MathematicsMathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of 'figures and numbers'. In the formalist view, it is the investigation of axiomatically defined abstract structures using logic and mathematical notation; other views are described in Philosophy of mathematics. Mathematics might be seen as a simple extension of spoken and written languages, with an extremely precisely defined vocabulary and grammar, for the purpose of describing and exploring physical and conceptual relationships.
Although mathematics itself is not usually considered a natural science, the specific structures that are investigated by mathematicians often have their origin in the natural sciences, most commonly in physics. However, mathematicians also define and investigate structures for reasons purely internal to mathematics, because the structures may provide, for instance, a unifying generalization for several subfields, or a helpful tool for common calculations. Finally, many mathematicians study the areas they do for purely aesthetic reasons, viewing mathematics as an art form rather than as a practical or applied science. Some mathematicians like to refer to their subject as "the Queen of Sciences".
Mathematics is often abbreviated to math (in American English) or maths (in British English).
Overview and history of mathematics
See the article on the history of mathematics for details.
The word "mathematics" comes from the Greek μάθημα (máthema) which means "science, knowledge, or learning"; μαθηματικός (mathematikós) means "fond of learning".
The major disciplines within mathematics arose out of the need to do calculations in commerce, to measure land and to predict astronomical events. These three needs can be roughly related to the broad subdivision of mathematics into the study of structure, space and change.
The study of structure starts with numbers, first the familiar natural numbers and integers and their arithmetical operations, which are recorded in elementary algebra. The deeper properties of whole numbers are studied in number theory. The investigation of methods to solve equations leads to the field of abstract algebra, which, among other things, studies rings and fieldss, structures that generalize the properties possessed by the familiar numbers. The physically important concept of vectorss, generalized to vector spaces and studied in linear algebra, belongs to the two branches of structure and space.
The study of space originates with geometry, first the Euclidean geometry and trigonometry of familiar three-dimensional space (also applying to both more and less dimensions), later also generalized to non-Euclidean geometries which play a central role in general relativity. Several long standing questions about ruler and compass constructions were finally settled by Galois theory. The modern fields of differential geometry and algebraic geometry generalize geometry in different directions: differential geometry emphasizes the concepts of functions, fiber bundles, derivatives, smoothness and direction, while in algebraic geometry geometrical objects are described as solution sets of polynomial equations. Group theory investigates the concept of symmetry abstractly and provides a link between the studies of space and structure. Topology connects the study of space and the study of change by focusing on the concept of continuity.
Understanding and describing change in measurable quantities is the common theme of the natural sciences, and calculus was developed as a most useful tool for doing just that. The central concept used to describe a changing variable is that of a function. Many problems lead quite naturally to relations between a quantity and its rate of change, and the methods to solve these are studied in the field of differential equations. The numbers used to represent continuous quantities are the real numbers, and the detailed study of their properties and the properties of real-valued functions is known as real analysis. For several reasons, it is convenient to generalise to the complex numbers which are studied in complex analysis. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions, laying the groundwork for quantum mechanics among many other things. Many phenomena in nature can be described by dynamical systems and chaos theory deals with the fact that many of these systems exhibit unpredictable yet deterministic behavior.
In order to clarify and investigate the foundations of mathematics, the fields of set theory, mathematical logic and model theory were developed.
When computers were first conceived, several essential theoretical concepts were shaped by mathematicians, leading to the fields of computability theory, computational complexity theory, information theory and algorithmic information theory. Many of these questions are now investigated in theoretical computer science.
Discrete mathematics is the common name for those fields of mathematics useful in computer science.
An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis and prediction of phenomena and is used in all sciences. Numerical analysis investigates the methods of efficiently solving various mathematical problems numerically on computers and takes rounding errors into account.
Topics in mathematics
An alphabetical and subclassified list of mathematical topics is available. The following list of subfields and topics reflects one organizational view of mathematics.
In general, these topics and ideas present explicit measurements of sizes of numbers or sets, or ways to find such measurements.
- Number -- Natural number -- Pi -- Integers -- Rational numbers -- Real numbers -- Complex numbers -- Hypercomplex numbers -- Quaternions -- Octonions -- Sedenions -- Hyperreal numbers -- Surreal numbers -- Ordinal numbers -- Cardinal numbers -- p-adic numberss -- Integer sequences -- Mathematical constants -- Number names -- Infinity -- Base
These topics give ways to measure change in mathematical functions, and changes between numbers.
- Arithmetic -- Calculus -- Vector calculus -- Analysis -- Differential equations -- Dynamical systems and chaos theory -- List of functions
These branches of mathematics measure size and symmetry of numbers, and various constructs.
- Abstract algebra -- Number theory -- Algebraic geometry -- Group theory -- Monoids -- Analysis -- Topology -- Linear algebra -- Graph theory -- Universal algebra -- Category theory -- Order theory
These topics tend to quantify a more visual approach to mathematics than others.
- Topology -- Geometry -- Trigonometry -- Algebraic geometry -- Differential geometry -- Differential topology -- Algebraic topology -- Linear algebra -- Fractal geometry
Topics in discrete mathematics deal with branches of mathematics with objects that can only take on specific, separated values.
- Combinatorics -- Naive set theory -- Probability -- Theory of computation -- Finite mathematics -- Cryptography -- Graph theory -- Game theory
Fields in applied mathematics use knowledge of mathematics to real world problems.
- Mechanics -- Numerical analysis -- Optimization -- Probability -- Statistics -- Financial mathematics
Famous theorems and conjectures
These theorems have interested mathematicians and non-mathematicians alike.
- Fermat's last theorem -- Goldbach's conjecture -- Twin Prime Conjecture -- Gödel's incompleteness theorems; -- Poincaré conjecture; -- Cantor's diagonal argument -- -- Four color theorem -- Zorn's lemma -- Euler's identity -- Scholz Conjecture -- Church-Turing thesis
These are theorems that have changed the face of mathematics throughout history.
- Riemann hypothesis -- Continuum hypothesis -- P=NP -- Pythagorean theorem -- Central limit theorem -- Fundamental theorem of calculus -- Fundamental theorem of algebra -- Fundamental theorem of arithmetic --Fundamental theorem of projective geometry -- classification theorems of surfaces -- Gauss-Bonnet theorem
Foundations and methods
Such topics are approaches to mathematics, and influence the way mathematicians study their subject.
- Philosophy of mathematics -- Mathematical intuitionism -- Mathematical constructivism -- Foundations of mathematics -- Set theory -- Symbolic logic -- Model theory -- Category theory -- Theorem-proving -- Logic -- Reverse Mathematics -- Table of mathematical symbols
History and the world of mathematicians
- History of mathematics -- Timeline of mathematics -- Mathematicians -- Fields medal -- Abel Prize -- Millennium Prize Problems (Clay Math Prize) -- International Mathematical Union -- Mathematics competitions -- Lateral thinking
Mathematics and other fields
- Mathematics and architecture -- Mathematics and education -- Mathematics of musical scales
- List of mathematical coincidences
Referring to the axiomatic method, where certain properties of an (otherwise unknown) structure are assumed and consequences thereof are then logically derived, Bertrand Russell said:
This may explain why John Von Neumann once said:
- Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.
About the beauty of Mathematics, Bertrand Russell said in Study of Mathematics:
- In mathematics you don't understand things. You just get used to them.
- Mathematics, rightly viewed, possesses not only truth, but supreme beauty -- a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can sho
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The STIMULUS Project
STIMULUS - peer assisted learning (PAL) through visits to schools. University students visit schools to share with school students their enthusiasm for science, engineering, computers, mathematics or medical sciences and, guided by experienced teachers, to assist the pupils in their school work.
An amusement park of mathematics. Puzzles and number problems, fractals, geometry, calculus, algebra, online games, online calculators, and links.
NRICH-Online maths club
An online maths club for bright young Mathematicians. Publishing a monthly magazine on the web with problem sets, childrens solutions to previous months problems, games, and news of events.
Interactive Math Server
Interactive exercises, online calculators and plotters, mathematical recreations. Available in French and English.
Web page for the newsgroup k12.ed.math. Contains the charter, FAQ and other information about the newsgroup.
A mathematics-related, one-stop website for interactive lessons, puzzles, educational issues, terminology, help, projects and lessons.
Glossary of mathematical mistakes; quotes; puzzles and problems; links.
Piguy's Math Page
This page has dozens of math calculators, making all sorts of math, from algebra to arithmetic to geometry, easier. Also, 50,000 decimals of pi and a logarithm table.
Visual Mathematics is a forum for the dialogue between artists and scientists.
QELP Web Site
Quantitative Environmental Learning Project (QELP) at Seattle Central Community College. Resources to integrate math and environmental science in college classrooms.
The Precalculus Algebra TI-83 Tutorial
Includes lessons for using the TI-83 graphing calculator in a college algebra or precalculus algebra course.
Curious and Useful Math
Tricks and rules for quickly calculating certain types of problems. Also has some entertaining trivia and facts that are nice to slip into casual conversation.
Math Forum: Search mam-list
April is Math Awareness Month (MAM). For more information about MAM, consult the postings of the MAM-list.
Participating students from all over the world work on open-ended word problems called MathWorld Interactive Challenges. During the cycle, students discuss, share ideas, meet each other, and ask questions on the he MathWorld Interactive Message Boards.
Activities for a Pi Day, March 14 (3.14) celebration including pi math problems, history, songs, stories, and greetings.
We will use the amazing properties of the Pythagorean Theorem to explore the world around us.
Mathematics Reference: Math Tables, Facts and Formulas
A mathematics reference collection of "K through 14" math tables, facts, definitions, formulas and explanations ... from general math through college calculus.
Books, games, hints, tricks.
Meridian Creative Group
Creates educational print and software programs that teach math concepts with real-life applications.
MathMol (Mathematics and Molecules) is designed to serve as an introductory starting point for K-12 students and teachers interested in the field of molecular modeling and its application to mathematics.
The Student Support Centre (UK) Ltd
Home based learning materials to supplement classroom work using Maths and English videos for ages 5-18. With over 40,000 students the results speak for themselves.
For teachers and parents to download practice worksheets and tests to improve children's achievement. Competitions, children's pages and children's contributions.
Homework help, tutoring, and formulas.
Mathematics Experiences Through Image Processing (METIP)
Materials that promote digital image processing as a means of motivating K-12 students in mathematics and as a means of teaching mathematical concepts.
The Mathematics of Cartography
Learning math through maps.
Using tools of yesterday, and tools specially created for this site, Ptolemy Ptools will give you geometry projects to do with your 3D computer games as well as projects to take outside and explore the world.
Helps teachers, students and parents find software, videos and supplies to lesson plans, maps, news and books. Also provides services such as forums, online teacher training, sites of the week and web tools.
Mathematics Help Central
Devoted to helping the searching math student, this site includes topics concerning all aspects of mathematics. Also includes class notes.
Extracting Square Roots by Means of Napier Rods
Illustrated, step-by-step tutorial.
Visitors can test their math skills and do research. Also, offered are math supplies like: books, calculators, and software.
Students use real time weather maps and track real planes flying in the U.S. skies to learn about vectors and then put their knowledge to the test by taking the Pilot Test Flight. Here they will solve an authentic real world problem using science and mathematics.
The Living Mathematics Project
Working to apply recent advances in computer programming languages and the technology of the World Wide Web to construct a new medium for the communication of Mathematics. Includes links to several visualization sites. [Java.]
Answering Mathematics Helper
Exercise tool to practice addition, subtraction, multiplication, division, trigonometry and fuzzy logic problems.
Get To Mathematics
Mathematics teaching and learning resources including quiz, game, tutorials, animations and Geometric's Sketchpad files.
2 plus 2: The Home of Mathematically Correct
Devoted to the concerns raised by parents and scientists about the invasion of our schools by the New-New Math and the need to restore basic skills to math education.
A collaborative project, using the concept of PI as the focus, is a home page on the WWW. This multidisciplinary project which includes math, history, English and thinking skills is designed for fifth through eighth graders.
An Invitation To Mathematics
A free tutorial that explains difficult algebra, trigonometry and calculus concepts to beginning middle/high school students in a simplified way that they can understand and use.
Complex Numbers Review
A review based on Virginia State Standards of Learning objectives covering operations with the imaginary unit i and applying complex equations to real-world scenarios. Developed by high school students.
Introduction to Numeracy
Provides some ideas, findings from research, resources and links to people interested in numeracy particularly as it relates to the use of mathematics outside of the school mathematics classroom.
Mr. Heitmann's Mega Math Resource Page
Links to hundreds of Math Terms, Mathematicians, and math games. Help in Algebra, Geometry and more
Mathematics, Statistics and Operational Research
Information, resources and links to learning and teaching.
Activities For Pi Mathematics
Includes projects, history and videos to help learn about this unique constant.
Mathematics Learning and Assessment
Sharing Innovative Practices. A report on innovations at various UK universities.
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An educational project of Los Alamos National Laboratory.
To create and then develop an environment for learning mathematics. Articles, software tools and links.
A Mathematical Fair in Kazakhstan
Puzzles and educational projects in maths.
Mathematics for Mechanics
Topics: Algebra, Geometry, Trigonometry, Analytical Geometry, Calculus, Vectors. Teaching material and tests. Pages created by Mehrdad Negahban and the University of Nebraska.
GCSE mathematics explanations, examples and exercises. Also, English football club statistics.
A method of doing basic arithmetic using the fingers that is attributed to the Korean tradition.
The National Curriculum for England - Mathematics
Resources for teaching and learning all required math courses.
Solid Gold Gnarly Math
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Solve It Math Program
Problem solving program for grades 4th through 8th.
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21st Century Problem Solving
Sound problem solving concepts and examples that apply to grade school through graduate school and beyond.
Mazes from Newton's Apple
What does mathematics have to do with mazes? What is the history behind mazes? Are there different kinds of mazes? What is topology? David solves a maze by applying the principles of mathematics.
A personal account of the mathematical learning disability, Dyscalculia.
Steve's Math Page
Dedicated to students who need help with their math courses. It contains a brief description of each branch of mathematics.
Problem Of the Week
Creative and challenging math problems appropriate for high school students, updated weekly. Questions are archived.
The Math Page
Arithmetic and trigonometry courses taught at Borough of Manhattan Community College, The City University of New York.
Offers assistance in algebra, geometry, trigonomety and calculus. Contains real-world examples, detailed example problems, and interactive features. Internet Explorer 5.0+ on a Windows PC is required.
Study Skills Online
An electronic guide to some of the best ways to study by Martin Greenhow, Brunel University.
Refresher programme at Loughborough University.
Math Forum Quick Reference
Resources for teaching all levels of K-12 math.
K-12 telecommunications project motivates students to use computer technology while increasing problem-solving strategies and communications skills. Posts challenges in four categories (k-3, 4-6, 7-9 and 10-12) to trigger each registered team to pair up with another team and engage in a problem-solving dialog.
Those interested and involved in adult mathematical literacy may find helpful input at the numeracy discussion list.
Teachers' Lab - Patterns in Mathematics
Teachers explore how patterns occur in numbers and in words in this interactive lab.
Provides regular newsletters, survey data, a discussion group and conference details, as well as resource lists.
Online Math Applications
Free stock market game to learn how the market works. Using real time data from Wall street.
Topics in High School Mathematics
Several topics in high school mathematics presented so as to engage interest of students. Includes description of slide rule to find missing side of right triangle and method of calculating times for Earth to rotate and for the moon to revolve around Earth.
Trigonometry identities and calculus rules for integration and differentiation.
Web-Based Math Resources for the College Student
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Irish school mathematics review, with answers.
Resources for teachers and students, including 'webquests', interesting math facts, bulletin boards and links.
Features games, puzzles and competitions for all ages. Details of events, an online maths museum, newspaper and magazines. Resources for parents and teachers.
The Wonders of Vedic Mathematics
An alternative way of learning mathematics based on Sanskrit religious texts.
Learn to multiply by counting with fingers. Fee based service.
Contains modules for probability and statistics, discreet math, linear systems and advanced calculus. Developed by the Rensselaer Polytechnic Institute.
Covers forms, numbers and functions. Introduces high school students to basic to advanced math principles and reasoning.
Quick reference for basic algebra, trigonometry, geometry, calculus, and physics formulas. Includes online calculators.
For high school teachers and students. Products include an interactive textbook, class video clips, projects, glossary, and resource links.
MCA's Free Online Math Lessons
Features selected learning modules for arithmetic, algebra, geometry, trigonometry, statistics and probability. Includes some resource links.
Vectors by the University of Guelph
Explanations notations, arithmetic operations, rules and the relationship to Cartesian graphs.
Contains tutorials covering algebra, geometry, calculus, differential equations, matrices, and complex variables. Reviews the most important results, techniques and formulas. Presented in worksheet format and require active participation. Includes practice quizzes and forum board.
Learning disability (Math Dyslexia) that inverts numeral digits, making it difficult to do basic math functions. Teaching and learning tips for memorizing formulas; working on exercises and taking tests; articles on learning disorders; links to related resources.
MSc/PGDip Computer Enhanced Maths Education
Offered by Napier University, Scotland. Includes information on the program and student handbook.
Common mathematical and logical errors.
Math in Daily Life
Virtual exhibit looks at the ways people use math every day, from cooking to planning for retirement. Offers teachers labs and workshops.
Primary Arithmetic, or Math
Four booklets, and more, that you can easily copy. One of math games for ages 4-8., two about teaching early math, and one that is more general. Some worksheets, number charts.
The Cornell Historical Math Book Collection
Digital images of many famous and important math texts Indexed by author and title.
A collection of resource tables, facts and formulas.
Gelatinous Vector Components
Meant for Grade 7-9 (age 12-14), younger and older. This experiment demonstrates the x and y components of a vector force.
Shelley's Mathematics Articles
Notes and afterthoughts from teaching for UMUC European Division. Essays on arithmetic, algebra, geometry, trigonmetry, and assorted other topics.
A resource bank of teaching material for lecturers and students in mathematics at UK universities. Free upload and download of material. Registration required.
Math as Language
A brief discussion of math lingo and how it is different from regular communications. Includes additional resource links.
Cut the Knot: Interactive Mathematics Miscellany and Puzzles
Features instructional games and puzzles based on a variety of math topics. Includes more serious articles about the language of math and related links.
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Simply Number Sense
A site dedicated to teaching basic mathematics and how to solve mathematical problems without reliance on the calculator
Cynthia Lanius: Fractals
Features class exercises and lessons or making fractals. Includes explanation of this unique image and types of formulas that are used to create it.
Tutorial covers trigonometry, vectors, lines,complex numbers, cubic equations sequences and limits, continuity derivatives, minimum and maximum values, exponential and logarithmic functions, hyperbolic functions, finding roots of equations, integration, polar coordinates and integration.
A network of teachers, educationalists and students interested in supporting and developing children's early mathematical graphics.
Mathematics real-time video-conferencing for schools.
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Covers the use of chalk boards, geometrical models, rulers, and, calculators and computers to assist the teaching of mathematical concepts. Explains the overseas influence in education and America's additional contribution to the use of these tools. Article is from the Smithsonian Institute.
Reaching the Second Tier: Learning and Teaching Styles in College Education
Discusses the study conducted by Sheila Tobias about two groups of science students that are separated by personal math class experience. Also, explains ways to discover a student's strong point as far as assimilating and processing knowledge.
Free mathematics support materials for UK university professionals and students.
Articles on mathematics, problem of the month, software and computing projects, and other resources.
Cornell Library Historical Mathematics Monographs
Consists of digital surrogates for materials that were part of a joint study involving Digital Preservation between Cornell University and the Xerox Corporation.
GCSE Maths Revision
Features brief reviews of basic math concepts, from numbers and fractions to probability and statistics. Also, includes formula page and help forum.