Glossary of areas of mathematics

Glossary of Areas of Mathematics

This glossary provides definitions and explanations of various areas of mathematics, serving as a resource for students and educators alike. Mathematics is a vast field with numerous branches, each with its own unique focus and applications.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating those symbols. It is a unifying thread of almost all of mathematics and includes everything from solving elementary equations to studying abstractions such as groups, rings, and fields.

• Elementary Algebra: The study of basic operations and their properties, including solving equations and inequalities.
• Abstract Algebra: The study of algebraic structures such as groups, rings, and fields.
• Linear Algebra: The branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces.

Analysis

Analysis is the branch of mathematics dealing with limits and related theories, such as differentiation, integration, measure, sequences, and series. It is a major area of mathematics that has applications in many fields.

• Real Analysis: The study of real numbers and real-valued functions, focusing on limits, continuity, and integrals.
• Complex Analysis: The study of functions of a complex variable, including complex differentiation and integration.
• Functional Analysis: The study of vector spaces with limits and the linear operators acting upon these spaces.

Geometry

Geometry is the branch of mathematics concerned with the properties and relations of points, lines, surfaces, and solids. It has a rich history and is fundamental to many areas of mathematics and science.

• Euclidean Geometry: The study of plane and solid figures based on axioms and theorems employed by the Greek mathematician Euclid.
• Non-Euclidean Geometry: Any form of geometry that is based on a set of postulates that differ from those of Euclidean geometry, such as hyperbolic and elliptic geometry.
• Differential Geometry: The study of geometry using calculus and linear algebra, focusing on curves, surfaces, and manifolds.

Topology

Topology is the branch of mathematics that studies the properties of space that are preserved under continuous transformations. It is sometimes referred to as "rubber-sheet geometry."

• General Topology: The study of topological spaces and the basic concepts of continuity, compactness, and connectedness.
• Algebraic Topology: The study of topological spaces with algebraic methods, focusing on concepts such as homotopy and homology.
• Differential Topology: The field dealing with differentiable functions on differentiable manifolds.

Number Theory

Number theory is the branch of mathematics devoted to the study of the integers and more generally to objects built out of them. It is sometimes called "the queen of mathematics" because of its foundational place in the discipline.

• Elementary Number Theory: The study of integers, divisibility, and the properties of prime numbers.
• Analytic Number Theory: The study of number theory using methods from mathematical analysis.
• Algebraic Number Theory: The study of the algebraic structures related to algebraic integers.

Probability and Statistics

Probability and statistics are branches of mathematics that deal with the analysis of random phenomena and data.

• Probability Theory: The branch of mathematics concerned with the analysis of random phenomena and the likelihood of different outcomes.
• Statistics: The study of the collection, analysis, interpretation, presentation, and organization of data.

Discrete Mathematics

Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. It includes topics such as graph theory, combinatorics, and logic.

• Graph Theory: The study of graphs, which are mathematical structures used to model pairwise relations between objects.
• Combinatorics: The branch of mathematics dealing with combinations of objects belonging to a finite set in accordance with certain constraints.
• Logic: The study of reasoning and the principles of valid inference and demonstration.

Applied Mathematics

Applied mathematics involves mathematical methods used in practical applications in science, engineering, business, and industry.

• Mathematical Physics: The application of mathematics to problems in physics and the development of mathematical methods suitable for such applications.
• Computational Mathematics: The study of algorithms and numerical methods for solving mathematical problems.
• Operations Research: The application of mathematical methods to decision-making, optimization, and the efficient allocation of resources.

See Also

• List of mathematical topics
• History of mathematics
• Mathematical notation