Maths: Concepts, Mathematics Formulas & Problem-Solving Techniques

History of Mathematics: Origins and Evolution

Mathematics, often called the language of numbers, has a rich and fascinating history that spans thousands of years. The word Mathematics comes from the Greek word “Mathema”, meaning “knowledge gained” or “that which is learned.” From ancient times, mathematical concepts were used for counting, calendar calculations, trade, land measurement, and construction planning. The history of mathematics shows how basic arithmetic slowly evolved into the advanced maths we study today.

Ancient Mathematics and Early Civilizations

The foundations of modern mathematics were laid by ancient civilizations like the Babylonians and Egyptians. They used arithmetic, algebra, and geometry for financial records, building monuments, and solving real-life problems. One of the earliest and most famous math formulas, the Pythagorean Theorem, emerged during this time, becoming a cornerstone of geometry and problem-solving techniques in mathematics.

Key Contributions in Ancient Mathematics

Given below are the contributions of the famous mathematicians:

Archimedes

Known as the Father of Mathematics, he developed methods to calculate surface area, volume of solids, and summation of series, shaping many modern mathematical concepts.

Apollonius of Perga

He introduced conic sections, which play a vital role in geometry and coordinate systems.

Hipparchus of Nicaea

A pioneer of trigonometry, his work became essential for navigation, astronomy, and trigonometric formulas.

Diophantus

Called the Father of Algebra, his contributions formed the base of algebraic equations and problem-solving strategies.

Mathematics During the Islamic Golden Age

Persian and Arab mathematicians greatly advanced algebra, trigonometry, and number systems. Their work was later translated into Latin, influencing European mathematics and helping shape the future of modern algebra and calculus.

Mathematics in the Modern Era

We have given below the evolution and technological enhancements in the modern time:

Development of New Math Formulas

The introduction of variables, logarithms, and coordinate geometry made complex calculations easier and more systematic.

Role of Isaac Newton

Newton’s discovery of calculus transformed mathematics and became a foundation for physics, engineering, and scientific research.

German Contributions to Modern Mathematics

German mathematicians expanded fields like differential geometry, matrix theory, number theory, algebra, and statistics, strengthening the structure of modern maths.

Evolution of Mathematics Today

Mathematics continues to grow with new problem-solving techniques, formulas, and applications in education, science, artificial intelligence, and technology. The history of mathematics proves that it is not just a subject, but a powerful tool that shapes the modern world.

Unit	Chapter	Topics
Algebra & Basics	Sets	Universal Set, Roster & Set Builder Form, Union of Sets, Intersection, Subsets, Complement, De Morgan's Laws, Power Set
	Relations & Functions	Cartesian Product, One-to-One Function, Domain & Range, Bijective Function, Inverse Functions, Modulus Function, Even and Odd Function
	Complex Numbers & Quadratic Equations	Quadratic Equation, Powers of Iota, Modulus of Complex Number, Argument of Complex Number, Quadratic Inequalities, Graphical Representation of Quadratic Equation
	Permutations & Combinations	Permutation, Combination, Applications of Permutations, Permutation vs Combination, Rank Of A Word, Derangement
	Binomial Theorem	Binomial Theorem - Formula, Expansion, Sum of Binomial Coefficients, General and Middle Terms
	Sequences & Series	Arithmetic Progression, Geometric Progression, Harmonic Progression, Sum of Common Series
	Matrices	Types of Matrices, Matrix Operations, Matrix Multiplication, Transpose of a Matrix
	Determinants	Minors and Cofactors, Adjoint and Inverse of a Matrix, Properties of Determinants, Cramer's Rule
Trigonometry	Trigonometric Functions	Trigonometric Ratios, Trigonometric Identities, Law of Sines, Law of Cosines, Graphs of General Trigonometric Functions
Trigonometry	Inverse Trigonometric Functions	Domain & Range of Inverse Functions, Sum & Difference, Graph of Inverse Functions
Calculus	Limits & Derivatives	Limits, Sandwich Theorem, Left-hand & Right-hand Limits, L’Hospital’s Rule, Differentiation
	Continuity & Differentiability	Differentiation Rules, Implicit Differentiation, Logarithmic Differentiation, Derivative of Inverse Trigonometric Functions
	Integrals	Definite Integral, Indefinite Integral Formulas, Trigonometric Integrals, Properties of Definite Integral
	Differential Equations	Homogeneous Differential Equation, Linear Differential Equation, Exact Differential Equation, Applications
Geometry & Vectors	Three Dimensional Geometry	Equation Of A Line In Three Dimensions, Family Of Planes, Section Formula In 3d, Shortest Distance Between Two Lines In 3d Space
	Vector Algebra	Dot Product Of Two Vectors, Cross Product, Vector Addition And Subtraction, Types Of Vectors
	Straight Lines	Distance Between Two Points, Centroid, Equation Of Straight Line
	Conic Sections	Circles, Ellipse, Parabola, Hyperbola
Statistics & Probability	Statistics	Measures Of Central Tendency, Measures Of Dispersion, Data Representation
Statistics & Probability	Probability	Conditional Probability, Bayes' Theorem, Random Variables, Bernoulli Trials And Binomial Distribution

Topic	Formulae
Algebra	$(a+b)^2 = a^2 + 2ab + b^2$ $(a-b)^2 = a^2 - 2ab + b^2$ $a^2 - b^2 = (a-b)(a+b)$ $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ (quadratic equation) Sum of roots of $ax^2+bx+c=0$: $\alpha + \beta = -\frac{b}{a}$ Product of roots of $ax^2+bx+c=0$: $\alpha \beta = \frac{c}{a}$ $(a+b)^n = \sum_{k=0}^{n} {n \choose k} a^{n-k}b^k$ (binomial theorem) $n$th term of AP: $a_n = a + (n-1)d$ Sum of first $n$ terms of AP: $S_n = \frac{n}{2}[2a + (n-1)d]$ $n$th term of GP: $a_n = ar^{n-1}$ Sum of first $n$ terms of GP: $S_n = a\frac{r^n - 1}{r-1}, r\neq1$ Sum to infinity of GP: $S_\infty = \frac{a}{1-r}, \|r\|<1$
Geometry	Area of triangle: $A = \sqrt{s(s-a)(s-b)(s-c)}, s=\frac{a+b+c}{2}$ Area of circle: $A = \pi r^2$ Circumference of circle: $C = 2\pi r$ Volume of sphere: $V = \frac{4}{3}\pi r^3$ Surface area of sphere: $A = 4\pi r^2$ Volume of cone: $V = \frac{1}{3}\pi r^2 h$ Pythagoras theorem: $a^2 + b^2 = c^2$
Trigonometry	$\sin^2\theta + \cos^2\theta = 1$ $1 + \tan^2\theta = \sec^2\theta$ $1 + \cot^2\theta = \csc^2\theta$ $\sin 2\theta = 2\sin\theta \cos\theta$ $\cos 2\theta = \cos^2\theta - \sin^2\theta$ $\tan 2\theta = \frac{2\tan\theta}{1-\tan^2\theta}$ $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ $\cos(A \pm B) = \cos A \cos B \mp \sin A \sin B$
Calculus	$d(x^n)/dx = nx^{n-1}$ $d(\sin x)/dx = \cos x$, $d(\cos x)/dx = -\sin x$ $d(\tan x)/dx = \sec^2 x$, $d(\cot x)/dx = -\csc^2 x$ $\int x^n dx = \frac{x^{n+1}}{n+1} + C, n\neq -1$ $\int \frac{dx}{x} = \ln\|x\| + C$ $\int \sin x dx = -\cos x + C$, $\int \cos x dx = \sin x + C$ $\frac{d}{dx}[uv] = u'v + uv'$ (Product rule) $\frac{d}{dx}[\frac{u}{v}] = \frac{u'v - uv'}{v^2}$ (Quotient rule)
Probability & Statistics	$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}}$ $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ $P(A \cap B) = P(A) \cdot P(B),$ if independent Mean: $\bar{x} = \frac{\sum x_i}{n}$ Variance: $\sigma^2 = \frac{\sum(x_i - \bar{x})^2}{n}$ Standard deviation: $\sigma = \sqrt{\sigma^2}$
Sequences & Series	Arithmetic progression (AP) $n$th term: $a_n = a + (n-1)d$ Sum of first $n$ terms of AP: $S_n = \frac{n}{2}[2a + (n-1)d]$ Geometric progression (GP) $n$th term: $a_n = ar^{n-1}$ Sum of first $n$ terms of GP: $S_n = a \frac{r^n - 1}{r-1}, r\neq 1$ Sum to infinity of GP: $S_\infty = \frac{a}{1-r}, \|r\|<1$ Harmonic progression (HP): $n$th term: $h_n = \frac{1}{a + (n-1)d}$
Matrices	Addition: $(A+B){ij} = A{ij} + B_{ij}$ Subtraction: $(A-B){ij} = A{ij} - B_{ij}$ Scalar multiplication: $(kA){ij} = k \cdot A{ij}$ Multiplication: $(AB){ij} = \sum_k A{ik} B_{kj}$ Transpose: $(A^T){ij} = A{ji}$ Inverse of 2×2 matrix: $A^{-1} = \frac{1}{ad-bc} \begin{pmatrix}d & -b \\ -c & a\end{pmatrix}$ Properties: $(AB)^T = B^T A^T, (A^T)^{-1} = (A^{-1})^T$
Determinants	2×2: $\|A\| = \begin{vmatrix}a & b \\ c & d\end{vmatrix} = ad - bc$ 3×3: $\|A\| = \begin{vmatrix}a & b & c \\ d & e & f \\ g & h & i\end{vmatrix} = a(ei − fh) − b(di − fg) + c(dh − eg)$ Minor: $M_{ij}$ = determinant of matrix after removing $i$th row and $j$th column Cofactor: $C_{ij} = (-1)^{i+j} M_{ij}$ Adjoint: $adj(A) = $ transpose of cofactor matrix Inverse using determinant: $A^{-1} = \frac{1}{\|A\|} adj(A)$
Vectors	Magnitude of vector: $\|\vec{A}\| = \sqrt{A_x^2 + A_y^2 + A_z^2}$ Direction cosines: $\cos \alpha = \frac{A_x}{\|\vec{A}\|}, \cos \beta = \frac{A_y}{\|\vec{A}\|}, \cos \gamma = \frac{A_z}{\|\vec{A}\|}$ Dot product: $\vec{A} \cdot \vec{B} = AB \cos \theta = A_xB_x + A_yB_y + A_zB_z$ Cross product: $\vec{A} \times \vec{B} = \|\vec{A}\|\|\vec{B}\|\sin\theta \hat{n}$ Scalar triple product: $\vec{A} \cdot (\vec{B} \times \vec{C})$ Vector addition: $\vec{R} = \vec{A} + \vec{B}$ Unit vector: $\hat{A} = \frac{\vec{A}}{\|\vec{A}\|}$
Conic Sections	Circle: $(x-h)^2 + (y-k)^2 = r^2$ Parabola: $y^2 = 4ax$ or $x^2 = 4ay$ Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ Eccentricity: $e = \frac{\sqrt{a^2-b^2}}{a}$ (ellipse), $e = \frac{\sqrt{a^2+b^2}}{a}$ (hyperbola) Latus rectum: $4a$ (parabola), $2b^2/a$ (ellipse), $2b^2/a$ (hyperbola)

Book Title	Author / Publisher	Description
NCERT Mathematics	NCERT	Core syllabus for algebra, calculus, trigonometry, vectors, coordinate geometry. Essential for board exams and foundation for competitive exams.
Mathematics for Class 11 & 12	R.D. Sharma	Comprehensive theory, solved examples, and extensive practice problems for conceptual clarity and exam preparation.
Higher Algebra & Trigonometry	S.L. Loney	Classic reference for deep understanding, analytical thinking, and advanced problem-solving (JEE, Olympiads).
Trigonometry, Calculus & Coordinate Geometry	S.L. Loney	Focused advanced problems for JEE Main, JEE Advanced, and other competitive exams.
Objective Mathematics	R.D. Sharma / Arihant	Designed for competitive exams, provides objective-type questions, shortcuts, and problem-solving practice.
Cengage Mathematics Series	G. Tewani / Cengage	Detailed theory and practice questions, ideal for JEE and other engineering entrance exams.
IIT Mathematics	M.L. Khanna	Advanced problems and solutions for competitive exams, strong conceptual emphasis.

Maths: Concepts, Formulas & Problem-Solving Techniques

History of Mathematics: Origins and Evolution

Ancient Mathematics and Early Civilizations

Key Contributions in Ancient Mathematics

Archimedes

Apollonius of Perga

Hipparchus of Nicaea

Diophantus

Mathematics During the Islamic Golden Age

Mathematics in the Modern Era

Development of New Math Formulas

Role of Isaac Newton

German Contributions to Modern Mathematics

Evolution of Mathematics Today

Why is Mathematics Important?

How to Learn Mathematics Effectively and Easily

Start with Strong Fundamentals

Practice Mathematics Regularly

Focus on Conceptual Understanding

Learn from Solved Examples

Fundamentals of Mathematics: From Basic to Advanced

Mathematics Topics List 2026: Detailed Chapterwise Syllabus for Algebra, Calculus, Geometry & Statistics

Maths Important Formulae List

Mathematics Problem-Solving Techniques for Faster Speed, Higher Accuracy, and Top Exam Scores

Concept Identification Before Solving

Smart Use of Maths Formulas and Shortcuts

Visual Representation for Better Understanding

Practice High-Quality and Diverse Questions

Error Analysis and Performance Improvement

Recommended Books for Mathematics

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