Formal sciences – branches of knowledge that are concerned with formal systems. Unlike other sciences, the formal sciences are not concerned with the validity of theories based on observations in the real world, but instead with the properties of formal systems based on definitions and rules.
Arithmetic – oldest and most elementary branch of mathematics, involving the study of quantity, especially as the result of combining numbers. The simplest arithmetical operations include addition, subtraction, multiplication and division.
Algebra – branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures.
Analysis/Calculus – branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.
Category theory – branch of mathematics examining the properties of mathematical structures in terms of collections of objects and arrows
Discrete mathematics – study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic – do not vary smoothly in this way, but have distinct, separated values.
Combinatorics – branch of mathematics concerning the study of finite or countable discrete structures.
Geometry – one of the oldest branches of mathematics, it is concerned with questions of shape, size, relative position of figures, and the properties of space.
Topology – developed from geometry, it looks at those properties that do not change even when the figures are deformed by stretching and bending, like dimension.
Trigonometry – branch of mathematics that studies triangles and the relationships between their sides and the angles between these sides. Trigonometry defines the trigonometric functions, which describe those relationships and have applicability to cyclical phenomena, such as waves.
Logic – formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science.
Other mathematical sciences – academic disciplines that are primarily mathematical in nature but may not be universally considered subfields of mathematics proper.
Statistics – study of the collection, organization, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments.
Regression analysis – techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. More specifically, regression analysis helps one understand how the typical value of the dependent variable changes when any one of the independent variables is varied, while the other independent variables are held fixed.
Probability – way of expressing knowledge or belief that an event will occur or has occurred. The concept has an exact mathematical meaning in probability theory, which is used extensively in such areas of study as mathematics, statistics, finance, gambling, science, artificial intelligence/machine learning and philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems.
Theoretical computer science – a division or subset of general computer science and mathematics that focuses on more abstract or mathematical aspects of computing and includes the theory of computation.