Maths: Concepts, Formulas & Problem-Solving Techniques

Mathematics is a universal language of numbers, encompassing the study of patterns, structures, and shapes. It involves logical, analytical, and critical thinking, along with problem-solving skills. Mathematics has applications across various domains, including medicine, computer science, engineering, architecture, and finance. While many students find mathematics challenging, it can be an enjoyable subject. As the saying goes, “Practice makes perfect.” With consistent practice, anyone can find mathematics easier and more enjoyable.

History of Mathematics

The word “Mathematics” comes from the Greek word “Mathema” which means “that which is learnt” or “what one gets to know”. Ancient people used mathematics for counting objects, calculating the days and months etc. The use of modern mathematics started from the Babylonians and Egyptians who used Geometry, Algebra and Arithmetics for the financial calculations and construction of Buildings etc. Pythagorean Theorem (anciently known as ‘Pythagorean Triplets’) is the ancient mathematical concept after basic algebra and arithmetic.

The Greek Mathematician “Archimedes” is considered as the Father of Mathematics. He formulated the concept of surface area, volume of solids and most importantly using summation of numbers for calculating the area under an arc. Many Greek Mathematicians have contributed to different topics of Mathematics, namely, Apollonius of Perga in conic sections, Hipparchus of Nicaea in Trigonometry and Diophantus in Algebra. After the Greeks, it took a turn to the Islamic period (medieval period) by Persians contributing to the further development of Algebra and Spherical Trigonometry. After this, the mathematical texts in Arabic and Greek were translated into Latin making it available for the Europeans.

In the early modern period, Western Europeans started to introduce variables and logarithms which made the calculations simple. Other developments include astronomy, marine navigation and the introduction of coordinates. One of the notable developments is the development of calculus by Isaac Newton. German Mathematicians contributed to the development of various modern mathematical concepts like Differential Geometry, Algebra, Matrix Theory, Number Theory and Statistics. Mathematics is being developed even today by many mathematicians.

Why is Mathematics Important?

Mathematics plays a significant role in the existence of the things around us. Anything without Mathematics is impossible. Mathematics is one of the foundational subjects for most subjects like Physics, Chemistry, Biology, Computer Science, Economics, etc. Apart from these subjects, Mathematics plays an important role, especially in our day-to-day life. From waking up with an alarm, cooking, travelling, planning your day, and household budgets, knowing the weather, everything we see, everything around us is of Mathematics like probability, statistics, geometry, algebra, data analytics, etc. Learning Mathematics helps you improve your numerical, analytical, logical and problem-solving skills.

How to learn Mathematics?

Mathematics is not just about numbers, rules, equations and theorems. It is a puzzle to be solved. Mathematics is an interesting and enjoyable subject. It is not a subject to just read. It is about practical understanding. Don’t just drive away to advanced topics. Start by mastering the fundamentals. By regular and consistent practice, Mathematics would feel easy to learn and enjoy. We provide detailed explanations for the mathematical concepts with solved examples for better understanding.

The fundamentals of mathematics start with the basic arithmetic operations like addition(+), subtraction (-), multiplication (x) and division (/). Following these, there are some more basic and advanced topics of mathematics with explanations listed below.

Sets

A set is a collection of well-defined objects. These objects in the set are called elements. Sets are fundamentals of almost every branch of mathematics. Sets are especially used to define the relations and functions. The topics under sets are,

Relations and Functions

Relations can be defined as the relationship between two different sets of information and Functions can be defined as the relation between two different sets where every element in one set has a unique element in another. These functions and relations are used in various topics like trigonometry, algebra, calculus etc. The topics of Relations and Functions are,

Trigonometric Functions

Trigonometry is a branch of mathematics dealing with the angles to the length of the right-angled triangle. Trigonometric ratios are defined for acute angles as the ratio of the sides of a right-angled triangle. The extension of trigonometric ratios to any angle in terms of radian measure (real numbers) are called trigonometric functions. Trigonometric functions are fundamental in mathematics, particularly in geometry, calculus, and applied mathematics. Trigonometric Functions include

Complex Numbers and Quadratic Equations

The numbers that are not real are imaginary numbers. When we square an imaginary number, it gives a negative number. The complex number system is an extension of the real number system with imaginary unit i. Complex numbers are useful in representing a phenomenon that has two parts varying at the same time. It has many real-life applications like signal processing, control theory, electromagnetism, fluid dynamics, quantum mechanics, cartography, vibration analysis, etc. The topics included in Complex Numbers and Quadratic Equations are,

Permutations and Combinations

A permutation is an arrangement in a definite order of a number of objects taken some or all at a time. A combination is a process of selecting from a set. The combination is a type of permutation where the order of the selection is not considered. Combinatorics are largely used in the counting problems of Network communications, Cryptography, Network Security, Probability theory, etc. The topics under permutations and combinations are

Binomial Theorem

An algebraic expression consisting of only two terms is called a Binomial Expression. The Binomial Theorem is an important concept of algebra that helps to expand the binomial expressions. Binomial theorem is used in all branches of Mathematics and also in other Sciences. Binomial Theorem includes

Sequence and Series

A sequence is formed when terms are written in order such that they follow a particular pattern. A sequence can have any number of terms which can be finite or infinite. A series is when a sequence is added or subtracted. Sequence and Series extensively used in many branches of mathematics. Topics covered under sequence and series are

Matrices

Matrices are one of the most powerful tools in mathematics. A matrix is a rectangular array or arrangement of entries or elements displayed in rows and columns. The numbers or functions are called the elements or the entries of the matrix. The entries of a matrix may be real or complex numbers or functions of one variable or more variables or any other object. The knowledge of matrices is necessary in various branches of mathematics. The sub topics of Matrices are

Determinants

A matrix is a rectangular array or arrangement of entries or elements displayed in rows and columns. The determinant of a matrix A is a number that is calculated from the square matrix. Determinants are widely used in Algebra and some other branches of mathematics. The topics under Determinants are

Inverse Trigonometric Functions

The extension of trigonometric ratios to any angle in terms of radian measure (real numbers) are called trigonometric functions. Inverse trigonometric functions are the inverse operations of the trigonometric functions. Trigonometric functions give the value of the angles while the inverse trigonometric functions yield the angles to the respective trigonometric values. Inverse Trigonometric functions are fundamental in mathematics, particularly in geometry, calculus, and applied mathematics. Inverse Trigonometric Functions include

Limits and Derivatives

The limit of a function describes the value that a function approaches as the variable x approaches a particular value. The rate of change of a quantity y concerning another quantity x is called the derivative of y. Limits and derivatives are widely used in various domains like calculus, physics, engineering, etc. The topics under limits and derivatives are

Continuity and Differentiability

A function is said to be continuous if the graph of the given function does not have any breaks. And a function is said to be differentiable if it has a derivative. These concepts of Continuity and Differentiability have been broadly applied in branches of mathematics, physics, engineering, economics, and biology. Continuity and Differentiability includes

Integrals

The process of finding the integral is called Integration. Integration is the reverse process of differentiation. Integration is a topic under Calculus. It is one of the powerful tools in mathematics with applications in various fields like Mathematics, Physics, Engineering, Economics, etc. The sub topics of Integrals are

Three Dimensional Geometry

Three Dimensional Geometry is a branch of mathematics that deals with the geometrical properties of 3D structures. This is an important topic which has many applications in various fields like engineering, architecture, etc. Three Dimensional Geometry includes

Differential Equations

A differential equation is a mathematical equation that relates or equates a polynomial function with its derivatives. Differential equations describe the relationship between changing quantities and the rates at which these quantities change. Differential equations play a crucial role in various fields such as Mathematics, Physics, Engineering, Economics, Biology, etc. The topics under Differential Equations are

Vector Algebra

A vector is a quantity that is determined by both its magnitude and its direction and hence it is a directed line segment. The branch of Algebra about the vectors is called the Vector Algebra. Vector Algebra has many applications in fields of Physics, Engineering and Technology. The sub topics of Vector Algebra are

Mathematical Reasoning

Mathematical Reasoning is a branch of mathematics dealing with the concept of logic. Logic is the study of general patterns of reasoning, without reference to particular meaning or context. This topic mathematical reasoning is very important as it is a topic covered in many competitive examinations. The topics under Mathematical Reasoning are

Mathematical Induction

Mathematical induction is one of the methods which can be used to prove a variety of mathematical statements which are formulated in terms of n. Mathematical Induction is a fundamental mathematical technique used to prove statements or formulas that are asserted to be true for all natural numbers.

Statistics

Statistics is a science of counting. Statistics can be defined as the branch of mathematics that deals with collection, analysis and interpretation of numerical data. Statistics plays a major role in various fields like Economics, Business, Biology, Astronomy, etc. Statistics include

Probability

Probability is a branch of mathematics dealing with the possibility of an event to occur. Probability is defined as the ratio of the number of favourable outcomes to the total number of outcomes. It plays an important role in estimating the outcome or predicting the chances of that event. It has numerous applications in various fields like Statistics, Finance, Technology, etc.

The topics covered under Probability are

Straight Lines

A straight line is an infinite length between two points without any curves. Straight lines are under the topic of analytical geometry in Mathematics. Analytical Geometry has many applications. Straight lines include

Conic Section

Conic Sections are the curves obtained by the intersection of circular cones by a plane. The intersection of the plane with the cone can take place either at the vertex of the cone or at any other part of the nappe either below or above the vertex. The sub-topics of conic sections are