Applied Maths Tutors Near Me, Applied Mathematics Classes Near Me, Maths Tuition Classes Near Me :
Maths, Mathematics, Conventional Maths, Applied Mathematics, Engineering Mathematics Online/Home ( Tutor, Tuition Classes, Lesson, Coaching Classes, Private Tutor, Digital Education Vocational Skills Near Me, home math tutor near me offline, home Applied Mathematics Tutors Near Me offline, home Applied Maths Tutors Near Me offline, Applied Maths Online Tutors, Applied Mathematics Online Tutors, Applied Maths Home Tutors Near Me offline, offline Applied Mathematics Home Tutors Near Me offline, offline Applied Mathematics Home Tuition Classes Near Me offline, online Engineering Tuition Near Me offline, Applied Mathematics Online Tuition Classes Near Me, online Engineering Maths Tuition Classes Near Me offline, Applied Maths Lessons Near Me, Applied Maths Online Lessons Near Me, Applied Mathematics Online Lessons, Applied Maths Home Lessons Near Me, online math tutor near me online, offline Applied Mathematics Home Lessons Near Me offline, Applied Maths Courses Near Me, Applied Maths Online Courses Near me, offline Applied Maths Home Courses Near Me offline, Home Applied Maths Private Teachers Near Me offline, Applied Maths Tutoring Services Near Me, online Engineering Math Classes Near Me offline, Applied Maths Online Tutoring Services Near Me, online Engineering Maths tutor near me offline, offline math tutor near me at home, Applied Mathematics Online Tutoring Services near me, Applied Maths Home Tutoring Services Near Me, Applied Mathematics Home Tutoring Services Near Me, Applied Maths Olympiads Tutor Near Me, Applied Mathematics NTSE Classes Near Me, Applied Maths Workshops Near Me, Applied Maths Seminars Near Me ) for Class 6, Class 7, Class 8, Class 9, Class 10, Class 11, Class 12, College Level Students ( BA, BSc, BCom, BE, ME, BTech, MTech, BCA, MCA ), Recreational Adult Learners near me in Greater Noida, Faridabad, Gurgaon ( Gurugram ), Greater Noida West, Ghaziabad, Mumbai, Hyderabad, Bangalore ( Bengaluru ), Pune, Alwar ( Rajasthan ), Bhiwadi ( Rajasthan ), Jaipur ( Rajasthan ) and other major locations within India and Abroad ( Manitoba, British Columbia - BC, Canada - CA ) :
Conventional Maths & Classical Mathematics, Modern Math & Advanced Mathematic - Scope & Significance :
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, as well as 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 aggressively evolved 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, societal necessities, ease of learning, 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 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. 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 revolution, the adoption of computing tools, advanced softwares and Cloud based platforms 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 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, 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.
Maths Teachers Near Me, Maths Online Tuition Classes Near Me, Home Maths and Science Tuition Near Me :
Wise Turtle Academy offers expert tutoring/tuition services for teaching the subjects of various courses, viz., Conventional Mathematics, Applied Maths & Engineering Mathematics, including standard Maths and Mathematics. All prominent School Level Educational Statutory Boards, like Central Board of Secondary Education CBSE, Indian Certificate of Secondary Education ICSE, International Baccalaureate IB, International General Certificate of Secondary Education IGCSE, UP Uttar Pradesh Board of Secondary and Senior Secondary Education, Rajasthan Board of Secondary Education, MP Madhya Pradesh Board of Secondary and Senior Secondary Education, National Institute of Open Schooling NIOS, Indira Gandhi National Open University IGNOU, Private Candidates, Patrachaar Students, Distance Education Students, Correspondence Courses, Online Courses, Other State Boards, Intermediate, Matriculation, etc., catering to both Indian and International curriculum are reasonably covered under our tutoring services. All Classes, viz, class 1st, 2nd, 3rd, 4th, 5th, 6th, 7th, 8th, 9th, 10th, 11th, 12th are looked into. Our Maths & Mathematics classes are delivered according to the respective Boards' stipulations. These classes are delivered through Home & Online Tuitions, Online & Home Tutors, Home & Online Tuition Classes, Private Teachers, Live Virtual Classes & related Tutoring Services near me in Greater Noida, Noida, Greater Noida West, Noida Extension, Gurgaon, Ghaziabad, Faridabad, Delhi ( Kailash Colony, East of Kailash Blocks A B C D E F G H I, Sant Nagar, Sreenivaspuri, Okhla Phase 1 2 3 4 5 6,Nehru Place ), Alwar, Jaipur, Rajasthan and all across India.
Wise Turtle Academy - Geographical Presence & Scope :
Services' Coverage - Greater Noida and Greater Noida West, Uttar Pradesh, India
Core Areas :
Pari Chowk, Omaxe NRI City, Eldeco Greens, Unitech Habitat, Ace Infrastructure, Super Tech Czar Suites Omicron 1/2/3, IFS Society Villas Pari Chowk, SDS NRI Residency Pari Chowk, The Palms Pocket P 7, ATS Pristine, Jaypee Greens ( Sun Court 1, Crescent Court 3 ) Pari Chowk, Metro Line, Sectors Alpha 1, Alpha Commercial Belt, Beta 1, Mu 1, Alpha 2 Main Market, Mu 2, Ansals Golf Links, Eldeco Meadows, Mu 3, Paramount Golf Foreste Studio Apartments, Mu 4, Beta 2, Gama2, Shisham Estate Gama 1 ( Officer's Colony ), Kadamba Estate, Gamma 1 ( Pocket A Officer's Colony, Pocket B , C, D, E, F, G ), Omega 1, Eta 1, Gamma 2, Omega 2, Eta 2, Chi 1, Omega 3,Eta 3, Chi 2, Omega 4, Chi 3, Eta 4, Xu 1, Phi 1, Xu 2, Phi 2, Xu 3, Sigma 1, Phi 3, Sigma 2, Jalvayu Vihar Society, Sigma 3, Army Welfare Housing Organisation AWHO Twin Towers Societies, CGEWHO, Gamma 2, Sigma 4, Zeta 1, Builder's Area P 1 2 3 4 5 6 7 8 9, Eachhaar,
Swarna Nagari ( Pockets A, B, C, D, E, F ), Phi 4, Zeta 2, Zeta 3, HIG Apartment Omicron 1, Chi 4, Xi 4, Zeta 4, Pi 1, Xu 4, Pi 2, Omicron 2, Ashiana Orchids, Pi 3, MSX Alpha Homes, Rail Vihar, Omicron 3, Pi 4, Swarna Nagri, Knowledge Park 1 ( KP 1 ) KP I, Pi 5, Rampur Jagir, Advocate's Colony, The King's Reserve, Purvanchal Heights, Site 1, Ecotech 1, Site 2, Ecotech 2, Site 3, Ecotech 3, Site 4, ATS Paradioso, Ecotech 4, Site A, The Oasis, IFS Villas, Shivalik Residency, Site B, Oasis Venetia Heights, UPSIDC Site C, Uttar Pradesh, Stellar Mi Legacy, Site D,
Silver City 1 2 3, Migsun Green Mansion, Site E, DesignArch, Site 5, Eachhaar, Site 6, Makora, Site 7, Knowledge Park 2 ( KP 2 ) KP II, Rail Colony, Jal Vayu Vihar, Knowledge Park 3 ( KP 3 ) KP III, Ashirwaad Apartments, Tughalpur Village, NTPC Colony, The Palms, Jagat Farm, Kulesra, Officer's Colony, NRI City 1 2 3, NRI Colony , LG Chowk, Surajpur, Kasna Village, Cherry County, Alistair Meadows, The Oasis, AVJ Heightss, ACE Platinum, Alpha Homes ), Greater Noida West ( Gaur City I - 1, Gaur City II - 2, Gaur Chowk )
Pari Chowk, LG Chowk, Gamma 1, Gama 1, Gamma 2, Gama 2, Alpha 1, Xu 1, Mu 1, Pi 1, Chi 1, Sigma 1, Surajpur Site 1, Tau 1, KP 1, Omega 1, Delta 1, Blocks A B C D E F G H I J K L M, Ecotech 1, Eachhaar, Alpha 2, Beta 1, Beta 2, Xi 1, Xi 2, Phi 1, Phi 2, Omicron 1, Omicron 2 A, Omicron 3, Zeta 1, Zeta 2, Eta 1, Eta2, Delta 1, Delta 2, Knowledge Park 1, Knowledge Park 2, Knowledge Park 3, Omaxe Connaught Place Mall, Rampur Jagir Chowk, Alpha Commercial Belt, Surajpur, Sharda Hospital, Sector 150, ATS Pristine Sector 150, Alpha 2, Xu 2, Mu 2, Pi 2, Chi 2, Sigma 2, Surajpur Site 2, Tau 2, KP 2, Omega 2, Sector 144, Sector 143, Sector 27, AWHO, CGEWHO, Swarna Nagari ( Pockets A B C D E F G H I J K L ),
Tughalpur Village, Kasna, Greater Noida Expressway, Sector 31, Sector 32, Sector 33, NRI City, Blocks, Sectors, Silver City 1 2 3 4 5, Sector 34, Alpha 3, Xu 3, Mu 3, Pi 3, Chi 3, Sigma 3, Surajpur Site 3, Tau 3, KP 3, Omega 3, Delta 3, Sector 35, Rampur, Sector 36, Alpha 4, Xu 4, Mu 4, Pi 4, Chi 4, Sigma 4, Surajpur Site 4, Tau 4, KP 4, Omega 4, Delta 4, Sector 37, sector 38, sector 39, Sector 76, Ace Platinum, The King's Reserve, Sector 78 ( Aditya Urban Casa Towers A B C D E F G H I, Hyde Park, UPSIDC Site B, Surajpur, Greater Noida, Uttar Pradesh, Block C, Amrapali Princely Estate, Civitech Stadia, Mahagun Mirabella, Mahagun Moderne ), Migsun Mansion Greens, Shivalik,
Jaypee Greens ( Narmada Gate Star Court Tower 1 2 3 4 5 6 7 8 9 10 ), Paramount Golf Foreste, Purvanchal Heights, AVJ Heights, MSX Alpha Homes, Rail Vihar, Rail Colony, Jal Vihar, Judge Society, IFS Villas, Ashirvad Apartments, Sector 40, IRDO Colony Apartment, Builder's Area P 1 2 3 4 5 6 7 8 9, Alpha 5, Xu 5, Mu 5, Pi 5, Chi 5, Xi 5, DesignArch, Shivalik, Makora, Tilapta, AVJ Heights, Sigma 5, Surajpur Site 5, Tau 5, KP 5, Omega 5, Delta 5, Sector 41, Sector 42, Noida Extension ( Sector 1 Arihant Arden ), Sector 43, Ace Platina, Ace Platinum, Sector 45, Greater Noida West ( Gaur Chowk, Gaur City 1, Gaur City 2, Saya Zoin ).
Peripheral Areas - Inner :
Migsun Green Mansion, Stellar MI Legacy, Ratan Vihar, Defence Empire I, Tilpata Golchakkar, Devla, Village Tilpatta, UPSIDC Site C, Site F, Site G, Site H, Site I, Site J, Site K, Site L, Site M, Surajpur Industrial Area, Tata Enclave, Anand Ashray Complex, Chorisia Speciosa Estate, NHPC Society, CGEWHO Project, Kendriya Vihar, Nirman Kunj CPWD Society, Unitech Heights, Vrinda City Apartment, Shri Kripa Kunj, Icon Apartments, Adarsh Vihar Society, Ottoman Turkish Baklava, Green Noida Vertical Garden, Purvanchal Royal City Phase 1, Earthcon Casa Grande 2, Sampada Livia, Express Park View 1 2 Apartments, Durva Greens, Chuharpur Market, Lohiya Enclave, Red Building, Lal Building,
Omicron 1A Kali Building, Omicron 1B, Omicron 1C, Omicron 1D, Possession Office, Omaxe Orchid Avenue, Omaxe Society, EWS Society, Stellar MI Citihomes, Ropan, Eldeco Residency Greens, Cassia Estate Society, BSF Housing Society, Aichhar, Parsvnath Estate, Swarn Nagri ( Pockets A B C D E F G H I J K L ), Ambey Bharti Apartment, LG Cooperative Housing Society, Jamia Apartment, Khushboo Apartments, Khushboo Sahkari Awas Samiti, Sun Twilight Villas, Jyoti Kiran Society, White House Apartments, Unitech Cascades Apartment, Tower 1, Tower 15, Tower 18, Plumeria Garden Estate, Fairway Apartment Tower D, Nambardar Residence, Ajju Prajapati Town, Anjana Welfare Society,
Paradise Dream City, Defence Empire 2, Eden Golf Group, Victory Enclave, Shyam Enclave, Mahamaya Enclave, Royal Paradise, Prithvi Greens, Bhoomi Greens Phase 2, Shiv Enclave, Kartik Nagar, Himalaya Hi Tech City, Ajnara City, Galactic City, Ace City, Ace Divino, ATS Destinaire, Arihant Ambar, Flora Heritage, The Palm Valley, Akshardham Colony, Noor Colony, Vidyapati Nagar, Jalpura, Jamia Nagar Colony, Tusiana Village, Tusyana, Supertech, Amrapali West, Udyog Vihar, Brahmpur Rajraula, Nawada, Parsvnath Privilege, Parsvnath Edens.
Peripheral Areas - Outer :
NTPC Society, Nirman Vihar, Techoma Estate, Cassia Fistula Estate, Rasoolpur Rai, Jaitpur Village, Ekanki Enclave ( Block G ), Vimal Sadan Society, Cassia Sigma, Sigma Group Housing Society, Chorosia Estate, Grand Forte, BSNL Society, Kyampur, Ecotech Extension 1, Migsun Ultimo Sun 3, Omaxe Palm Greens, KKS Homes, Ghodi Bachheda, Ghori Bachhera, MamaPikin Suya House, Ebony Estate, Austonia Estate, ATS Paradiso, Lagerstroemia Estate, Cassia Nodosa Estate, Chakrasia Estate, Himsagar Apartment, Mitra Enclave, Surajpur Site 4, Godrej Golf Links, Ansal Golf Links 1, Surajpur Site 1, Surajpur Site 2, Surajpur Site 3, Surajpur Site 4, Surajpur Site 5,
Surajpur Site 6, Block A, Block B, Block C, Block D, Block E, Block F, Block H, Block I, Block J, Block K, Block L, Block M, 1st Cross Street, 2nd Cross Street, 3rd Cross Street, 4th Cross Street, 5th Cross Street, 6th Cross Street, 7th Cross Street, 8th Cross Street, 9th Cross Street, 10th Cross Street, Service Road, First Avenue, Second Avenue, Third Avenue, Fourth Avenue, Fifth Avenue,1st Avenue, 2nd Avenue, 3rd Avenue, 4th Avenue, 5th Avenue, 6th Avenue, 7th Avenue, 8th Avenue, 9th Avenue, 10th Avenue,
ATS Dolce, Amrapali Grand Apartment, Migsun Wynn, SKA Metro Ville, Cluster ETA 2, 1st Cross Avenue, 2nd Cross Avenue, 3rd Cross Avenue, 4th Cross Avenue, 5th Cross Avenue, 6th Cross Avenue, 7th Cross Avenue, 8th Cross Avenue, 9th Cross Avenue, 10th Cross Avenue, 11th Cross Avenue, Main Road, Tilpata Karanwas, Luharli, Ajayabpur, Eachachhar, Accher, Habibpur, Gujarpur, Jhatta, Gulavali, Malakpur, Judge Society, Amit Nagar, NTPC Anandam Society, Purvanchal Silver City 2,
Unitech Horizon, Alistonia Estate, Sector 34, Gulistanpur Village, Gulmohar Estate, Theta 1, Theta 2, Theta 3, Theta 4, Theta 5, Delhi Police Housing Society, Gaur Atulyam, Eldeco Mystic Greens, Palash Estate, Pocket 4, Khadar Ke Marhiya, Dadha, Sadar Tehsil, GNIDA BHS 16 Housing Scheme, Bironda, Haier Industrial Park, Sakipur, Makora, Tugalpur, NSG Society, Gurjinder Vihar.
Services' Coverage - International ( Overseas & Abroad )
Canada ( British Columbia ( BC ), Canada ( Ca ) & Manitoba ( University of Manitoba ), Canada ( Ca ) ), Australia, Mexico, London, Singapore, Hong-Kong, United States ( Florida, Carolina, New Jersey, Washington ), United Kingdom, Abu Dhabi, Sri Lanka, Bhutan, Nepal, Burma, Malaysia, Bangladesh, Dubai, Africa ( South Africa ), Netherlands, Denmark, Korea, Japan, Asia-Pacific ( APAC ), Americas ( AMER ), Europe, Warsaw, Poland, Russia, France, Germany, Spain, Greece, Belgium, Switzerland and other countries.
Services' Coverage - Pan India ( Across India - National )
Faridabad, Ghaziabad, Delhi ( Kalkaji, Okhla ( Phase - 1, 2, 3 ), Nehru Place, Hauz Khas, South Delhi ( East of Kailash, Kailash Hills, Lajpat Nagar, Okhla Phase 1, Okhla Phase 2, Okhla Phase 3, Srinivaspuri, Kalkaji, Nehru Place, Sant Nagar ), North Delhi, West Delhi, East Delhi ), Gurgaon, Pune, Mumbai, Bangalore, Hyderabad, Andhra Pradesh, Secunderabad, Ahmedabad, Alwar ( Rajasthan ), Jaipur ( Rajasthan ), Bhiwadi ( Rajasthan ), Gurugram, New Delhi, Greater Noida West, Faridabad, Ghaziabad, Gurgaon, Gurugram, Delhi, Noida, Telangana, Bhopal, Gandhinagar, Lucknow, Shimla, Mangalore, Chennai, Noida ( viz., ATS Pristine, Sector 150, Sector 144, Sector 148, Sector 125, Sector 76 Metro Station, Aditya Urban Casa Towers A B C D E F G H I J K L, Sector 78, Mahagun Moderne, Sports City, Hyde Park, Amrapali Princely Estate, Mahagun Mirabella, Civitech Stadia, Sectors 76, 77, 79, 80, 70, 71, 72, 73, 74, 75 ) and other nearby locations.
Maths Tutorials Near Me Online, Maths Home Tuition Near Me Offline, Physics Chemistry Maths Tutor Near Me :
Wise Turtle Academy has very good experience in delivering "Maths & Mathematics Home Tuitions, Maths Online Tutors, Maths Home & Online Tuition Classes, Maths Home & Online Tutoring Services, online Mathematics Lessons Near Me offline, Mathematics Online Tuition Near Me" for many English medium schools and boards, namely, Central Board of Secondary Education CBSE, Indian Certificate of Secondary Education ICSE, International Baccalaureate IB, International General Certificate of Secondary Education IGCSE, Rajasthan State School Board of Secondary and Senior Secondary Education, Uttar Pradesh State School Board of Secondary and Senior Secondary Education, National Institute of Open Schooling NIOS, Indira Gandhi National Open University IGNOU, Patrachaar Students, Private Board Students, Correspondence Courses, Distance Education Courses and more. We also cover quality oriented, contemporary, educational & learning support related Study Notes, solved assignments, crash courses for Boards, Home Work Help and every other way to assist our students & clients. Our Maths & Mathematics learning support services are provided by best, experienced and result oriented Online & Home Tutors, as well as, Private Teachers in Greater Noida West. Right from Math - Magic ( Shapes & Space, Numbers From One To Nine, Addition, Subtraction, Numbers from Ten to Twenty, Time, Measurement, Numbers From Twenty-one to Fifty, Data Handling, Patterns, Numbers, Money, How Many, Teacher's Note, Shape Kit ), Ganit Ka Jaadu, Trigonometry, Algebra, Probability, Statistics, Geometry, Mensuration, Constructions, Functions, Sets, Matrices, Determinants, Inverse Trigonometric Functions, Differentiation, Integration, Integral Calculus, Differential Equations, Continuity & Differentiability, Linear equations, etc., to college level Maths & mathematics, like Linear Programming, Game Theory, PERT Analysis, Decision Research Frameworks, Decision Support Systems, Operations Research, Marketing Research, etc., are widely covered. We also cover un-conventional mathematics, including Applied Mathematics, Statistics, Probability, Linear Programming and Engineering Mathematics that focus upon various college level subjects, courses, classes and university syllabus.
Maths and Science Home Tutor Near Me, Home Tutors For Maths and Science Near Me, Maths Home Tutor Near Me Offline :
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
Maths Tuition Near Me Class 12, Online Maths Coaching Near Me Online, Home Maths Private Tutor Near Me, English and Maths Tuition Near Me :
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 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,
Congruence of Triangles, Criteria for Congruence of Triangles, Some Properties of a Triangle, Some More Criteria for Congruence of Triangles, Inequalities in a Triangle,
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 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,
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,
Collection of Data, Presentation of Data, Graphical Representation of Data, Measures of Central Tendency,
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
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
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
Introduction, Tangent to a Circle, Number of Tangents from a Point on a Circle
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
A1.2 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,
Appendix A2 : Mathematical Modelling
A2.2 Stages in Mathematical Modelling,
A2.3 Some Illustrations,
A2.4 Why is Mathematical Modelling Important?
Class 11 - XI ( Class 11th )
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
Introduction, Measures of Dispersion, Range, Mean Deviation, Variance and Standard Deviation, Analysis of Frequency Distributions
Introduction, Random Experiments, Event, Axiomatic Approach to Probability
Appendix 1 : Infinite Series
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.2.3 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
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
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.1.2 What is a Proof?
Appendix 2: Mathematical Modelling
A.2.2 Why Mathematical Modelling?,
A.2.3 Principles of Mathematical Modelling
Class 12 - XII ( Class 12th ) - Part - 2 ( II )
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
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, Social Sciences, and many more. It has been found that the Mathematics syllabus in senior secondary grades designed for Science topics 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 ) :
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.
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.
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.
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.
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)
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.
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.
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)
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, 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 the aligned mathematical analytical treatments, we can reflect upon the following related as well as critical Mathematical sub - disciplines :
Inverse Trigonometric Functions,
Continuity & Differentiability,
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 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 ( 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.
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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 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.
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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.
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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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, 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 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.
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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.
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" ).
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.
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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