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Mathematics

A guided introduction to algebra, calculus, geometry, linear algebra, statistics, and game theory.

For: Learners at different starting points, including people reviewing foundations and learners pursuing a focused topic.

After this page you can: Choose a suitable learning path, explain key ideas across six mathematical areas, and solve a representative problem in each.

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About Mathematics

Mathematics studies patterns, quantities, structures, and relationships through precise definitions and logical reasoning. Its areas include algebra, geometry, calculus, statistics, linear algebra, and game theory. Algebraic functions describe relationships; geometry concerns size, shape, and space; calculus uses limits, derivatives, and integrals; statistics includes inference from data; linear algebra includes matrices and vectors; and game theory models strategic interaction. These areas overlap: functions are part of the study of calculus, matrices can represent systems of equations, and probability and statistics are studied together in introductory courses. This hub offers entry points into each area so you can start with a familiar question and build outward.

Mathematics is studied by moving between explanations, examples, representations, and problems. You can express a relationship in words, as an equation, in a table, or on a graph, then consider which representation helps answer the question. A worked example can show a method; trying a similar problem can help you practice using it. If a step is unclear, return to the definition or prerequisite idea and try a simpler case. The resources here include open textbooks, university courses, and interactive study material; they differ in level and format, so choose one that fits your aim. You do not need to follow every branch in order: review algebra before calculus if useful, or begin with geometry, statistics, or game theory.

Start here

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I’m building my foundations

Start with a textbook whose opening chapters cover prerequisites and equations before its chapters on algebraic functions.

Start with: College Algebra 2e

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I’m returning to calculus

Use the free web textbook to revisit functions, limits, derivatives, and integration in a structured sequence.

Start with: Calculus Volume 1

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I want a probability and statistics course

Follow MIT OpenCourseWare’s introductory probability and statistics materials, including its topics in inference and regression.

Start with: Introduction to Probability and Statistics

Learn the foundations

1.Algebra and functionsAlgebraic functions represent relationships.

Algebraic functions represent relationships. OpenStax introduces function notation and how to evaluate a function from a formula.

Function

A relation in which each possible input value leads to exactly one output value.

Domain

The set of input values for a function or relation.

Try it

For f(x) = 2x + 1, find f(2).

Substitute 2 for x: f(2) = 2(2) + 1 = 5.

2.Calculus: limits and changeCalculus includes limits, derivatives, and integrals.

Calculus includes limits, derivatives, and integrals. Begin with functions and limits, then connect the derivative to instantaneous rate of change.

Limit

The value a function approaches as its input approaches a specified value.

Try it

For f(x) = x², calculate the average rate of change from x = 1 to x = 3.

The change in output is 9 − 1 = 8 and the change in input is 3 − 1 = 2, so the average rate of change is 8/2 = 4.

3.Geometry and shapeGeometry is concerned with size, shape, and space.

Geometry is concerned with size, shape, and space. The Open University course explores angles, shapes, symmetry, area, and volume through interactive activities.

Conic

A shape formed by intersecting a plane with a right circular cone.

Try it

Name a familiar round plane figure and identify one geometric measurement associated with its boundary.

A circle is a round plane figure; its boundary length is called its circumference.

4.Linear algebra: vectors and matricesMIT OpenCourseWare’s linear algebra course covers systems of equations, vector spaces, determinants, and eigenvalues.

MIT OpenCourseWare’s linear algebra course covers systems of equations, vector spaces, determinants, and eigenvalues. A matrix can represent a system of equations.

Matrix

A rectangular arrangement of entries that can represent a system of equations.

Try it

Solve the system 2x + y = 5 and x − y = 1.

Adding the equations gives 3x = 6, so x = 2; substituting gives y = 1.

5.Statistics and dataMIT’s introductory course covers probability distributions, inference, confidence intervals, and regression.

MIT’s introductory course covers probability distributions, inference, confidence intervals, and regression.

Median

A value that separates ordered data into two halves.

Try it

Find the mean of 2, 4, and 6.

The sum is 12 and there are 3 values, so the mean is 12/3 = 4.

6.Game theory and strategic choicesGame theory models strategic interaction.

Game theory models strategic interaction. MIT’s syllabus includes strategic-form games, Nash equilibrium, and applications.

  • Game Theory

    The lecture notes introduce simultaneous-move games, Nash equilibrium, and mixed strategies.

Nash equilibrium

A profile of strategies in which each player’s strategy is a best response to the strategies of the others.

Try it

In a two-player payoff table, explain what you would check to determine whether a strategy pair is a Nash equilibrium.

Check whether each player’s chosen strategy is a best response to the other player’s choice.

Explore subtopics

Algebra

Build fluency with algebraic expressions, equations, functions, graphs, and polynomials.

0 subtopicsCalculus

From limits and continuity through single-variable and multivariable differentiation and integration.

0 subtopicsGeometry

Learn how geometry describes space, shape, measurement, transformation, and structure.

0 subtopicsLinear Algebra

Core ideas and methods for systems of equations, vector spaces, linear transformations, and matrix analysis.

0 subtopicsStatistics

Introductory statistical reasoning: data, probability, inference, and models.

0 subtopicsGame Theory

Learn how mathematical models describe strategic choices, information, timing, and cooperation.

0 subtopics

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Books

Glossary

Every free Mathematics course and book on RedLectern

Free Mathematics online courses

  1. A Vision of Linear AlgebraMIT RES.18-010 · Browse materials · Video
  2. Advanced AlgorithmsMIT 6.854J · Browse materials
  3. Advanced AlgorithmsMIT 6.854J · Browse materials
  4. Advanced Analytic Methods in Science and EngineeringMIT 18.305 · Browse materials
  5. Advanced Calculus for EngineersMIT 18.075 · Browse materials
  6. Advanced Complexity TheoryMIT 18.405J · Browse materials
  7. Advanced Partial Differential Equations with ApplicationsMIT 18.306 · Browse materials
  8. Advanced Topics in CryptographyMIT 6.876J · Browse materials
Show all 40 listed coursesShow fewer courses
  1. Algebra IMIT 18.701 · Browse materials
  2. Algebra IIMIT 18.702 · Browse materials
  3. Algebraic CombinatoricsMIT 18.212 · Browse materials
  4. Algebraic GeometryMIT 18.725 · Browse materials
  5. Algebraic GeometryMIT 18.725 · Browse materials
  6. Algebraic GeometryMIT 18.726 · Browse materials
  7. Algebraic Topology IMIT 18.905 · Browse materials
  8. Algebraic Topology IIMIT 18.906 · Browse materials
  9. Algorithmic Aspects of Machine LearningMIT 18.409 · Browse materials
  10. Analysis IMIT 18.100B · Browse materials
  11. Analysis IIMIT 18.101 · Browse materials
  12. Ancient Greek Philosophy and MathematicsMIT ES.113 · Browse materials
  13. Automata, Computability, and ComplexityMIT 6.045J · Browse materials
  14. Behavior of AlgorithmsMIT 18.409 · Browse materials
  15. Calculus of Several VariablesMIT 18.022 · Browse materials
  16. Calculus Revisited: Complex Variables, Differential Equations, and Linear AlgebraMIT RES.18-008 · Browse materials · Video
  17. Calculus Revisited: Multivariable CalculusMIT RES.18-007 · Browse materials · Video
  18. Calculus Revisited: Single Variable CalculusMIT RES.18-006 · Browse materials · Video
  19. Calculus with TheoryMIT 18.014 · Browse materials
  20. Category Theory for ScientistsMIT 18.S996 · Browse materials
  21. Combinatorial AnalysisMIT 18.314 · Browse materials
  22. Combinatorial OptimizationMIT 18.433 · Browse materials
  23. Combinatorial Theory: Hyperplane ArrangementsMIT 18.315 · Browse materials
  24. Combinatorial Theory: Introduction to Graph Theory, Extremal and Enumerative CombinatoricsMIT 18.315 · Browse materials
  25. Commutative AlgebraMIT 18.705 · Browse materials
  26. Complex Variables with ApplicationsMIT 18.04 · Browse materials
  27. Complex Variables with ApplicationsMIT 18.04 · Browse materials
  28. Computational Science and Engineering IMIT 18.085 · Browse materials · Video
  29. Computational Science and Engineering IMIT 18.085 · Browse materials
  30. Design and Analysis of AlgorithmsMIT 6.046J · Browse materials
  31. Design and Analysis of AlgorithmsMIT 6.046J · Browse materials · Video
  32. Differential AnalysisMIT 18.155 · Browse materials

40 of 208 shown. Search the Catalog for Mathematics →

How these courses are chosen

Listed as Mathematics courses:

  • those whose department’s name includes “Mathematics”, or
  • those whose title contains “Mathematics” or “mathematical”, or
  • those tagged “Mathematics” or “mathematical” as one of at most 2 topic tags, or as the first of 3.

Free Mathematics textbooks and books

Show all 60 listed booksShow fewer books

Search all 352 Mathematics books on the Books page →

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