“Mathematics is a language God has written the universe in.” - Galileo Galilei.
Mathematics can be defined as a science involving numbers, space and quantities either as abstract concepts or as applied to other subjects like engineering and physics. It has several branches, like arithmetic, algebra, geometry and number theory. The foundational branches have paved the way for discovering others within the field. Though the study of mathematics was limited in the past, it gradually expanded into a broad subject as time passed by. The developments in mathematics have greatly influenced today’s technology, earning its title of the Queen of Science. Throughout history, mathematics has evolved from its humble beginnings with number systems to contemporary research areas like computational sciences and probability. The expansion of the subject’s scope and utility led to the need for classifying its various branches.
Branches of Mathematics
There are two main branches of mathematics- 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!
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:
- Algebra
- Number Theory
- Geometry
- Arithmetic
- Topology
- Combinatorics
- Mathematical Analysis
Applied Mathematics
This branch of mathematics combines mathematical concepts with specialised knowledge from various disciplines to solve practical problems. Here are the branches of applied mathematics:
- Statistics and Probability
- Set Theory
- Calculus
- Trigonometry
Main branches of mathematics
- Geometry
- Algebra
- Arithmetic
- Calculus
- Topology
- Trigonometry
- Number theory
- 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 as a discipline includes the study of angles, shapes, sizes, figures, or any geometrical object in both the 2D and 3D dimensions. The functioning of points, lines, line segments, angles, coordinate planes, types of surfaces, shapes and other solids are some of the primary terms linked with the subject. A few great examples of the practical application of geometry are seen in GPS for providing directions for navigation, planning the layout of buildings, creating animations etc. Thus, Geometry is also known as the most practical branch of mathematics!
2. Algebra
“An ounce of Algebra is worth of a ton of verbal argument” - John B. S. Haldane.
The branch of mathematics studying symbols, numbers, variables and the rules for controlling it is known as Algebra. This branch presents problems or situations in mathematical expressions. Algebra comprises variables like x,y, and z and expressions such as addition, subtraction, multiplication or division. It helps derive unknown quantities such as interest, proportions, percentages, etc and solves mathematical questions. Algebra is used to find out 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. It has many practical applications in various fields of study, such as financial planning, designing structures, creating software solutions, analysing trends, protecting sensitive information etc. Hence, Algebra 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 the branch of mathematics which consists of numbers and various operations, including addition, subtraction, multiplication and division, alongside 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 most advanced and hardest branches of mathematics and 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.
It is utilised for solving complex shape areas, evaluating survey data, ensuring vehicle safety and discovering the fluctuating conditions that affect systems, among other uses. Calculus 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 is a recently developed branch of mathematics which examines the features of spaces that stay consistent despite undergoing continuous deformations. It involves the stretching, crumpling, twisting and bedding of different geometrical shapes and is therefore also known as 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 a 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 ancient Egypt and Mesopotamia, it 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. It 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 pure 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 is often overlooked among the various branches of mathematics, even though it holds significant importance. It is an abstract discipline which uses mathematical concepts to predict events that are likely to happen in the future and organises, analyses and interprets a collection of data. Probability and Statistics also involve a set of rules and formulas that controls numerical data and random events to predict ahead. It has applications in both scientific and social sciences and has thus grown more indispensable today. It is 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 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. Given below is a list of the top 10 universities offering top-notch mathematics programs according to the QS World University Rankings by Subject 2023: Mathematics
10 Best Careers in Mathematics
If you are fascinated by numbers 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 mathematics better and clear up any fear or confusion in your mind. Even though it may seem hard to figure out the different branches of math 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!