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**Mathematics** is the most significant of all existing branches of science, and no other branch of pure science be it physics, **chemistry**, or biology, can be studied without an elemental knowledge of mathematics. It is one of the fundamental subjects introduced to kids at school, and throughout their academic career, the knowledge of mathematics is built upon. To put it in most simplistic terms, mathematics deals with numbers and it attempts to explain the logics that work behind quantity, shape, size, change, structure, and arrangement. It would be quite impossible to comprehend the world around us without mathematics, just as it would be impossible to communicate without some form of language. Mathematics help in counting, measurement, calculation, and in understanding such concepts as change and motion. To understand the pivotal role that mathematics has played in the development of human civilization, one needs to focus on the history of mathematics.

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Modern mathematics is generally categorized into **Pure Mathematics** and **Applied Mathematics**. Pure mathematics is defined as the study of mathematics for its own sake, without any consideration about the application of mathematics in the outside world. The knowledge of pure mathematics is nurtured for the sake of knowledge itself, even when the research might be inspired by real life concerns. The theories or principles thus developed might eventually find practical application, but such applications are not the leading stimulus behind the research. Research in pure math attempts to solve mathematical problems logically and to arrive at scientific solutions. Pure mathematics might study abstract concepts as well.

Applied mathematics, however, is the application of mathematics to solve practical problems which arise in different fields of sciences such as physics, medicine, engineering, computer sciences, biology, business, industries, and even in the fields of social sciences. Probability, **approximation theory**, differential equation, are only some of the examples of applied mathematics. The very purpose of applied mathematics is to develop mathematical models that can provide solutions for practical mathematical problems. Apart from providing solution to existing questions, applied mathematics quests for new problems and questions which can be solved to smoothen the practical procedures. Such use of applied mathematics in the fields of engineering cannot be emphasized enough. An applied mathematician would be required to possess the knowledge of math along with different sciences, and some of the most explicit uses of applied mathematics in the present world in found in aerospace engineering, civil engineering, finance, control theory, mathematical physics, mathematical biology, and so on.

Here, it is pertinent to highlight that pure mathematics and applied mathematics are not mutually exclusive, but they are generally affected by one another. Researches in pure mathematics are inspired by practical needs, and while in pure math the focus is on advancing knowledge, the knowledge thus derived can be used to solve practical problems where it becomes applied mathematics.

**Arithmetic** studies numbers and basic numerical operations such as addition, subtraction, multiplication, and division. The word ‘arithmetic’ originates from the Greek word ‘arithmos’ which means numbers. This is usually the very first branch of mathematics that learners are introduced to since it helps them to classify numbers into basic categories and perform simple numerical operations which are associative and commutative, while allowing to find sums, products, quotients, and differences. Some of the concepts and calculations that fall under the category of basic arithmetic are natural numbers, odd and even numbers, fractions, decimals, percentages, integers, whole numbers & variables.

In **algebra**, symbols are introduced, and these symbols work together with numbers to facilitate calculations. The symbols presently in use are Greek and Latin alphabets, and they are used to represent variables, that is, quantities for which fixed values are not known, and whose fixed values are to be estimated by solving the mathematical equations. Algebra introduces a number of formulas which helps in the rapid calculation of said variables and play a fundamental role in solving problems. These formulas are rules which dictate how the variables are to be treated. Algebra is usually divided into elementary algebra and abstract algebra. Learners study elementary algebra in school and the same finds application in various sciences, engineering, economics and study of medicine. Abstract algebra is, however, more advanced and is studied only by professional mathematicians.

The term **geometry** is the amalgamation of two Greek words, which are ‘geo’ meaning earth and ‘metron’ meaning measurement. Remembering the name itself helps to understand the basic function of geometry, which is to measure space. Geometry is concerned with the shape, size, position, and properties of space. It introduces the learners to various shapes and proposes scientifically proven methods to carry out calculations with respect to those shapes, including the estimation area, diameter, angles, volumes, depth, length, breath, relative position, and so on. Positioning a point at a specific location on a graph, identifying different types of lines and their relations, finding the angles of different shapes and deriving related values from the same are some of the elemental lessons in geometry. Notably, the knowledge of geometry is consequently used in advanced areas of study including astronomy itself. For instance, It would be difficult to understand the shape of earth, earth’s rotation on its axis and revolution around the sun without geometry; defining the trajectory of rockets carrying astronauts to the moon would be impossible without geometry; all existing speculations and theories derived about time and space would also not be possible without geometry.

This branch of mathematics is particularly concerned with the shape of triangle. It deals with the measurement of angles and different sides of a triangle, particularly right-angled triangles. **Trigonometry** can be useful in finding the missing side of a triangle by calculating the available ratios. Sine, Cosine, and Tangent are three significant functions in trigonometry and measurements are carried out in terms of degrees or radians. The use of trigonometry is ubiquitous in architecture, civil engineering, and mechanical engineering.

**Calculus** measures continuous change and two main divisions of calculus are integral calculus and differential calculus. Differential calculus breaks down the given equation into smaller pieces to find how it changes during the course of the breakdown. Integral calculus joins the small pieces to show how they add up over the course of integration. Calculus can be used to plot graphs based on specific functions and formulas, and this has a number of practical applications, especially in association with other sciences such as physics for instance. Use of calculus helps in understanding concepts of motion, dynamics, light, heat, electricity, and astronomy among others. Complex theories in physics such as relativity and electromagnetism are based on calculus. In chemistry, it helps to measure the rates of radioactive decay and reaction rates. In biology, it helps in estimation of birth and mortality rates. It also has applications in fields of economics, statistics, and other branches of mathematics like algebra.

**Statistics and Probability** is the branch of mathematics that performs two different jobs. Statistics is concerned with the collection, arrangement, description, and interpretation of numerical data. Probability, on the other hand, predicts the chances of the occurrence of events, basically focussing on random events. Probability is therefore used to make prediction about future based on mathematical chance, and statistics is closely related to it because statistics analyses available data to try and figure existing patterns which could be used to make predictions for future events. Both statistics and probability find application in the diverse fields including but not limited to the study of weather, genetics, sociological research, political research, economic research, medical research, and so on.

**Number theory** is a subdivision of pure mathematics which is exclusively concerned with the study of natural/ whole numbers and integers. Arithmetic is regarded as the elemental version of number theory, which is an advanced area of study. Number theory examines and establishes the relationships between various sets of numbers, and it is partly theoretical, partly experimental. Number theory is called the ‘Queen of Mathematics’ because it sets down the rules and relations followed by numbers, which are observed in all other branches of Mathematics.

This branch of mathematical logic concerns itself with the sets or groups of variables that seem to belong together. The sets are usually well-defined collections, and objects contained within the sets are called elements/ members of the set. The sets in question can be finite or infinite. Some of the elemental concepts of **set theory** are union, intersection, set difference, symmetric difference, cartesian product, and power set. It is another sub-division of pure mathematics that is explored.

**Order theory** studies and elaborates the intuitive concept of order by placing numbers and figures into hierarchical orders where their positions and values are fixed. The three basic concepts used in order theory are ‘greater than,’ ‘lesser than,’ and ‘equal to.’ To introduce order among the values, the order theory defines that certain values precede or succeed others. Like Mathematics, order theory is used in many other scientific fields, most notably in computer science. The initial introduction to order theory is provided through natural numbers where one number is defined as greater than the other (such as 6 is greater than 5, and 6 succeeds 5 in traditional arrangement of numbers). The concept of this ‘order’ can then be extended to real numbers, integers, and to abstract concepts as well.

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