ( 1773 ) dealt with determinants of the second and third order and applied it to issues with elimination theory; he demonstrated numerous special examples of generic identity. The following step was taken by Gauss ( 1801 ). Like Lagrange, he used determinants extensively in his theory.

He coined the term "determinant" ( Laplace had used "resultant" ), but not in its current sense ; rather, he used it to refer to a quantic's discriminant. Gauss also came quite close to the multiplication theorem and came up with the idea of reciprocal ( inverse ) determinants.

The next significant author is Binet ( 1811 – 1812 ), who officially established the multiplication theorem for the exceptional case of m = n in addition to the theorem pertaining to the product of two matrices with m columns and n rows. Cauchy delivered a paper on the same topic on the same day ( November 30, 1812 ) as Binet did for the Academy. The Cauchy - Binet formula is used.

In this, he adopted the term "determinant" in its modern sense, collected and clarified the information that was available at the time, refined the notation, and provided a more convincing demonstration for the multiplication theorem than Binet's. With him, the theory's broad application gets underway. The functional determinant, which Sylvester later referred to as the Jacobian, was used by Jacobi 1841.

He specifically addresses this topic in his memoirs in Crelle's Journal for 1841, along with the group of alternating functions Sylvester has dubbed alternants. Sylvester ( 1839 ) and Cayley started working on their projects at the same time as Jacobi's final memoirs. Vertical bars were first used by Cayley in 1841 to represent the determinant in contemporary notation.

The conclusion of the general theory has naturally led to the study of special types of determinants. Lebesgue, Hesse, and Sylvester studied axisymmetric determinants; Sylvester and Hankel studied persymmetric determinants; Catalan, Spottiswoode, Glaisher, and Scott studied circulants; Cayley studied skew determinants and Pfaffians in connection with the theory of orthogonal transformation; Sylvester studied continuants; Christoffel and Frobenius studied Wronskians ( referred to as Wronskians by Muir ); Spottiswoode's was the first textbook on the topic. Treatises were published in America by Hanus ( 1886 ), Weld ( 1893 ), and Muir / Metzler ( 1933 ).

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Inverse Trigonometric Functions :

In mathematics, the inverse trigonometric functions are the opposite of the trigonometric functions. They are also sometimes referred to as arcus functions, anti trigonometric functions, or cyclometric functions. They are used to calculate an angle from any of the angle's trigonometric ratios and are specifically the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions. In engineering, navigation, physics, and geometry, inverse trigonometric functions are frequently employed.

There are numerous notations for inverse trigonometric functions. Inverse trigonometric functions are typically named with an arc- prefix, such as arcsin(x), arccos(x), arctan(x), etc. Certain geometric relationships lead to this notation: An arc with a length of r * x corresponds to an angle x of radians when measured in radians, where r is the radius of the circle. As the length of the circle's arc in radii and the measurement of the angle in radians are the identical, "the arc whose cosine is x" and "the angle whose cosine is x" are equivalent in the unit circle. The inverse trigonometric functions are frequently referred to by their abbreviated names in computer programming languages.

While certain high-level programming languages, like as Python ( including SymPy and NumPy ), Matlab, and MAPLE, employ capitalization for the common trig functions, many others, including Wolfram's Mathematica and University of Sydney's MAGMA, utilise lower case. As a result, the ISO 80000-2 standard only specifies the "arc" prefix for inverse functions as of 2009.

All six of the trigonometric functions must be constrained in order to have inverse functions because none of them are one - to - one. As a result, the domains of the original functions are proper ( i.e. strict ) subsets of the result ranges of the inverse functions.

The Pythagorean theorem and definitions of the trigonometric ratios can be used to quickly derive them from the geometry of a right-angled triangle with one side of length 1 and another side of length x. Derivations that are only algebraic are longer. It is important to keep in mind that the figure assumes that x is positive for arcsecant and arccosecant, thus the result must be adjusted using absolute values and the signum ( sgn ) operation.

The inverse trigonometric functions can also be calculated using power series, similarly to the sine and cosine functions. The series for arcsine can be obtained by expanding its derivative, as a binomial series, then integrating term by term.

The arcsecant and arccosecant functions can have both negative and positive values, therefore the absolute value is required to account for both. The absolute values in the derivatives of the two functions lead to two distinct solutions for positive and negative values of x, which is another reason why the signum function is required. The inverse hyperbolic functions' logarithmic definitions can be used to further simplify them.

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The arcosh function is identical to the signum logarithmic function since the absolute value in its parameter generates a negative half of its graph. The inverse trigonometric functions can be extended from the real line to the complex plane since they are analytic functions. Functions with numerous sheets and branch points are produced as a result. Moreover, complex logarithms can be used to express functions. This logically expands their scopes to the complex plane. The principal values of the functions, including their branch cuts, are true anywhere they are defined.

As every inverse trigonometric function produces a right triangle angle, it is possible to generalise them by forming a right triangle in the complex plane using Euler's formula. By entering the lengths of each side into the final equation, we can determine the angle of the triangle in the complex plane. With a total of six equations, we can construct a formula for one of the inverse trig functions by setting one of the three sides to 1 and one of the remaining sides to our input z. We must employ the Pythagorean Theorem relation to translate the final side of the triangle into terms of the other two because the inverse trig functions only need one input.

The specific form of the reduced formulation is important in order to match the principal branch of the natural log and square root functions to the typical principal branch of the inverse trig functions.

In this way, the complex-valued log function and all of the inverse trig functions are special instances of one another. The definitions can be used to further describe the inverse hyperbolic functions because they accept hyperbolic angles as outputs for z. Simple proofs of the relationships can also be made by expanding trigonometric functions to exponential forms.

When the lengths of the sides of a right triangle are known, inverse trigonometric functions can be used to determine the remaining two angles.

Differentiation :

Differentiation is the action of locating a derivative. Antidifferentiation is the term for the opposite process. Calculus's fundamental theorem connects anti differentiation and integration. The two core operations of single-variable calculus are differentiation and integration.

The derivative of a function of a real variable in mathematics quantifies the sensitivity of the function's value ( output value ) to changes in its argument ( input value ). Calculus's core tool is the derivative. The velocity of an item, for instance, is the derivative of its position with respect to time; it quantifies how quickly the object's position varies as time passes.

When it occurs, the slope of the tangent line to the function's graph at a given input value is the derivative of a function of a single variable. The function closest to that input value is best approximated linearly by the tangent line.

Because of this, the derivative is frequently referred to as the "instantaneous rate of change," which is the ratio of the instantaneous change in the dependent variable to that in the independent variable.

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The use of derivatives as functions of multiple real variables can be generalised. In this version, the derivative is reinterpreted as a linear transformation whose graph is the best linear approximation of the graph of the original function after an appropriate translation. The matrix that depicts this linear transformation in relation to the basis provided by the selection of independent and dependent variables is known as the Jacobian matrix. The partial derivatives with regard to the independent variables can be used to calculate it. The gradient vector replaces the Jacobian matrix for a real-valued function with several variables.

In conclusion, a continuous function is one that has a derivative, albeit there are also continuous functions without a derivative. In actuality, most functions have derivatives at all points or nearly all points. Several mathematicians believed that a continuous function was differentiable at the majority of its points early in the history of calculus. This is accurate in simple cases ( such as when the function is monotone or a Lipschitz function ). Weierstrass did discover the first instance of a function that is continuous everywhere but differentiable nowhere in 1872, though. The Weierstrass function is the modern name for this illustration.

Stefan Banach established in 1931 that the set of continuous functions that have a derivative at some point is a small set. This essentially means that very few random continuous functions have a derivative at any point.

The object's velocity is represented by x's first derivative. The acceleration is x's second derivative. The jerk is x's third derivative. Finally, snap, crackle, and pop are the fourth through sixth derivatives of x, which are most relevant to astrophysics.

Every polynomial function is endlessly differentiable on the real line. A polynomial of degree n becomes a constant function according to normal differentiation rules if it is differentiated n times. The derivatives that come after it are all identically zero. In particular, polynomials are smooth functions since they exist.

Polynomial approximations to a function near a point x are given by a function's derivatives at that point.

An inflection point is the location at which the sign of a function's second derivative changes. The second derivative may be zero at an inflection point, as in the case of the inflection point x = 0.

A dot is placed over the function name in Newton's differentiation notation, often known as the "dot notation," to signify a time derivative. The only derivatives that employ this notation are those that relate to time or arc length. Usually, it is applied to differential equations in differential geometry and physics. However, the dot notation is inapplicable to several independent variables and high-order derivatives of order 4 or more.

In theory, it is possible to calculate a function's derivative from its definition by thinking about the difference quotient and figuring out its limit. After the derivatives of a few simple functions are known, it is usually simpler to compute the derivatives of other functions using the formulas for deriving the derivatives of more complex functions from simpler ones.

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The fact that the derivative is the limit of difference quotients illustrates the fact that it is the best linear approximation in one variable. Nevertheless, because it is typically impossible to divide vectors, the standard difference quotient does not make sense in higher dimensions. In particular, the difference quotient's numerator and denominator are not even in the same vector space.

In contrast to the one-variable situation, the total derivative of a function does not yield another function. This is due to the fact that a multivariable function's total derivative must store a lot more data than a single-variable function's derivative. Instead, a function from the tangent bundle of the source to the tangent bundle of the target is provided by the total derivative.

The natural analogue of second, third, and higher-order total derivatives is not a function on the tangent bundle, is not constructed by continually taking the total derivative, and is not a linear transformation. Higher-order derivatives represent subtle geometric information, such as concavity, which cannot be described in a linear transformation, hence the analogue of a higher-order derivative, known as a jet, cannot be a linear transformation.

As the tangent bundle only has area for the base space and the directional derivatives, it cannot be a function on it. Jets accept additional coordinates that represent higher-order changes in direction as parameters since they capture higher-order information. The jet bundle is the area defined by these additional coordinates. The relationship between a function's kth order jet and its partial derivatives of order less than or equal to k is analogous to the relationship between a function's total derivative and partial derivatives.

There are numerous different contexts in which the derivative concept might be applied. The derivative of a function at a point acts as a linear approximation of the function at that point, which is the shared characteristic.

Functions between differentiable or smooth manifolds are a further generalisation. Such a manifold M is, intuitively, a space that may be approximated by a vector space known as its tangent space near any point x.

Between the tangent bundles of M and N, the derivative function transforms into a map. This definition is crucial to differential geometry and has numerous applications.

Maps between infinite dimensional vector spaces like Banach spaces and Fréchet spaces can also be defined as differentiating. Both the differential and the directional derivative, known as the Gateaux and Fréchet derivatives, have a generalisation.

The fact that a large number of functions cannot be differentiated is one shortcoming of the classical derivative. However, it is possible to extend the idea of the derivative in order to distinguish any continuous functions as well as many other functions using a concept known as the weak derivative. The concept is to merely need that a function is differentiable "on average" and embed the continuous functions in a wider space known as the space of distributions.

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Several comparable objects in algebra and topology have been introduced and studied as a result of the features of the derivative. Finite differences are differentiation's discrete counterpart. In time scale calculus, the study of differential calculus and the calculus of finite differences are combined.

Calculus is a branch of mathematics that focuses on limits, functions, derivatives, integrals, and infinite series. In its early years, it was known as infinitesimal calculus. Calculus was independently developed by Gottfried Leibniz and Isaac Newton in the middle of the 17th century. But in a furious argument that lasted until their deaths, each inventor insisted the other had stolen his ideas.

The mathematical field of differential calculus, which is covered in this article, is very well-established and has a wealth of references.

Number Systems :

A numeral system is a way of writing numbers; it's a way of mathematically notating a collection of numbers by utilising a consistent set of digits or other symbols.

In several numeral systems, the same set of symbols may represent various numbers. For instance, "11" stands for the numbers eleven in the decimal numeral system, which is currently the most widely used system worldwide, three in the binary numeral system, which is used in modern computers, and two in the unary numeric system that is frequently utilised in tallying scores.

The value of a numeral is the number it stands for. The same set of numbers cannot be represented by all number systems; for instance, the number 0 cannot be represented by Roman numerals but by the Hindu - Arabic numeral system.

A number system should ideally :

Depict a valuable collection of numbers ( e.g. all integers, or rational numbers )

Give each integer a distinct representation ( or at least a standard representation )

Describe the number's arithmetic and algebraic structure.

The standard decimal representation, for instance, assigns each non-zero natural number a distinct representation as a finite sequence of digits that starts with a non-zero digit.

Numeral systems are occasionally referred to as number systems, however this term is misleading because it can apply to a variety of number systems, including the real number system, the complex number system, the p - adic number system, etc.

Decimal is the most widely used numeral system. The Hindu-Arabic numeral system, which represents the integer version, was created by Indian mathematicians. Aryabhata of Kusumapura invented the place-value notation in the 5th century. A century later Brahmagupta introduced zero as a mathematical symbol. Due to their military and commercial engagements with India, nearby regions like Arabia experienced a delayed development of the system. According to a treatise written in 952-953 by the Syrian mathematician Abu'l-Hasan al-Uqlidisi, Middle Eastern mathematicians expanded the system to include negative powers of 10 ( fractions ). Sind ibn Ali, who also authored the earliest treatise on Arabic numerals, also introduced the decimal point notation.

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Due to trade between merchants, the Hindu - Arabic numeral system later extended to Europe. The digits used in Europe are known as Arabic numerals since they were acquired from the Arabs.

The unary numeral system, in which each natural number is represented by a corresponding number of symbols, is the most basic. Tally markers are one such mechanism that is still widely used. The unary system is only useful for tiny numbers, yet it plays an essential role in theoretical computer science. Elias gamma coding, which is frequently used in data compression, uses unary to denote the length of a binary numeral in order to express arbitrary sized integers.

A positional system, commonly referred to as place value notation, is more elegant. The Hindu - Arabic numeral system is a positional base 10 system that was developed in India and is now used all over the world.

Positional systems make math more simpler than the older additive ones, and they also use various symbols for the various powers of 10 than additive systems do.

Nowadays, all human writing uses the positional decimal system. The base 1000 is also used but not commonly, by clustering the digits and treating a sequence of three decimal digits as a single digit.

The binary numeral system, which has two binary digits, 0 and 1, is the foundation for the primary numeral systems used in computers. Commonly used positional systems are those created by multiplying binary digits by three called the octal numeral system or four called the hexadecimal numeral system. Bases 232 or 264 for grouping binary digits by 32 or 64, which is the length of the machine word are used for exceptionally big numbers.

The unary coding system is used in some biological systems. The neural networks that generate bird songs use unary numerals. The HVC is the songbird brain region that is involved in both the learning and creation of bird song, which is a high vocal center. The command signals for various notes in the birdsong come from various locations within the HVC. Due to its inherent simplicity and resilience, space coding, which is how this coding operates, is an effective method for biological circuits.

Arithmetic numerals ( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 ) and geometric numerals ( 1, 10, 100, 1000, 10000, ... ) are two categories of numerals that are used when writing numbers using digits or symbols, respectively. Only geometric and arithmetic numbers are used in sign-value and positional systems, respectively. Except for the Ionic system, sign - value systems do not require arithmetic numerals because they are created by repetition, and positional systems do not require geometric numerals since they are created by position. Yet, both geometric and mathematical numerals are used in spoken language.

Bijective numeration, a modified base k positional system, is employed in various fields of computer science. Digits 1, 2,..., k ( where k is greater than or equal to 1 ), and zero are represented by empty strings. This prevents the non-uniqueness brought on by leading zeros by creating a bijection between the set of all such digit-strings and the set of positive integers. To distinguish p-adic numbers from bijective base-k numeration, another name for it is k-adic notation. Unary and bijective base 1 are equivalent.

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Depending on the frequency of occurrence of numbers of different sizes, the flexibility in selecting threshold values enables optimization for the number of digits. Bijective numeration is the scenario where all threshold values are equal to 1, and zeros are used as separators between numbers that have non-zero digits.

Here, numerical systems are divided into groups according to whether they employ positional notation, also known as place-value notation, and further divided into groups according to radix or base. The Roman and Greek roots of the common names are mixed together somewhat arbitrarily, sometimes resulting in a name that has elements of both languages. Several standardisation proposals though have been made.

From the use of fingers and tally marks, maybe more than 40,000 years ago, to the employment of sets of glyphs capable of accurately representing any imaginable number, number systems have advanced. Around 5000 or 6000 years ago in Mesopotamia, the first unambiguous notations for numerals are known to have appeared.

At least forty thousand years ago, tally marks formed by carving notches in wood, bone, and stone have been discovered in the archaeological record. These tally markings may have been used to keep track of quantities such as the number of animals or other valuable goods, or to count time such as the number of days or lunar cycles. However, there is currently no diagnostic method that can accurately pinpoint the social function or application of ancient linear marks engraved on surfaces, and modern anthropological examples demonstrate that similar objects are created and used for non-numerical purposes.

A baboon fibula with engraved marks known as the Lebombo bone was found in the Lebombo Mountains, which are situated between South Africa and Eswatini. The bone has a 42,000-year age estimate. The 29 notch Lebombo bone suggests that "it may have been employed as a lunar phase counter, in which case African women may have been the first mathematicians, because keeping track of menstrual cycles needs a lunar calendar," according to The Universal Book of Mathematics. The 29 notches may only be a piece of a longer series, though, because the bone is visibly shattered at one end. Comparable artefacts from current countries, like those of Australia, also imply that such notches can serve mnemonic or customary roles, rather than meaning numbers.

The Ishango bone is a piece of artefact with a pointy quartz chunk attached to one end, possibly for carving. It is thought to have existed 25,000 years ago. The object's three rows of carved symbols that have been interpreted as tally marks run the length of the object, leading to the initial assumption that it was a tally stick. A second row appears to add and remove 1 from 10 and 20 ( i.e., 9, 19, 21, and 11 ), while the third row contains values that might be halves and doubles although these are inconsistent. The first row has been interpreted as the prime numbers between 10 and 20 ( i.e., 19, 17, 13, and 11 ).

Researchers like Jean de Heinzelin have stated that the notch groupings reveal a mathematical understanding far beyond mere counting, pointing out the statistical chance of getting such numbers by mistake. It has also been proposed that the marks may have been made for non-mathematical reasons or for a practical cause, such as improving the handle's grip. There is ongoing discussion in academic literature about the significance and use of the notches.

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A system of accounting based on tiny clay tokens gave rise to the earliest recorded writing for record keeping. The earliest items that have been identified as tokens are from two sites: Tell Abu Hureyra in Syria, which dates to the 10th millennium BCE, and Ganj-i-Dareh Tepe in Iran, which dates to the 9th millennium BCE. Two tokens, each denoting a single unit, were combined to indicate "two sheep" in a record. The way that various object kinds were counted also varied.

In a sexagesimal number system, higher groups of ten or six were represented by tokens of various sizes and shapes. Various combinations of token forms and sizes encoded the different counting methods. According to archaeologist Denise Schmandt-Besserat, elaborate tokens that indicated the goods being counted were employed in addition to the simple geometric tokens used for numbers. This intricate token was a flat disc imprinted with a quartered circle for ungulates like sheep. Yet, there are a variety of reasons why the alleged use of complicated tokens has been challenged.

Tokens were placed into clay bullae, hollow ball-shaped envelopes, to guarantee they weren't misplaced or changed in type or amount a bulla. On the surfaces of the bullae, which might also be kept plain, ownership and witness seals were imprinted. After the bulla containing the tokens was sealed, it had to be broken open if the tokens needed to be confirmed. Tokens started being pushed into a bulla's outside surface before being sealed inside around the middle of the fourth millennium BCE, possibly to prevent the need to open the bulla to see them. External impressions of the encased tokens' sizes, shapes, and numbers were made on the bullae surfaces as a result of this process.

At some point, it appears that the redundant nature of the impressions outside a bulla and the tokens inside a bulla was understood, and impressions on flat tablets replaced tokens as the favoured means of storing numerical data. Scholars like Piere Amiet were among the first to recognise and publish the correspondences between impressions and tokens as well as the chronology of forms they included.

The Sumerians possessed a sophisticated mathematics by the time the numerical impressions revealed information about ancient numbers.

Calculations were likely carried out using tokens, an abacus, or a counting board.

In the middle to late fourth millennium BCE, proto - cuneiform numerals imprinted onto clay with a round stylus held at various angles to make the various shapes used for numerical signs supplanted the numerical impressions made with bullae. Each number sign reflected the commodity being counted as well as the quantity or volume of that commodity, just like tokens and the numerical impressions on the outside of bullae. Soon after, little images identifying the item being counted were added to these digits. Several object types were tallied differently by the Sumerians.

There were more than a dozen different counting systems, as discovered through analyses of early proto-cuneiform notations from the city of Uruk, including a general system for counting the majority of discrete objects such as animals, tools, and people and specialised systems for counting cheese and grain products, grain volumes ( including fractional units ), land areas, and time. Object-specific counting is common and has been recorded for modern populations all across the world; such contemporary systems offer useful insight into how the ancient Sumerian number systems probably worked.

The reed stylus, which created the wedge-shaped impressions that give cuneiform signs their name, started to supplant the round stylus around 2700 BCE. Similar to tokens, numerical impressions, and proto-cuneiform numerals, modern cuneiform numerals occasionally have confusing numerical values. This ambiguity is caused in part by the lack of a convention, such as a decimal point, to distinguish integers from fractions or higher exponents from lower ones in the Sumerian number system, as well as in part by the fact that the base unit of an object-specified counting system is not always understood. Conversions between object-specific counting systems were made easier by the development, around 2100 BCE, of a common sexagesimal number system with place value.

Various creative measures were taken during old times to fulfill the needs for applicability of number systems. All were archaic in nature considering today's era leveraging of Multi Media Technology MM, Information & Communication Technology ICT, Macroeconomics Studies, Computers & Communication Technology CCT, Artificial Intelligence AI, Data Science DS, Biotechnology Studies BT, Library Information Science LIS, General Science GS, Web Applications WebApp, Mass Media Technology MMT, Political Science Pol Science, Economics Studies, Geospatial Technology GT, Python programming & coding, Electrical technology ET, Electronics Technology, Microeconomics Studies, Informatics Practices IP, computer science and allied information technology spheres.

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While it is less well known today than its sexagesimal counterpart, a decimal version of the sexagesimal number system that is known as Assyro-Babylonian Common developed in the second millennium BCE, reflecting the increased influence of Semitic peoples like the Akkadians and Eblaites; although it would eventually become the dominant system used throughout the region, especially as Sumerian cultural influence started to wane.

Tokens, number impressions, and proto-cuneiform numerical signs all used alternating bases of 10 and 6, but sexagesimal numerals maintained this pattern. Sexagesimal numbers were utilised for computations in astronomy and other fields as well as in commerce. Angles and time are still counted in Arabic numerals using the sexagesimal system with seconds per minute and minutes per hour degrees.

Around the middle of the first millennium BCE, the Roman numerals evolved from Etruscan symbols. While the symbols for 5 an inverted V shape and 50 an inverted V split by a single vertical mark were possibly derived from the lower halves of the signs for 10 and 100, there is no convincing explanation as to how the Roman symbol for 100, C, was derived from its asterisk.

Polynomials :

A polynomial is a mathematical statement made up of coefficients and indeterminates also known as variables that uses only the operations addition, subtraction, multiplication, and powers of positive integers of the variables.

In many branches of mathematics and science, polynomials are used. For instance, they are used to define polynomial functions, which show up in contexts ranging from fundamental chemistry and physics to economics and social science. They are also used in calculus and numerical analysis to approximate other functions. Polynomial equations are used to form these functions. Polynomial rings and algebraic varieties, two fundamental ideas in algebra and algebraic geometry, are constructed in advanced mathematics using polynomials.

A polynomial's x is frequently referred to as a variable or an indeterminate. When the polynomial is viewed as an expression, the symbol x is fixed and has no meaning. Its value is "indeterminate". But, if one thinks about the function that the polynomial defines, then x represents the function's argument and is referred to as a "variable." Both of these terms are frequently used in writing.

Often, a polynomial P in the indeterminate x is written as P or as P(x). Technically, the polynomial's name is P, not P(x), but the use of the functional notation P(x) stems from a time when it wasn't clear how to distinguish a polynomial from the related function.

Moreover, it is frequently helpful to define a polynomial and its indeterminate using the functional notation. Let P(x) be a polynomial, for instance, is short for "let P be a polynomial in the indeterminate x." However, many formulas are more simpler and easier to comprehend if the name( s ) of the indeterminate( s ) do not appear at every occurrence of the polynomial when it is not required to stress the name of the indeterminate.

It is possible to explicitly address the ambiguity of having two notations for the same mathematical item by taking into account the overall meaning of the functional notation for polynomials. P(a) denotes, by convention, the outcome if a represents a number, a variable, another polynomial, or, more broadly, any expression.

The Greek term poly, which means "many," and the Latin word nomen, which means "name," combine to form the word polynomial. By substituting the Greek root poly- for the Latin bi-, the term binomial was created. In other words, it refers to the accumulation of several phrases ( many monomials ). In the 17th century, the word "polynomial" was first employed.

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A polynomial expression is one that can be created from variables and other constants and symbols by adding, multiplying, and raising it to a non-negative integer power. The constants, which represent mathematical objects that can be added to and multiplied, are often numbers, but they can also be any expression that does not contain the indeterminates. When two polynomial expressions can be converted from one to the other using the standard addition and multiplication properties of commutativity, associativity, and distributivity, they are regarded as defining the same polynomial.

Thus, a polynomial may either be stated as the sum of a finite number of non-zero terms or it can be written as zero. Each term is made up of a limited number of indeterminates that have been raised to non-negative integer powers plus a number known as the coefficient of the term a.

The degree of an indeterminate in a term is the exponent on that indeterminate in that term; the degree of a polynomial is the maximum degree of any term with a nonzero coefficient. The degree of a term is the sum of the degrees of the indeterminates in that term.

A constant term is a term without any indeterminates, while a constant polynomial is a polynomial without any indeterminates. A constant term and a nonzero constant polynomial have a degree of zero. It is customary to treat the degree of the zero polynomial, which contains no terms at all, as undefined.

Lower degree polynomials have names that are distinctive. A constant polynomial, often known as a constant, is a polynomial of degree zero. Linear polynomials, quadratic polynomials, and cubic polynomials, respectively, are polynomials of degree one, two, or three. Although quartic polynomial ( for degree four ) and quintic polynomial ( for degree five ) are occasionally used, the precise terms for higher degrees are not frequently employed. The polynomial or its terms may be given the names for the degrees.

The zero polynomial is a type of polynomial that can be thought of as having no terms at all. Its degree is not 0 like other constant polynomials. The zero polynomial's degree is instead either explicitly left unspecified or defined as negative, either 1 or. The zero polynomial stands apart from all other polynomials in one indeterminate by having the only infinitely many roots. The x-axis is represented by the graph of the zero polynomial, f(x) = 0.

If all of a polynomial's non-zero terms have degree n, it is said to be homogeneous of degree n when applied to polynomials with more than one indeterminate. The degree of the zero polynomial is indeterminate because it is a homogeneous polynomial.

Terms can be rearranged using the commutative property of addition in any desired order. The terms of polynomials with one indeterminate are typically arranged in descending or ascending powers of x, with the term with the largest degree coming first, depending on the degree of the term.

"Like terms" or "similar terms" are two words with identical indeterminates raised to the same powers. These terms can be merged using the distributive rule to create a single term whose coefficient is the sum of the coefficients of the terms that were combined. This can result in the coefficient being 0. One-term polynomials are referred to as monomials, two-term polynomials as binomials, and three-term polynomials as trinomials depending on the number of terms with non zero coefficients. A four-term polynomial is occasionally referred to as a "quadrinomial."

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A polynomial with real coefficients is considered a real polynomial. The domain is less constrained when it is used to define a function. However, a function from the reals to the reals that is defined by a real polynomial is referred to as a real polynomial function. The same is true for complex polynomials, which have complex coefficients, and integer polynomials, which have integer coefficients.

A multivariate polynomial is one that has more than one indeterminate; a univariate polynomial is one that has only one indeterminate. Bivariate polynomials are defined as polynomials having two indeterminates.

These ideas are less about specific polynomials and more about the type of polynomials one typically works with. For example, when working with univariate polynomials, one does not necessarily exclude constant polynomials ( which may be obtained by subtracting non-constant polynomials ), even though strictly speaking, constant polynomials do not contain any indeterminates at all. Multivariate polynomials can be further categorised as bivariate, trivariate, and so on, depending on the permissible number of indeterminates. A study of tri variate polynomials typically enables bi variate polynomials, and so forth, so that the set of objects under examination is closed under subtraction.

A polynomial is evaluated by assigning a numerical value to each indeterminate and performing the corresponding additions and multiplications. By utilising Horner's approach, the evaluation of polynomials in a single indeterminate is typically more effective as it requires less arithmetic operations to complete.

Another polynomial is produced when two polynomials are combined together. Polynomial subtraction is comparable. In addition, polynomials can be multiplied. Using the distributive law repeatedly causes each term of one polynomial to be multiplied by each term of the other, expanding the product of two polynomials into a sum of terms.

A polynomial is always the result of polynomials. When two polynomials with different numbers of variables, f and g, are given, the composition f of g can be created by replacing each copy of the first polynomial's variable with a copy of the second polynomial. The polynomial multiplication and division rules can be used to expand a composition into a sum of terms. Another polynomial is created when two polynomials are combined.

A polynomial is not often a division of another polynomial. Instead, these ratios belong to a larger class of objects known as rational expressions, rational functions, or rational fractions, depending on the situation. This is comparable to how the ratio of two integers is a rational number and not always an integer.

There is a concept of Euclidean division of polynomials, which generalises the Euclidean division of integers, for polynomials in one variable. Any of a number of algorithms, including polynomial long division and synthetic division, may be used to calculate the quotient and remainder.

Every polynomial that has coefficients that can be factored just once like the integers or a field has a factored form that is represented as the sum of irreducible polynomials and a constant. Up to the order of the components and their multiplication by an invertible constant, this factored form is distinct. The irreducible factors in the case of the field of complex numbers are linear. They have a degree of either one or two when compared to real numbers.

The irreducible factors can be any degree over integers and rational numbers. The factored form calculation, also known as factorization, is typically too complex to be performed by hand-written computation. Nonetheless, the majority of computer algebra systems have effective polynomial factorization algorithms.

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When compared to other types of functions, calculating the derivatives and integrals of polynomials is exceptionally straightforward. The derivative formula can still be formally interpreted when the coefficient kak is understood to mean the sum of k copies of ak for polynomials whose coefficients come from more abstract settings.

Polynomial functions typically have complex coefficients, arguments, and values, unless otherwise stated. A function from the complex numbers to the complex numbers is specifically defined by a polynomial that is only allowed to have real coefficients. The resulting function is a real function that maps reals to reals if the domain of this function is also limited to the reals.

There may be expressions that categorically are not polynomials but nonetheless define polynomial functions, in accordance with the definition of polynomial functions. Each polynomial function is whole, continuous, and smooth.

A graph can serve as a representation for a polynomial function in a single real variable. As the variable rises forever, a polynomial function that is not constant tends to infinity in absolute value. The graph lacks an asymptote if the degree is greater than one. It has two vertically oriented parabolic branches, one branch for positive x and one for negative x. Calculus use intercepts, slopes, concavity, and end behaviour to evaluate polynomial graphs.

Algebraic equation is another name for a polynomial equation. The indeterminates variables of polynomials are also known as unknowns when referring to equations, and the solutions are the potential unknown values for which the equality is true, in general more than one solution may exist.

A polynomial equation differs from a polynomial identity in that the two expressions in the former represent the same polynomial in the latter, and any evaluation of both members results in a true equivalence.

The quadratic formula and other techniques are taught in elementary algebra to solve all first - and second - degree polynomial equations in one variable. Moreover, the cubic and quartic equations have formulae.

The Abel - Ruffini theorem states that a generic radical formula is impossible at higher degrees. Nonetheless, numerical approximations of the roots of a polynomial expression of any degree can be discovered using root - finding methods.

A polynomial equation with real coefficients can only have so many solutions. When the complex solutions are enumerated with their multiplicity, the number of solutions equals the degree. The basic theorem of algebra states this truth.

Every integer in the case of the zero polynomial is a zero of the associated function, and the idea of root is rarely taken into account.

If all complex roots are taken into account, the number of roots of a nonzero polynomial P, when tallied with their corresponding multiplicities, equals this degree and this is a consequence of the fundamental theorem of algebra. The formulae of Vieta relate the coefficients of a polynomial to its roots.

The fundamental theorem of algebra states that every non - constant polynomial has at least one root if the set of acceptable solutions is extended to complex numbers. Any polynomial with complex coefficients can be written as a constant, its leading coefficient times a product of these polynomial factors of degree 1, as can be seen by repeatedly dividing out factors x - a. As a result, the number of complex roots counted with their multiplicities is exactly equal to the degree of the polynomial.

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Mathematicians have attempted to express the solutions as algebraic expressions from the beginning of time, however this is generally impossible for equations of degree greater than one.

Only degrees one and two were successful in the past. The quadratic formula offers these expressions of the solutions for quadratic equations. For equations of degree three and four, comparable formulas including cube roots in addition to square roots have been known since the 16th century, however they are much more difficult. But for many centuries, academics couldn't come up with any formulas for degrees 5 and higher. There exist degree 5 equations whose solutions cannot be represented by a finite formula using solely arithmetic operations and radicals, as Niels Henrik Abel demonstrated in 1824.

Évariste Galois demonstrated in 1830 that the majority of equations with degrees greater than four cannot be solved by radicals and that for any equation, one may determine whether it can be solved by radicals and, if so, how. Galois theory and group theory, two significant subfields of contemporary algebra, were introduced as a result of this finding. Galois acknowledged that his method's calculations were impractical. Nonetheless, solvable equations of degrees 5 and 6 have published formulas.

The only way to solve it is to compute numerical approximations of the solutions when there is no algebraic expression for the roots or when one does exist but is too sophisticated to be usable.

There are numerous techniques available for doing that. Some are limited to polynomials while others can be used with any continuous function. The most effective algorithms provide computer-based solutions to polynomial problems with degrees greater than 1,000.

The combinations of values for the variables for which the polynomial function takes the value zero are typically referred to as zeros instead of "roots" for polynomials with more than one indeterminate. Algebraic geometry's focus is on the study of polynomial set zeros. There are techniques to determine whether a set of polynomial equations with multiple unknowns has a finite number of complex solutions, and if so, to compute the solutions.

A system of linear equations is the special case when all the polynomials are of degree 1, and there are a variety of various solutions available, including the traditional Gaussian elimination.

A Diophantine equation is a polynomial equation for which the only integer solutions are of interest. Diophantine equations are notoriously difficult to solve. There is no generic procedure that can be used to solve them or even determine whether the set of solutions is empty. Diophantine equations have a role in some of the most well - known mathematical puzzles that have been solved in the past fifty years, including Fermat's Last Theorem.

Polynomials with indeterminates used in place of some other mathematical objects are frequently thought of and occasionally given a particular name. A trigonometric polynomial is a finite linear combination of the functions sin( nx ) and cos( nx ), where n can take on any number of natural values. For functions with real values, the coefficients could be regarded as real numbers.

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A finite Fourier series and such a function are identical for complex coefficients. Many applications of trigonometric interpolation, such as the interpolation of periodic functions, make use of trigonometric polynomials. Additionally, within the discrete Fourier transform also, they are quite well utilised.

An equality between two matrix polynomials that holds for the particular matrices in question is known as a matrix polynomial equation. Matrix polynomial equations are known as matrix polynomial identities.

An equality between two matrix polynomials that holds for the particular matrices in question is known as a matrix polynomial equation. Any matrix polynomial equation that is true for all matrices A in a given matrix ring is known as a matrix polynomial identity.

An exponential polynomial is a bivariate polynomial in which the second variable stands in for the exponential function applied to the first variable.

The quotient ( also known as an algebraic fraction ) of two polynomials is a rational fraction. A rational function is any algebraic expression that may be expressed as a rational fraction.

A rational function is only defined for the values of the variables for which the denominator is not zero. This is very different and unlike polynomial functions. This is because polynomial functions are defined for all values of the variables.

The Laurent polynomials are included in the rational fractions, which do not restrict the denominators to powers of an indeterminate. The Laurent polynomials are similar to polynomials but permit the occurrence of negative powers of the variable(s).

Formal power series are similar to polynomials in that they can have an endless number of non-zero terms, which prevents them from having a limited degree. Similar to irrational numbers, they generally cannot be explicitly and completely written down, unlike polynomials, but the rules for manipulating their terms remain the same. Although non-formal power series can also generalise polynomials, they may not converge when multiplied by one another.

Important tools for creating new rings out of existing ones include the formation of the polynomial ring and factor rings by factoring out ideals.

The creation of finite fields is another example, which follows a similar process and begins with the field of integers modulo a prime number as the coefficient ring R.

The fact that many operations on polynomials, such as Euclidean division, require looking at what a polynomial is built of as an expression rather than evaluating it at some fixed value for x makes the distinction between polynomials and polynomial functions even more crucial.

F and g jointly define the identity of the polynomials q and r. This demonstrates that the ring is a Euclidean domain and is known as Euclidean division, division with remainder, or polynomial long division.

Prime polynomials, or irreducible polynomials, are similar in that they are non - zero polynomials that cannot be factored into the product of two non - constant polynomials. "Non - constant" should be changed to "non - constant or non - unit" when referring to coefficients in a ring. Each polynomial can be broken down into a product of irreducible polynomials and an invertible constant. This decomposition is unique up to the order of the components and the multiplication of any non - unit factor by a unit. It happens if the coefficients belong to a field or a distinct factorization domain and division of the unit factor by the same unit.

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There exist algorithms to determine whether the coefficients are irreducible and to calculate the factorization into irreducible polynomials when the coefficients are integers, rational numbers, or members of a finite field. These algorithms are available in any computer algebra system, however they are not practical for hand - written computation. In some circumstances, irreducibility can also be determined using Eisenstein's criterion.

The digits and their places in the representation of an integer, such as 45, are a shorthand notation for a polynomial in the radix or base in contemporary positional numbers systems, such as the decimal system. Polynomial approximations are a valuable tool for analysing generic functions because of their straight - forward nature.

The Stone - Weierstrass theorem, which states that every continuous function defined on a compact interval of the real axis can be approximated on the entire interval as closely as desired by a polynomial function. Also, Taylor's theorem, which roughly states that every differentiable function locally looks like a polynomial function, are both significant examples in calculus. Splines and polynomial interpolation are two useful approaches to approximation.

It is common practise to employ polynomials to encapsulate data about other objects. A matrix's or a linear operator's characteristic polynomial gives details on the eigenvalues of the operator. The simplest algebraic relation satisfied by an algebraic element is recorded by its minimum polynomial. The number of appropriate colorings for a graph is represented by its chromatic polynomial.

When used as an adjective, "polynomial" can refer to any quantity or function that can be expressed as a polynomial. For instance, the term "polynomial time" in computational complexity theory refers to an algorithm's execution time being constrained by a polynomial function of some variable, such as the amount of the input.

One of the first mathematical tasks is "solving algebraic equations," sometimes known as finding the roots of polynomials. But, the sophisticated and useful notation we employ today didn't emerge until the 15th century. Equations were previously spelled out in words. In Robert Recorde's The Whetstone of Witte, published in 1557, the equal sign is used for the first time.

In Michael Stifel's Arithemetica integra, published in 1544, the symbols + for addition, - for subtraction, and the usage of a letter for an unknown are all mentioned. The idea of the graph of a polynomial equation was first given by René Descartes in La géometrie, published in 1637. As seen above in the generic formula for a polynomial in one variable, where the a's signify constants and x denotes a variable.

He popularised the usage of letters from the beginning of the alphabet to denote constants and letters from the end of the alphabet to denote variables. Descartes also popularised the use of superscripts to represent exponents. The highest degree of a polynomial's monomials the individual terms with non - zero coefficients is referred to as the polynomial's degree in mathematics. A term's degree is a non - negative integer and is calculated by adding the exponents of all the variables that occur in it.

The highest exponent that occurs in a univariate polynomial is all that defines the polynomial's degree. Order used to be a synonym for degree, but today it can also apply to a number of other ideas. Nevertheless, as the degree of a product is equal to the sum of the degrees of its elements, this is not necessary when the polynomial is expressed as a product of polynomials in standard form.

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Roman ordinal numerals serve as the basis for names for degrees above three. The designations given to the number of variables, which are based on Latin distributive numbers, should be distinguished from this.

The degree of the input polynomials is closely related to the degree of the sum, product, or composition of two polynomials. The larger of the two polynomials' degrees is less than or equal to the degree of the sum ( or difference ) of the two polynomials.

The degree of a polynomial is equal to the product of the polynomial by a non-zero scalar. A vector space is made up of all the polynomials with coefficients from a particular field F whose degrees are less than or equal to a specific number n.

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