What are the Different Branches of Mathematics?

“Mathematics is a language God has written the universe in.” - Galileo Galilei.

Ever wondered how that perfect pizza gets sliced into equal portions? Or why does your phone battery seem to drain faster when you're playing games mysteriously? Believe it or not, the answers lie in the magical world of math! Math isn't just about numbers and equations - it's the secret sauce behind everything from baking to building bridges. So, grab your thinking cap and get ready to explore the different branches of mathematics that make our world tick (and those pizza slices perfectly symmetrical)!

What are the Branches of Mathematics?

Whether you love pure maths or use it to delve into other fields and branches like philosophy or medicine, a maths degree offers various career paths where your abilities are greatly appreciated.

There are two main branches of maths- pure and applied. Pure mathematics explores the abstract concepts and complex patterns that shape our reality, while Applied mathematics provides solutions to real-world problems. Now, let’s get ready to dig deeper into the intricacies of these branches!

1. Pure Mathematics

Pure mathematics is the study of mathematical concepts in their own right, solely focused on math itself and separate from external influences. Here is the list of branches of Pure Mathematics: 

1. Algebra

2. Number Theory

3. Geometry

4. Arithmetic

5. Topology

6. Combinatorics

7. Mathematical Analysis

2. Applied Mathematics

This is one of the branches of mathematics that combines mathematical concepts with specialized knowledge from various disciplines to solve practical problems. Here are the branches of applied mathematics: 

1. Statistics and Probability

2. Set Theory

3. Calculus

4. Trigonometry

Main Branches of Mathematics

Mathematics is like a superhero with different powers! You've got arithmetic to handle everyday tasks like shopping or splitting the bill with friends. Here are the main branches of mathematics:

1. Geometry

2. Algebra

3. Arithmetic

4. Calculus

5. Topology

6. Trigonometry

7. Number theory

8. Probability and Statistics

1. Geometry

“There is geometry in the humming of the strings, there is music in the spacing of the spheres.” - Pythagoras.

Concerned with the forms and sizes of objects, their relative positions and qualities of space, Geometry is a practical branch of pure mathematics. The word geometry is derived from the two ancient Greek words- ‘geo’ meaning earth, and ‘metric’ meaning measurement. Euclid, an esteemed mathematician from Greece, is widely regarded as the founder of geometry. 

Geometry is the branch of mathematics that includes the study of angles, shapes, sizes, figures, or any geometrical object in both the 2D and 3D dimensions. Some of the primary terms linked with the subject are the functioning of points, lines, line segments, angles, coordinate planes, types of surfaces, shapes, and other solids. 

A few great examples of the practical application of geometry are seen in GPS, which provides directions for navigation, plans the layout of buildings, creates animations, etc. Thus, Geometry is also known as the most practical branch of mathematics!

2. Algebra

“An ounce of Algebra is worth a ton of verbal argument” - John B. S. Haldane.

Algebra, a branch of mathematics, is not just about symbols, numbers, variables, and the rules for controlling. It's about presenting problems or situations in mathematical expressions, bringing clarity and precision to mathematical understanding.

Algebra comprises variables like x,y & z and expressions such as addition, subtraction, multiplication, or division. It helps derive unknown quantities such as interest, proportions, and percentages and solves mathematical questions. Algebra is used to find the unknown value in algebraic equations and includes other rules of operations. 

Some primary terms used in Algebra are variables, constants, coefficients, expressions, and equations. Algebra has many practical applications in various fields of study, such as financial planning, designing structures, creating software solutions, analyzing trends, protecting sensitive information, etc. Hence, It is considered one of the most essential branches of mathematics.

3. Arithmetic

“The hardest arithmetic to master is that which enables us to count our blessings” - Eric Hoffer.

Arithmetic is one of the hardest branches of mathematics. It consists of numbers and various operations, including addition, subtraction, multiplication, and division. It also includes advanced computations like percentages, logarithms, exponentiation, and square roots. 

The Greek word ‘arithmos’, meaning numbers, is the root of the word ‘arithmetic’. As stated in the ‘fundamental theorem of arithmetic’ proposed by Carl Freidrich (1801), any integer greater than 1 has a distinctive prime factorisation excluding the factor orders. 

It also incorporates higher-level mathematical ideas, including limits, exponents, and similar concepts. Arithmetics is considered the most essential and easiest branch of mathematics since it’s used in our day-to-day life activities.

4. Calculus

“Science is the Differential Calculus of the mind. Art the Integral Calculus; they may be beautiful when apart, but are greatest only when combined” - Ronald Ross.

Calculus is recognised as one of the branches of applied mathematics that displays distinct tiers such as pre-calculus, advanced calculus, accelerated multivariable calculus, differential calculus, integral calculus, integral calculus and beyond. Calculus is the mathematical study of continuous change and was therefore originally called ‘infinitesimal calculus’. 

Calculus was revolutionary because earlier math could only work on static objects; however, with this new branch of mathematics, mathematical principles began to be applied to objects in motion as well. Limits, functions, integrals and infinite series are the four main fundamental concepts in Calculus. 

Calculus is utilized for solving complex shape areas, evaluating survey data, ensuring vehicle safety, and discovering the fluctuating conditions that affect systems, among other uses. It is a language for economists, architects, medical experts, and statisticians alike. It is among the most fundamental branches of pure mathematics and finds extensive applications across various fields in both applied and pure mathematics. 

5. Topology

“Topology is precisely the mathematical discipline that allows the passage from local to global.” - Rene Thom.

Topology, a unique and relatively new branch of mathematics, focuses on the properties of spaces that remain unchanged under continuous deformations. Unlike other branches, it deals with the stretching, crumpling, twisting, and bending of various geometric shapes, earning it the nickname 'rubber-sheet geometry '. 

For example, a rectangle can be shaped into a circle without breaking it, but Figure X cannot. So, a rectangle is topologically equivalent to a circle; however, different from Figure X. Deformations like tearing and cutting are not included in topology. 

Within the field of topology, subfields such as general topology, combinatorial topology, algebraic topology and differential topology exist. Topology is used in numerous branches of mathematics, namely, differential equations, knot theory, dynamical systems and Reimann surfaces in complex analysis.

6. Trigonometry

“I was trying to unravel the complicated trigonometry of the radical thought that silence could make up the greatest lie ever told” - Pat Conroy.

Trigonometry is the branch of mathematics that deals with the study of angles and sides of triangles to measure the distance and length. Derived from the two Greek words “trigonon”, meaning triangle, and “metron”, meaning “measure”, reflecting its origins in the science of triangles and their measurements. 

In ancient civilisations of Egypt and Mesopotamia, trigonometry was used for practical purposes like land surveying and astronomy. Trigonometric functions, including sine, cosine, tangent, cotangent, secant, and cosecant, indicate the relationship between the triangle’s sides and angles. 

Trigonometry has applications in various fields, such as astronomy, architecture, physics, computer graphics, navigation, engineering, etc. This branch of mathematics models and solves problems related to angles, distances, periodic phenomena and waves.

7. Number Theory

“Numbers constitute the only universal language” - Nathanael West.

Number theory is one of the oldest branches of mathematics and is concerned with studying natural numbers and integers. Number theory is the study of the set of positive whole numbers which are usually known as the set of natural numbers. The focus of number theory is examining the set of natural numbers, which refers to positive whole numbers. 

This branch of mathematics is also known as ‘The Queen of Mathematics’ as it holds a foundational place in the discipline. Number theory forms a relationship between a set of real numbers, integers, whole numbers, and natural numbers. 

It investigates the properties of numbers, containing operations like addition, subtraction, multiplication and modulus, which also extends to associative and dissociative properties. The concepts of encryption, binary, cryptography, game theory etc., are also dealt with in the discipline. 

8. Probability and Statistics

“Probability theory is nothing but common sense reduced to calculation” - Pierre-Simon Laplace.

Probability and Statistics are the easiest branches of mathematics, and they are often overlooked among the various branches of mathematics, even though they hold significant importance. Mathematics is an abstract discipline that uses mathematical concepts to predict events that are likely to happen in the future and organizes, analyses, and interprets a collection of data. 

Probability and Statistics also involve a set of rules and formulas that control numerical data and random events to predict the future. They have applications in both the scientific and social sciences and have thus become more indispensable today. They are used in various fields, such as risk management, medical research, quality control, data analysis, weather forecasting, etc. 

Top Universities for Courses in Mathematics

Many renowned universities of the world offer lauded degrees in different types of mathematics. Being aware of the major picks available to one helps in reducing the dilemma that comes with choosing the university. Given below is a list of the top 10 universities offering top-notch mathematics branches programs according to the QS World University Rankings by Subject 2024: Mathematics

10 Best Careers in Mathematics

If you are fascinated by different branches of mathematics and find yourself solving problems in your mind, then a career in Mathematics is the right choice for you. Consider pursuing the following careers to make a living crunching numbers:

Are you intrigued by the different branches of mathematics? This blog will help you understand branches of math better and clear up any fear or confusion in your mind. Even though it may seem hard to figure out all branches of mathematics at first, students need to know about them to get a complete picture of the subject. Ready to dive in? Grab your pen and paper, and let’s jump into solving your first math problem!