Branches of Mathematics: Arithmetic, Algebra, Geometry, Calculus, Trigonometry, Topology, Probability and Statistics

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branches of mathematics

Shakuntala Devi rightly said that without mathematics, there is nothing you can do. To explore more, let’s first know the 5 main branches of mathematics, i.e. Algebra, Number Theory, Arithmetic and Geometry. In the past 2 decades or so, our modern world has introduced more branches like Probability and Statistics, Topology, Matrix Algebra, Game Theory, and Operations Research derived from these oldest branches of math. Breaking the myth of mathematics being called boring, we bring to you all about branches of mathematics with their list, tree and career options!

A Tree of Branches of Mathematics

Getting right to it, this is how the branches of the mathematics tree look like

Main Branches of Mathematics

Mathematics is a complex area of study and comprises interlinked topics and overlapping concepts. Additionally, extensive analysis of the branches of mathematics helps students in organizing their concepts clearly and develop a strong foundation. Being aware of the differences and uniqueness of branches of mathematics helps in methodically studying various concepts of maths and also guides students in deciding the branch they would like to pursue as a career.

Here are the main branches of mathematics:

  1. Foundations
  2. Arithmetic
  3. Algebra
  4. Geometry
  5. Trigonometry
  6. Calculus
  7. Probability and Statistics
  8. Number Theory 
  9. Topology
YT Source: Homework Minutes

Below explained are the important branches of mathematics with their unique features and distinct concepts. These branches are important for laying a strong foundation for mathematics. 

Fun Fact: -40 °C is equal to -40 °F.


“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.”

– Bertrand Russell

Arithmetic is one of the most basic branches of mathematics and deals with numbers and their applications in many ways. Addition, subtraction, multiplication, and division are used as the basic groundwork to solve a large number of questions and progress into more complex concepts like exponents, limits, and many other types of calculations. This is one of the most important branches because its fundamentals are used in everyday life for a variety of reasons from simple calculations to profit and loss computation.

Source: mathantics

Also Read: Word Problems on Arithmetic Operations


“The algebraic sum of all the transformations occurring in a cyclical process can only be positive, or, as an extreme case, equal to nothing.”

– Rudolf Clausius

In a broad field of mathematics, algebra deals with solving generic algebraic expressions and manipulating them to arrive at results. Unknown quantities denoted by alphabets that form a part of an equation are solved for and the value of the variable is determined. A fascinating branch of mathematics involves complicated solutions and formulas to derive answers to the problems posed. However, mastering algebra can often pose challenges, especially for beginners when dealing with a fascinating branch of mathematics involving complicated solutions and formulas to derive answers to the problems posed. In such cases, seeking guidance from the best algebra tutor becomes indispensable.

Fun Fact: The word “mathematics” only appears in one Shakespearean play, “The Taming of the Shrew”.


“The description of right lines and circles, upon which geometry is founded, belongs to mechanics. Geometry does not teach us to draw these lines, but requires them to be drawn.”

-Isaac Newton

Do you often wonder about the shapes and sizes of various objects? Then geometry is the branch you must explore. Dealing with the shapes, sizes, and volumes of figures, geometry is a practical branch of mathematics that focuses on the study of polygons, shapes, and geometric objects in both two-dimensions and three-dimensions. The congruence of objects is studied at the same time focusing on their special properties and calculation of their area, volume, and perimeter. The importance of geometry lies in its actual usage while creating objects in practical life.


Derived from the Greek words “trigonon” meaning triangle and “metron” meaning “measure”, trigonometry focuses on studying the angles and sides of triangles to measure the distance and length. Amongst the prominent branches of mathematics used in the world of technology and science to develop objects, trigonometry is a study of the correlation between the angles and sides of a triangle. It is all about different triangles and their properties!

Also Read: Trigonometry Formulas


“Calculus is the most powerful weapon of thought yet devised by the wit of man.”

– Wallace B. Smith

It is one of the advanced branches of mathematics and studies the rate of change. With the advent of calculus, a revolutionary change was brought about in the study of maths. Earlier, maths could only work on static objects but with calculus, mathematical principles began to be applied to objects in motion. Used in a multitude of fields, the branch can be further categorized into differential and integral calculus both starkly different from each other. A branch with mind-numbing questions, calculus is an interesting concept introduced to students at a later stage of their study in mathematics.

Fun Fact: “Forty” is the only number that is spelled with letters arranged in alphabetical order.

Probability and Statistics

The abstract branch of mathematics, probability and statistics use mathematical concepts to predict events that are likely to happen and organize, analyze, and interpret a collection of data. Amongst the relatively newer branches of mathematics, it has become indispensable because of its use in both natural and social sciences. The scope of this branch involves studying the laws and principles governing numerical data and random events. Presenting an interesting study, statistics, and probability is a branch full of surprises.

Number Theory 

“Mathematics is the queen of the sciences, and number theory is the queen of mathematics.”

–  Carl Friedrich Gauss

As the name suggests, Number theory is one of the oldest branches of Mathematics that established a relationship between numbers belonging to the set of real numbers. The basic level of Number Theory includes an introduction to properties of integers like addition, subtraction, multiplication, and modulus and builds up to complex systems like cryptography, game theory and more. 


“The basic ideas and simplest facts of set-theoretic topology are needed in the most diverse areas of mathematics; the concepts of topological and metric spaces, of compactness, the properties of continuous functions and the like are often indispensable.”

–  Pavel Sergeevich Aleksandrov

Topology is a much recent addition to the branches of Mathematics list. It is concerned with the deformations in different geometrical shapes under stretching, crumpling, twisting and bedding. Deformations like cutting and tearing are not included in topologies. Its application can be observed in differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis.

Fun Fact: Every odd number has an “e” in it.

Advanced Branches of Mathematics

There are a vast number of advanced branches which are a part of the major branches mentioned above. These branches are studied at an advanced level and involve complex concepts that need strong computational skills.

Here are the 10 branches of mathematics:

  1. Matrix Algebra
  2. Numerical Analysis
  3. Operations Research
  4. Complex Numbers
  5. Calculus
  6. Set Theory
  7. Game Theory
  8. Analysis
  9. Cartesian Geometry
  10. Combinatorics

Branches of Pure Mathematics

Pure mathematics can be simply described as the study of mathematical concepts purely based on maths and independent of any concept outside of mathematics. Here is the list of branches of Pure Mathematics:

  • Algebra
  • Number Theory
  • Geometry
  • Arithmetic
  • Topology
  • Combinatorics
  • Mathematical Analysis

Branches of Applied Mathematics

Applied Mathematics uses the applications of varied other disciplines and combines it with mathematical concepts. It is simply the combined application of mathematics with specialised knowledge.

Here are the branches of applied mathematics:

  • Statistics and Probability
  • Set Theory
  • Calculus
  • Trigonometry

Fun Fact: Archimedes [287 – 212 BC] is the father of Mathematics

Top Global Universities for Courses in Mathematics

For aspirants wishing to establish a career in mathematics, the first step is to pursue a course in your chosen branch of Mathematics. Many renowned universities of the world offer lauded degrees in mathematics and its varied branches. Being aware of the major picks available to one helps in reducing the dilemma that comes with choosing the university. Tabulated below are some of the top-class universities offering esteemed mathematics programs. 

Name of UniversityCountry
Imperial College LondonEngland
University of DundeeScotland
Jacobs University BremenGermany
University of PaviaItaly
Carleton UniversityCanada
Southern Utah UniversityUnited States
Cardiff UniversityWales
University of AlmeriaSpain
Clarkson UniversityUnited States
East-West UniversityUnited States

Top Indian Colleges for Mathematics

  1. Fergusson College
  2. Mount Carmel College
  3. Stella Maris College
  4. St. Xavier’s College
  5. Jadavpur University
  6. St.Stephen’s College
  7. Ramakrishna Mission Vivekananda College
  8. Christ University

Career Options in Mathematics

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If the number game attracts you and you can’t help but add or multiply any number you see, then you should pick up Mathematics as a career. You can become one of the following professionals if you choose a career in Mathematics:-

  • Teacher/Lecturer– You can teach mathematics at the high school level and even university level if you are qualified enough and even explore research via postgraduation.
  • Statistician- A statistician collects data, turns them into charts and graphs and makes interpretations which is an important aspect in all professions. 
  • Data Analyst/Scientist– A data analyst/scientist is similar to a statistician but with certain differences. In Data Science, you will use scientific methods of data collection, and comparison and give solutions to the organisation relatively.
  • Mathematician– A mathematician works in all fields of the profession. Whether it be inventing new business valuation methods or creating a formula that sends a rocket to the ISS, a mathematician does it all.
  • Cryptographer– A cryptographer is a person who encodes and decodes various symbols or modes of communication to decipher what it means. Additionally, they also create safe codes for the military or other organisations. 

Every profession needs the Branches of Mathematics as the subject ties all the broken pieces and creates uniformity.

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1. What are the four branches of Mathematics?

Algebra, Geometry, Calculus and Statistics & Probability are considered to be the 4 main branches of Mathematics.

2. What is the hardest branch of Maths?

Algebra is the hardest branch of Maths. Abstract algebra particularly is the most difficult portion as it includes complex and infinite spaces

3. .Who is the father of Mathematics?

Archimedes [287 – 212 BC] is the father of Mathematics

4. Who invented Math?

It is difficult to trace back the exact origins of Maths. But it is estimated that in the early 6th century, Pythagoreans invented Maths. After that Euclid introduced the axiomatic method consisting of the definition, axiom, theorem, and proof.

Are you intrigued by numbers and have a flair for doing difficult numeric calculations? Then, Mathematics is an all-encompassing science that deals with quantity, space and numbers and abstract concepts as well as applied maths in other fields. While at first glance it might be difficult to identify the branches of mathematics, it is imperative to possess knowledge about them for any student wishing to gain a holistic idea about the subject.

We hope this blog has presented a comprehensive list of different areas of study and branches of mathematics. Planning to study a degree in Mathematics? Study Abroad Experts at Leverage Edu can help you find the best course and university as well as assist you throughout the admission process to ensure that you send a winning application! Sign up for a free session today!

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