A Guide to the different Fields of Mathematics

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Mathematics is defined as the following: the study of quantity, structure, space, change, and anything related to patterns and form. This is a science of seeking out patterns. So far, this is something that is not limited to numbers. Mathematics is a science that can be applied to just about anything and everything. At the same time, mathematics could be considered as an art as well. There are many aspects of mathematics that are considered to be intriguing pieces of art.

With that being said of mathematics not being limited to numbers, there are many different fields of mathematics. Again, mathematics does not necessarily focus only on the numbers. This could be one of the biggest misconceptions in regards to the application of mathematics. It is something that can be applied to normal everyday life without having to resort to numbers that much.

The five major fields of mathematics are: quantity, space, change, structure, foundations & philosophy, discrete mathematics, and applied mathematics.


The study of quantity would be your basic mathematics. It would be the type that you learn in elementary and middle/junior high school.
This is the type of mathematics that revolves around numbers.

The types of mathematics that make up the study of quantity are: natural numbers, integers, rational numbers, real numbers, and complex numbers.

Natural numbers would be: "1," "2," "3," and so forth.

Integers would be: -2, -1, 0, 1, 2

Rational numbers would be a mix of negative numbers, fractions and decimals.

Real numbers would be a mix of rational numbers, irrational numbers, and so forth. It would be a combination of these studies of mathematics.

Complex numbers would be real numbers with imaginary units. A good portion of algebra revolves around the use of complex numbers.
This can be applied to addition, subtraction, multiplication, and division.


This field of mathematics deals in shapes and space. Mainly, this deals with the different shapes. The types of mathematics that make up space are: geometry, trigonometry, differential geometry, topology, and fractal geometry.

Geometry is defined as the type of mathematics that deal with sizes, shapes, positions, and so forth.

Trigonometry is an offshoot of geometry. This is a type of spatial mathematics that deals with triangles. It goes more in-depth with right triangles. If you are interested in triangles, this is the field of mathematics you go into.

Differential geometry is using calculus combined with geometry. This plays on the plane theory of spaces curves and surfaces. There is a lot that revolves around three-dimensional space.

Topology is a combination of geometry and set theory.

Fractal Geometry is the study of geometric shapes that can be split into parts.


This is a type of mathematics that takes one into the direction of the sciences. If you are going into natural sciences, you will probably take up these disciplines of mathematics.

Calculus goes into limits, derivatives, integrals, and infinite series.

Vector Calculus goes into differences and integration of vector fields.

Differential Equation is the unknown function for several variables for the function itself.

Dynamical system focuses on mathematics for any "fixed rule" in ambient space.


Structure is the field of mathematics that revolves around internal structure. It is made up of the following: number theory, abstract algebra, group theory, and order theory.

Number theory deals strictly with numbers. So far, there are many different fields of number theory: elementary, analytic, algebraic, geometry, combinational, computational, modular forms, and arithmetic algebraic geometry.

Abstract algebra studies algebraic structures.

Group Theory goes into groups of algebraic structures.


This focuses on the base of mathematics. The fields that make up this area consist of: mathematical logic, set theory, and category theory.

Mathematical Logic uses mathematics to study logic and formal logic. This can be applied to other disciplines of mathematic.

Set Theory studies a collection of objects called "sets."

Category Theory uses abstraction to deal with the relationships of mathematical structures.


This type of mathematics is mainly about computational mathematics. It consists of: combinatorics, theory of computation, cryptography, and graph theory.

Combinatronics studies discrete objects.

Theory of Computation uses model of computation to solve problems.

Cryptography is the type of mathematics that uses codes to hide information.

Graph Theory focuses on graphs.


Perhaps applied mathematics would be the type that we employ for everyday life. It consists of: mathematical physics, mathematical fluid dynamics, numerical analysis, optimization, probability theory, statistics, financial mathematics, and game theory.

Probability theory focuses on random phenomena.

Statistics focuses all about data.

Mathematical finance focuses on the financial markets.

Game Theory focuses on social sciences, political science, biology, engineering, computer science, philosophy, and so forth. Perhaps this is the one type of applied mathematics that we use almost all of the time.

Overall, these are the different fields and sub-fields of mathematics. When you look at everything, mathematics is not solely about numbers. The fields of mathematics are incredibly and diverse from one another. With all of that said, mathematics is not solely about the numbers.


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