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What are the Branches of Mathematics?


According to Merriam Webster, “Mathematics is defined as the science of numbers and their operations, interrelations, combinations, generalizations and abstractions and of space configurations and their structure, measurement, transformations and generalizations”. Mathematics is a complex course of interlinked concepts which are extended across different overlapping topics.

In order to do extensive analysis on various topics, the general mathematics has been branched into a number of areas which are helpful in developing strong basis for conceptual understanding.

Although Algebra, Number Theory, Geometry and Arithmetic are considered as the main branches of mathematics, other branches are also explored on the basis of these which developed mathematics into a enormous and diverse course.

  1. Arithmetic

Bertrand Russell once said, “Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.” This branch of mathematics is considered to be the most ancient and elementary branch which deals with the numeric and basic operations called as “arithmetic operations” such as “addition, subtraction, multiplication and division”.

These are devised to solve the complex concepts of exponents, proportions, limits and for real life profit/loss calculations etc.

  1. Algebra

The broadest branch of mathematics is considered as an arithmetic in nature which requires unknown values along with numbers to proceed with calculations. These unknown quantities are sometimes called as variables, denoted by English alphabet or symbols (e.g., x, y, z, k etc.). The use of unknown values with the help of symbols helps in the generalization of the basic principles and specific formula as well as in finding the solutions to the problems even in the absence of one or two values with the help of generic algebraic expressions.

  1. Geometry

One of the most practical branches of mathematics is known to be Geometry which deals with shapes, sizes and volumes of various geometrical figures and their geometric properties. Th most basic elements of this branch are points, line segments, angles and dimensions etc. and majorly focus on the study of polygons, congruence of objects and various measurements of area, volume and perimeter etc.

  1. Trigonometry

Trigonometry is derived from the Greek words “trigonon” meaning “triangle” and “metron” which means “measure”. This branch studies the angles and sides of a triangle in order to measure the distance and length of sides and diagonals. It is considered as one of the most prominent branches on mathematics in the modern high-tech world which works on scientific principles. Thus, trigonometry reads the relationship among the sides and angle of different triangles in order to develop objects.

  1. Calculus

This branch of mathematics is considered to be one of the most advanced branches which deals with the study of rate of change. This field brought a revolutionary change in the approaches of early mathematics which could focus solely on static objects. However, with the advent of Calculus, the principles of mathematics were made applicable to the dynamic objects or objects in motion. It has multiple application in a range of fields with the introduction of its two strikingly different subtypes; differential and integral calculus.

  1. Probability and Statistics

Probability and Statistics is one of a few branches of mathematics with abstract features of using mathematical concepts to draw inferences about the events which are likely to happen, with the help of descriptive data. It is relatively new but indispensable branches of mathematics because of its high usability in both natural and social sciences. It is because of its involvement in studying the laws and principles which govern the numerical or theoretical data bringing interesting interpretations.

  1. Number Theory

It is said by Carl Friedrich Guess: “Mathematics is the queen of the sciences, and number theory is the queen of Mathematics”. This is also one of the oldest branches of mathematics which is named for establishing the relationship among numbers, belonging to real numbers. It basically comprises of introduction to the integer properties related to arithmetic operations and functional in building complex systems such as “cryptography”, “game theory” etc.

  1. Topology

Topology is another modern branch of mathematics which involves deformation of various geometrical figures through “stretching, twisting, crumpling and bedding”. Here, deformations do not mean cutting or tearing the figures but its applications are observed with the help of “differentiable equations, knot theory, and dynamical systems” etc.


  1. Analysis

This branch of the mathematics is the composite of other two branches of Algebra and Calculus which primarily focuses on limits, infinity series in order to analyze the classes of equations and functions with generic properties. The branch was built by Leibniz and Newton during their exploration on the applications of the concepts such as derivatives and integral.

  1. Game Theory

This branch belongs to the division of applied mathematics, devised for the analysis of situations in which interplay between the parties are observed. These parties may have similar, opposed or mixed kind of interests. The field typically presents the competition with fixed principles and game proceeds with strategic approach in which average or expected outcome is predicted for each player. Usually only one solution is designed for a given play.


  • Hence, Mathematics is a vast field which comprises of different sub-fields, thereby, categorized under the broad domains of advanced mathematics, pure mathematics and applied mathematics. Interestingly, one branch can overlap with the other divisions of mathematics.


Advanced Mathematics

Pure Mathematics

Applied Mathematics

These branches are to be studied on advance level comprising of complex concepts which need robust computation. Some of them are given below:

·       Matrix Algebra

·       Operations Research

·       Numerical Analysis

·       Cartesian Geometry

·       Calculus

·       Game theory

This division comprises of those branches which are entirely mathematical in their concepts with no concept other than the subject itself. Some of them are given:

·       Algebra

·       Number Theory

·       Geometry

·       Topology

·       Arithmetic

The branches under this division use the practical applications of other disciplines or special fields and make the composite with mathematics. Some are given below:

·       Statistics and Probability

·       Trigonometry

·       Calculus

·       Set Theory


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