• Set theory: mathematical theories of sets, in varying consistency strengths.
  • Category theory: algebraic properties of collections of transformations between mathematical objects of the same type.

Pure mathematics:

  • Number theory: properties of integers, especially primes and prime factorization.
  • Algebra: numerical quantities and attempts to solve equations.
    • Abstract algebra;
  • Geometry: figures, objects, and their relationships to each other.
    • Differential geometry: distance and curvature on surfaces and manifolds
  • Topology: invariant property of objects under continuous transformations.
  • Analysis: real- and complex-valued continuous functions.
  • Dynamical systems: description of how a complex system changes over time.
  • Discrete math: objects that can assume only distinct, separated values.
    • Graph;
    • Combinatorics;

Applied mathematics:

  • Probability and statistics;
  • Computation: tasks that are theoretically possible with computing machines; the relative difficulty and complexity of these tasks.
  • Optimization and Game theory;


Table: Development of Common Mathematical Structures

Structure Origin/motivation New Operations Feature Abstraction
\( \mathbb{N}\) Counting \( +, * \)
\( \mathbb{Z}\) Closed inversion of \(+\) \( - \)
\( \mathbb{Q}\) Closed inversion of \( * \) \( / \) Polynomial (rational function)
\( \mathbb{R}\) Closure/completeness ^ Analysis metric, topology
\( \mathbb{C}\) root of negative numbers Trigonometrics, Fourier analysis
\( \mathbb{F}^n\) Product space \( (\cdot,\cdot) \) inner product, norm
\( \mathbb{F}^{\infty}\) Discrete process
\( L_p \) Approximation of functions

MathWorld Classroom

🏷 Category=Topics