📜Prologue: The Universal BlueprintPg 5▶
📘1. Introduction & How To Use This BookPg 30▶
How the book is organised and the best way to work through every method inside.
🧠2. Vedic Mathematics — All 16 Sutras & 13 Sub-Sutras (Deep Dive)Pg 32▶
A complete, worked-example deep dive into every Vedic Sutra, each one mapped to its closest global equivalent.
2.1 The 16 Main Sutras — Complete Guide With Global ComparisonsPg 32
2.2 Deep Dive: Each Sutra With Examples From All TraditionsPg 35
Sutra 1: Ekadhikena Purvena — "By one more than the previous one"Pg 35
Global equivalent: Trachtenberg ×11 method
A) Multiplying Any Number by 11 (the most famous Vedic trick!) — Pg 35
B) Multiplying by 111, 1111, etc. — Pg 37
C) Squaring Numbers Ending in 5 (the most useful Vedic trick!) — Pg 37
D) Finding Recurring Decimals (1/19, 1/29, etc.) — Pg 38
Sutra 2: Nikhilam Navatascaramam Dasatah — "All from 9 and the last from 10"Pg 40
Global equivalent: Complement Method (Abacus), Base Method (Trachtenberg)
A) Multiplication When Both Numbers Are Below the Base — Pg 40
B) Multiplication When One Number Is Above the Base — Pg 42
C) Division by 9, 99, 999… (Nikhilam Division) — Pg 44
D) Subtraction From Powers of 10 (Instant!) — Pg 46
Sutra 3: Urdhva-Tiryagbhyam — "Vertically and crosswise"Pg 47
Global equivalent: Cross Multiplication (Japan), FOIL Method (Modern Algebra)
A) 2-Digit × 2-Digit Multiplication — Pg 47
B) 3-Digit × 3-Digit Multiplication — Pg 49
C) Multiplying Numbers With Different Digit Counts (e.g., 2-Digit × 3-Digit) — Pg 50
D) Polynomial Multiplication — Pg 51
Sutra 4: Paravartya Yojayet — "Transpose and apply"Pg 53
Global equivalent: Egyptian Division, Long Division Shortcut
A) Division Using Paravartya Yojayet — Pg 53
B) Solving Linear Equations Using Paravartya Yojayet — Pg 55
C) Partial Fractions Using Paravartya Yojayet — Pg 56
Sutra 5: Sunyam Saamyasamuccaye — "When the sum is the same, that sum is zero"Pg 56
A) Solving Equations Where Sum of Numerators = Sum of Denominators — Pg 56
B) Factorization Using Sunyam Saamyasamuccaye — Pg 57
Sutra 6: Anurupye Sunyamanyat — "If one is in ratio, the other is zero"Pg 58
Global equivalent: Rule of Three (Ancient India, Greece)
A) Solving Proportion Problems — Pg 58
B) Solving Equations Using Ratios — Pg 58
Sutra 7: Sankalana-Vyavakalanabhyam — "By addition and subtraction"Pg 59
Global equivalent: Elimination Method (Modern Algebra)
A) Solving Simultaneous Equations — Pg 59
B) Finding HCF and LCM Using Sankalana-Vyavakalanabhyam — Pg 60
Sutra 8: Puranapuranabhyam — "By completion or non-completion"Pg 61
Global equivalent: Completing the Square (Babylonian, Greek)
A) Completing the Square — Pg 61
B) Solving Quadratic Equations — Pg 62
C) Factorization Using Puranapuranabhyam — Pg 62
Sutra 9: Calana-Kalanabhyam — "Differences and similarities"Pg 62
Global equivalent: Method of Differences (Newton)
A) Differentiation (Calculus) — Pg 62
B) Quadratic Factorization — Pg 63
Sutra 10: Yavadunam — "By the deficiency"Pg 63
Global equivalent: Deficiency Method (Trachtenberg)
A) Squaring Numbers Near a Base — Pg 64
B) Cubing Numbers Near a Base — Pg 65
Sutra 11: Vyastisamastih — "Whole as one and one as whole"Pg 66
A) Factorization of Polynomials — Pg 66
B) Grouping Terms — Pg 67
Sutra 12: Sesanyankena Caramena — "The remainders by the last digit"Pg 67
Global equivalent: Osculator Method (Modern Number Theory)
A) Divisibility by 7 — Pg 67
B) Divisibility by 13 — Pg 68
C) Divisibility by 17 — Pg 68
Sutra 13: Sopantyadvayamantyam — "The ultimate and twice the penultimate"Pg 69
A) Solving Quadratic Equations of the Form x² + ax + b = 0 — Pg 69
B) Finding Square Roots — Pg 69
Sutra 14: Ekanyunena Purvena — "By one less than the previous one"Pg 70
Global equivalent: Subtract from 10 (Trachtenberg)
A) Multiplying by 9 — Pg 71
B) Multiplying by 99 — Pg 72
C) Multiplying by 999 — Pg 72
Sutra 15: Gunitasamuccayah — "The product of the sum is the sum of the product"Pg 72
A) Verifying Factorization — Pg 72
B) Checking Algebraic Expansions — Pg 73
Sutra 16: Gunakasamuccayah — "The factors of the sum equal the sum of the factors"Pg 73
A) Advanced Factorization — Pg 73
B) Polynomial Identities — Pg 74
2.3 The 13 Sub-Sutras — Quick Reference With ApplicationsPg 74
2.4 Vedic vs. Trachtenberg vs. Abacus vs. Soroban: A ComparisonPg 76
1. Overview of Mental Math Systems — Pg 76
2. Vedic Mathematics — Pg 77
3. Trachtenberg System — Pg 77
4. Abacus (Traditional Chinese) — Pg 78
5. Soroban (Japanese Abacus) — Pg 78
6. Which Method Should You Use? — Pg 79
7. Hybrid Approach — Pg 79
⚡3. Mental Arithmetic & Lightning Calculation (Global Systems)Pg 81▶
Speed techniques drawn from Vedic, Trachtenberg, Abacus, Soroban, and other traditions — organised by operation.
3.1 Lightning Multiplication Tricks (All Cultures)Pg 81
Multiplying by 5, 25, 50, 125, 500, 625 (The Easy Multipliers) — Mental Arithmetic, used worldwide — Pg 81
Multiplying Two Numbers With Same Tens Digit & Unit Digits Summing to 10 — Vedic: Antyayordasake'pi — Pg 82
Multiplying Near 50, 200, etc. (Anurupyena — Proportional Base) — Vedic: Anurupyena / Global: Base Method — Pg 84
Multiplying by 15, 150 — The "Add Half" Trick — Pg 85
Fast Multiplication When One Number Is All 9's — Vedic: Ekanyunena Purvena — Pg 86
3.2 Lightning Squaring TechniquesPg 87
Squaring Numbers 26–100 (Using 50 as Base) — Vedic: Yavadunam — Pg 87
A) Squaring Numbers 26–50 — Pg 87
B) Squaring Numbers 51–75 — Pg 89
C) Squaring Numbers 76–100 — Pg 90
Duplex Method (Dvandva Yoga) — Squaring Any Number (Vedic) — Pg 91
Squaring by (a+b)² = a² + 2ab + b² (Mental Decomposition) — Algebraic Identity — Pg 94
3.3 Lightning Division ShortcutsPg 96
Dividing by 9, 99, 999 Using Patterns — Vedic: Nikhilam — Pg 96
A) Division by 9 — Pg 96
B) Division by 99 — Pg 98
3.4 Percentage TricksPg 99
X% of Y = Y% of X — The Symmetry Trick (used worldwide) — Pg 99
Quick Percentage Conversions to Memorize — Pg 99
Successive Percentage Change (Net Effect) — Pg 101
3.5 Trachtenberg System: Complete GuidePg 102
A) Trachtenberg Multiplication — Pg 102
B) Trachtenberg Division — Pg 104
C) Trachtenberg Addition and Subtraction — Pg 105
3.6 Abacus & Soroban Mental Math TechniquesPg 105
A) Soroban Addition — Pg 106
B) Soroban Subtraction — Pg 107
C) Soroban Multiplication — Pg 107
3.7 Other Mental Math ShortcutsPg 108
1. Russian Peasant Multiplication (Universal) — Pg 108
2. Fibonacci's Finger Multiplication (Medieval Europe) — Pg 109
3. Japanese Line Multiplication (Visual Method) — Pg 109
4. Chinese "Gui Gu" Division (Ancient China) — Pg 109
🔢4. Number System & Basic Operations (Expanded)Pg 111▶
The foundational rules and formulas every other shortcut in the book is built on.
4.1 Divisibility Rules — Complete List (1 to 100)Pg 111
Divisibility Rules for 1 to 20 (Most Common) — Pg 111
Divisibility Rules for 21 to 100 — Pg 112
4.2 HCF & LCM — Shortcut Formulas & Advanced MethodsPg 115
HCF (Highest Common Factor) & LCM (Least Common Multiple) — Pg 115
A) Finding HCF — Pg 115
B) Finding LCM — Pg 117
HCF & LCM for More Than Two Numbers — Pg 118
4.3 Surds & Indices — Key Shortcuts & SimplificationsPg 119
Surds (Roots) & Indices (Exponents) — All Properties & Shortcuts — Pg 119
B) Rationalizing the Denominator — Pg 120
C) Comparing Surds — Pg 120
E) Laws of Exponents With Variables — Pg 122
4.4 Logarithms — Essential Values, Properties & ApplicationsPg 122
Logarithms — All Properties & Shortcuts — Pg 122
B) Solving Logarithmic Equations — Pg 123
C) Change of Base Formula — Pg 123
4.5 Number of Zeros at the End of n! (Factorial)Pg 124
Trailing Zeros in Factorials — Instant Method — Pg 124
4.6 Recurring Decimals to Fractions — Instant MethodsPg 125
Converting Recurring Decimals to Fractions — Pg 125
A) Pure Recurring Decimals (e.g., 0.3, 0.142857) — Pg 125
B) Mixed Recurring Decimals (e.g., 0.16, 0.1234) — Pg 127
4.7 Binary, Hexadecimal & Number Base ConversionsPg 128
Number Bases — Binary, Octal, Hexadecimal & Conversions (Computer Science) — Pg 128
A) Decimal to Binary Conversion — Pg 129
B) Binary to Decimal Conversion — Pg 130
C) Decimal to Hexadecimal Conversion — Pg 130
D) Hexadecimal to Decimal Conversion — Pg 131
E) Binary to Hexadecimal Conversion — Pg 131
F) Hexadecimal to Binary Conversion — Pg 132
4.8 Prime Numbers — Sieve of Eratosthenes & ShortcutsPg 133
Prime Numbers — All Properties & Shortcuts — Pg 133
A) Checking if a Number Is Prime — Pg 133
B) Sieve of Eratosthenes (Finding All Primes Up to n) — Pg 134
C) Prime Factorization — Pg 134
4.9 Roman Numerals & Ancient Number SystemsPg 136
Roman Numerals & Ancient Number Systems (Ancient Rome, Egypt, Babylon) — Pg 136
A) Roman Numerals — Pg 136
B) Egyptian Numerals — Pg 138
C) Babylonian Numerals (Base 60) — Pg 138
➗5. Algebra — Polynomials, Equations & Inequalities (Expanded)Pg 140▶
Every major algebra topic covered with complete shortcut methods, from factoring to advanced inequalities.
5.1 Factoring Quadratics Instantly (Adyamadyenantya-mantye & More)Pg 140
Vedic: Adyamadyenantya-mantye
Factoring Quadratics — All Methods — Pg 140
A) Factoring Quadratics Where a = 1 — Pg 140
B) Factoring Quadratics Where a ≠ 1 — Pg 141
5.2 Cubic & Higher Degree Factoring (All Methods)Pg 143
Factoring Cubic & Higher Degree Polynomials — Pg 143
A) Factor Theorem — Pg 143
B) Rational Root Theorem — Pg 143
C) Sum and Difference of Cubes — Pg 144
D) Sophie Germain Identity — Pg 144
E) Factoring by Grouping — Pg 145
5.3 Quadratic Equations — All Shortcuts & Special CasesPg 145
Quadratic Equations — All Methods & Shortcuts — Pg 145
A) Solving by Factoring — Pg 145
B) Solving by Quadratic Formula — Pg 146
C) Solving by Completing the Square — Pg 146
D) Special Cases — Pg 147
5.4 Solving Linear Equations — Cramer's Rule & MorePg 148
Linear Equations — All Methods & Shortcuts — Pg 148
A) Solving Single Linear Equations — Pg 148
B) Solving Simultaneous Linear Equations (Two Variables) — Pg 148
C) Cramer's Rule — Pg 150
5.5 Inequalities — Wavy Curve Method & ModulusPg 151
Inequalities — All Methods & Shortcuts — Pg 151
A) Linear Inequalities — Pg 151
B) Quadratic Inequalities (Wavy Curve Method) — Pg 152
C) Absolute Value Inequalities — Pg 153
5.6 AM-GM, Cauchy-Schwarz & All Inequalities for JEEPg 155
Key Inequalities — AM-GM, Cauchy-Schwarz & More — Pg 155
A) AM-GM Inequality — Pg 155
B) Cauchy-Schwarz Inequality — Pg 156
C) Triangle Inequality — Pg 157
D) Bernoulli's Inequality — Pg 157
E) Jensen's Inequality — Pg 157
5.7 Polynomial Identities & Symmetric SumsPg 158
Polynomial Identities & Symmetric Sums — Pg 158
A) Common Polynomial Identities — Pg 158
B) Symmetric Sums — Pg 159
5.8 Partial Fractions — Cover-Up & Advanced MethodsPg 160
Partial Fractions — All Methods — Pg 160
A) Cover-Up Method (for Linear Factors) — Pg 160
B) Partial Fractions With Repeated Factors — Pg 161
C) Partial Fractions With Quadratic Factors — Pg 161
5.9 Instant Quadratic Formula (Babylonian Method)Pg 163
5.10 Sum of Cubes Factorization (Instant)Pg 163
Solving Cubic Equations (Cardano's Method Simplified) — Pg 163
📐6. Geometry — Triangles, Circles & Theorems (Expanded)Pg 165▶
Every core geometry theorem and formula, from triangles and circles to conic sections and 3D geometry.
6.1 Triangles — All Key Theorems & ShortcutsPg 165
Triangles — All Properties & Theorems — Pg 165
A) Pythagorean Theorem — Pg 165
B) Area of a Triangle — Pg 167
C) Apollonius Theorem (Median Length) — Pg 167
D) Angle Bisector Theorem — Pg 168
E) Stewart's Theorem — Pg 168
F) Basic Proportionality Theorem (Thales' Theorem) — Pg 169
G) Congruency & Similarity — Pg 169
6.2 Circles — Power of a Point, Chords & TangentsPg 170
Circles — All Theorems & Properties — Pg 170
A) Area and Circumference of a Circle — Pg 170
B) Length of an Arc and Area of a Sector — Pg 171
C) Power of a Point — Pg 171
D) Tangent-Secant Theorem — Pg 172
E) Chord Theorems — Pg 172
F) Inscribed Angle Theorem — Pg 172
G) Cyclic Quadrilaterals — Pg 173
6.3 Trigonometry in Geometry — Sine, Cosine & Area FormulasPg 173
Trigonometry in Geometry — All Formulas — Pg 174
A) Sine and Cosine Rules — Pg 174
B) Area of a Triangle Using Trigonometry — Pg 174
C) Trigonometric Identities in Geometry — Pg 175
6.4 Coordinate Geometry Shortcuts (All Levels)Pg 175
Coordinate Geometry — All Formulas & Shortcuts — Pg 175
A) Distance Formula — Pg 175
B) Section Formula — Pg 175
C) Midpoint Formula — Pg 176
D) Slope of a Line — Pg 176
E) Equation of a Line — Pg 177
F) Angle Between Two Lines — Pg 177
G) Distance From a Point to a Line — Pg 178
H) Area of a Triangle Using Coordinates — Pg 178
6.5 3D Geometry — Lines, Planes & DistancesPg 178
3D Geometry — All Formulas & Shortcuts — Pg 178
A) Distance Between Two Points in 3D — Pg 178
B) Equation of a Line in 3D — Pg 179
C) Equation of a Plane — Pg 179
D) Angle Between Two Planes — Pg 180
E) Distance From a Point to a Plane — Pg 180
F) Shortest Distance Between Two Skew Lines — Pg 180
6.6 Conic Sections — Parabola, Ellipse, HyperbolaPg 181
Conic Sections — All Properties & Formulas — Pg 181
A) Circle — Pg 182
B) Parabola — Pg 182
C) Ellipse — Pg 183
D) Hyperbola — Pg 184
📏7. Mensuration — 2D & 3D Master Formulas (Expanded)Pg 186▶
7.1 2D Shapes — Complete Area & PerimeterPg 186
7.2 3D Solids — Volume, CSA, TSA & ShortcutsPg 189
7.3 Scaling Rules & Similarity ShortcutsPg 191
A) Similar Triangles — Pg 191
B) Similar Polygons — Pg 191
C) Similar Solids — Pg 192
📊8. Trigonometry — Identities, Equations & Tricks (Complete)Pg 193▶
8.1 The ASTC Rule & Angle Values (Hand Trick + Unit Circle)Pg 193
A) Unit Circle Values for Common Angles — Pg 195
8.2 Complete Trig Identity Bank (200+ Identities)Pg 196
8.3 Conditional Identities (A + B + C = π or 180°)Pg 200
8.4 General Solutions — Never Forget ThesePg 201
8.5 Max/Min of a sinθ + b cosθ & All Trig OptimizationsPg 202
A) Finding the Phase Shift φ — Pg 203
B) Applications of Trig Optimization — Pg 204
8.6 Trigonometric Equations — All Types SolvedPg 204
A) Linear Trigonometric Equations — Pg 204
B) Quadratic Trigonometric Equations — Pg 204
C) Equations With Multiple Angles — Pg 205
D) Equations With Sum or Difference of Angles — Pg 205
8.7 Inverse Trigonometric Functions — Properties & ShortcutsPg 206
A) Properties of Inverse Trigonometric Functions — Pg 207
B) Solving Equations With Inverse Trigonometric Functions — Pg 207
8.8 Trig in Complex Plane — Euler's Formula & De MoivrePg 207
A) Euler's Formula — Pg 208
B) Polar Form of Complex Numbers — Pg 208
C) De Moivre's Theorem — Pg 209
D) Roots of Complex Numbers — Pg 209
∫9. Calculus — Limits, Continuity & Differentiability (Expanded)Pg 211▶
9.1 Standard Limits (Memorize Absolutely!)Pg 211
9.2 L'Hôpital's Rule — When & How to UsePg 212
9.3 Squeeze (Sandwich) Theorem & Advanced Limit TricksPg 214
A) Other Limit Tricks — Pg 214
9.4 Continuity & Differentiability — All TestsPg 215
A) Continuity — Pg 216
B) Differentiability — Pg 217
d/dx10. Differentiation — Rules, Shortcuts & Higher Orders (Expanded)Pg 219▶
10.1 Complete Derivative Table (100+ Functions)Pg 219
10.2 Logarithmic Differentiation — Advanced CasesPg 221
10.3 Parametric & Implicit DifferentiationPg 222
A) Parametric Differentiation — Pg 222
B) Implicit Differentiation — Pg 223
10.4 Higher Order Derivatives & Leibniz RulePg 224
A) Leibniz Rule — Pg 225
10.5 Differentiation of Vectors & MatricesPg 226
A) Differentiation of Vectors — Pg 226
B) Differentiation of Matrices — Pg 227
📈11. Applications of Derivatives — Tangents, Maxima & Minima (Expanded)Pg 229▶
11.1 Maxima & Minima — First & Second Derivative TestsPg 229
A) Critical Points — Pg 229
B) First Derivative Test — Pg 230
C) Second Derivative Test — Pg 230
11.2 Rolle's Theorem & Lagrange's Mean Value TheoremPg 232
A) Rolle's Theorem — Pg 232
B) Lagrange's Mean Value Theorem (MVT) — Pg 232
11.3 Curve Sketching — All Steps & ShortcutsPg 233
11.4 Optimization Problems — All Types SolvedPg 235
∮12. Indefinite Integration — All Methods (Expanded)Pg 238▶
12.1 Standard Integral Formulas (200+)Pg 238
12.2 Integration by Parts (ILATE Rule & Extensions)Pg 239
12.3 Partial Fractions — All Cases & ShortcutsPg 241
A) Linear Factors (Distinct) — Pg 241
B) Repeated Linear Factors — Pg 242
C) Irreducible Quadratic Factors — Pg 242
12.4 The eˣ(f + f') Trick & All Exponential ShortcutsPg 243
12.5 Trigonometric Integrals — All TypesPg 245
A) Integrals of sinⁿx and cosⁿx — Pg 246
B) Integrals of tanⁿx and cotⁿx — Pg 247
C) Integrals of Products of Sine and Cosine — Pg 248
D) Reduction Formulas — Pg 248
12.6 Integration of Rational Functions — All MethodsPg 249
A) Polynomial Long Division — Pg 249
B) Partial Fractions — Pg 250
∫₀13. Definite Integration — Properties & Tricks (Expanded)Pg 251▶
13.1 King Property (The Life-Saver!)Pg 251
13.2 Walli's Formula (Complete Guide)Pg 252
13.3 Gamma Function & Beta Function ConnectionPg 254
A) Gamma Function — Pg 254
B) Beta Function — Pg 255
C) Connection to Definite Integrals — Pg 255
13.4 Even-Odd & Periodic PropertiesPg 255
A) Even and Odd Functions — Pg 256
B) Periodic Functions — Pg 256
13.5 Leibniz Rule & Integration Under the Integral SignPg 257
A) Leibniz Rule — Pg 257
B) Differentiation Under the Integral Sign — Pg 257
dy/dx14. Differential Equations — All Types (Expanded)Pg 259▶
14.1 Complete Classification & Solving MethodsPg 259
A) First-Order Differential Equations — Pg 259
B) Second-Order Linear Differential Equations — Pg 260
14.2 First Order DEs — All Types & ShortcutsPg 260
A) Variable Separable Equations — Pg 260
B) Homogeneous Equations — Pg 261
C) Linear First-Order Equations — Pg 261
D) Exact Differential Equations — Pg 262
E) Bernoulli Equations — Pg 262
14.3 Second Order Linear DEs — All CasesPg 263
A) Homogeneous Equations — Pg 263
B) Non-Homogeneous Equations — Pg 264
14.4 Laplace Transforms — Complete Table & ApplicationsPg 266
A) Laplace Transform Table — Pg 266
B) Properties of Laplace Transforms — Pg 267
C) Solving Differential Equations Using Laplace Transforms — Pg 267
14.5 Partial Differential Equations (PDEs) — Basics for JEEPg 268
A) Classification of PDEs — Pg 269
B) Separation of Variables — Pg 269
🧭15. Vectors & 3D Geometry (Expanded)Pg 271▶
15.1 Vector Algebra — All Operations & ShortcutsPg 271
A) Vector Addition and Subtraction — Pg 271
B) Scalar Multiplication — Pg 271
C) Dot Product (Scalar Product) — Pg 272
D) Cross Product (Vector Product) — Pg 272
E) Magnitude of a Vector — Pg 273
F) Scalar Triple Product — Pg 273
G) Vector Triple Product — Pg 274
15.2 3D Geometry — Lines, Planes & Distances (Complete)Pg 274
A) Equation of a Line in 3D — Pg 274
B) Equation of a Plane — Pg 275
C) Distance From a Point to a Plane — Pg 275
D) Angle Between Two Planes — Pg 275
E) Shortest Distance Between Two Skew Lines — Pg 276
F) Coplanarity of Vectors — Pg 277
15.3 Vector Calculus — Grad, Div, Curl & TheoremsPg 277
A) Gradient — Pg 277
B) Divergence — Pg 278
C) Curl — Pg 278
D) Fundamental Theorems of Vector Calculus — Pg 278
15.4 Geometry ShortcutsPg 279
1. Heron's Formula (Area of Triangle Without Height) — Pg 279
2. Brahmagupta's Formula (Area of Cyclic Quadrilateral) — Pg 279
3. Napoleon's Theorem (Geometric Construction) — Pg 280
4. Pick's Theorem (Area of Lattice Polygons) — Pg 280
ℂ16. Complex Numbers — Polar Form & Geometry (Expanded)Pg 281▶
16.1 Fundamentals & Polar Form (All Representations)Pg 281
A) Basic Operations — Pg 281
B) Polar Form — Pg 282
C) Multiplication and Division in Polar Form — Pg 283
D) De Moivre's Theorem — Pg 283
16.2 Cube Roots of Unity & nth Roots (Complete)Pg 284
A) Cube Roots of Unity — Pg 284
B) nth Roots of Unity — Pg 285
C) Finding nth Roots of a Complex Number — Pg 285
16.3 Geometry in Complex PlanePg 286
A) Lines and Circles — Pg 286
B) Rotation and Translation — Pg 287
C) Parametric Representation — Pg 287
16.4 Complex Analysis Basics — Cauchy, Taylor & LaurentPg 287
A) Analytic Functions — Pg 288
B) Cauchy's Integral Theorem — Pg 288
C) Cauchy's Integral Formula — Pg 288
D) Taylor Series — Pg 289
E) Laurent Series — Pg 289
F) Residue Theorem — Pg 290
🧮17. Matrices & Determinants — Properties & Cramer's Rule (Expanded)Pg 291▶
17.1 Matrix Properties & Shortcuts (All Rules)Pg 291
A) Matrix Addition and Subtraction — Pg 291
B) Scalar Multiplication — Pg 291
C) Matrix Multiplication — Pg 292
D) Transpose of a Matrix — Pg 292
E) Inverse of a Matrix — Pg 293
F) Properties of Matrix Operations — Pg 293
17.2 Determinant Tricks & Advanced PropertiesPg 293
A) Determinant of a 2×2 Matrix — Pg 294
B) Determinant of a 3×3 Matrix (Sarrus Rule) — Pg 294
C) Properties of Determinants — Pg 294
D) Minors and Cofactors — Pg 294
E) Adjugate and Inverse — Pg 295
17.3 Cramer's Rule & Matrix Inversion ShortcutsPg 296
A) Cramer's Rule for 2×2 Systems — Pg 296
B) Cramer's Rule for 3×3 Systems — Pg 296
C) Matrix Inversion Shortcuts — Pg 298
17.4 Eigenvalues & Eigenvectors — Complete GuidePg 299
A) Finding Eigenvalues — Pg 299
B) Finding Eigenvectors — Pg 299
C) Properties of Eigenvalues and Eigenvectors — Pg 300
D) Diagonalization — Pg 301
🔗18. Set Theory, Relations & Functions (Expanded)Pg 303▶
18.1 Counting with Sets (All Formulas)Pg 303
A) Basic Set Operations — Pg 303
B) Counting Formulas — Pg 303
C) Power Set — Pg 304
D) Cartesian Product — Pg 304
18.2 Relations & Functions — Counting FormulasPg 304
A) Types of Relations — Pg 305
B) Types of Functions — Pg 305
C) Counting Relations and Functions — Pg 306
18.3 Binary Operations & Group Theory BasicsPg 307
A) Properties of Binary Operations — Pg 307
B) Group Theory Basics — Pg 307
C) Subgroups and Homomorphisms — Pg 308
🎲19. Probability & Statistics — All Distributions (Expanded)Pg 310▶
19.1 Probability — Key Formulas & Bayes' TheoremPg 310
A) Basic Probability Rules — Pg 310
B) Bayes' Theorem — Pg 310
C) Odds — Pg 311
19.2 Binomial, Poisson & All Discrete DistributionsPg 311
A) Binomial Distribution — Pg 312
B) Poisson Distribution — Pg 312
C) Geometric Distribution — Pg 313
D) Negative Binomial Distribution — Pg 313
19.3 Statistics — Shortcut Formulas & Advanced MethodsPg 314
A) Measures of Central Tendency — Pg 314
B) Measures of Dispersion — Pg 315
C) Empirical Relation Between Mean, Median, and Mode — Pg 315
D) Variance (Step Deviation Method) — Pg 315
E) Correlation Coefficient — Pg 316
19.4 Normal, Exponential & All Continuous DistributionsPg 318
A) Normal Distribution — Pg 318
B) Exponential Distribution — Pg 319
C) Uniform Distribution — Pg 319
🔀20. Permutations & Combinations — Counting Mastery (Expanded)Pg 321▶
20.1 Complete Formula Reference (All Cases)Pg 321
A) Permutations — Pg 321
B) Combinations — Pg 322
C) Permutations with Repetition — Pg 322
D) Circular Permutations — Pg 322
E) Permutations of Objects in a Circle (Clockwise vs. Counter-Clockwise) — Pg 323
F) Combinations with Repetition — Pg 323
20.2 Common Patterns & PitfallsPg 324
A) Common Patterns — Pg 324
B) Common Pitfalls — Pg 326
20.3 Advanced Combinatorics — Inclusion-Exclusion & MorePg 327
A) Inclusion-Exclusion Principle — Pg 327
B) Pigeonhole Principle — Pg 327
C) Catalan Numbers — Pg 328
D) Stirling Numbers — Pg 328
Σ21. Binomial Theorem & Mathematical Induction (Expanded)Pg 330▶
21.1 Binomial Theorem — All Key Facts & ExtensionsPg 330
A) Binomial Coefficients — Pg 330
B) General Term — Pg 330
C) Middle Term(s) — Pg 331
D) Greatest Term — Pg 331
E) Sum of Binomial Coefficients — Pg 332
21.2 Multinomial Theorem & Advanced ExpansionsPg 333
21.3 Mathematical Induction — All Types & TemplatesPg 333
A) Standard Induction — Pg 334
B) Strong Induction — Pg 334
C) Induction with Multiple Base Cases — Pg 335
📶22. Sequences & Series — AP, GP, HP & Special Sums (Expanded)Pg 337▶
22.1 Arithmetic Progression (AP) — All FormulasPg 337
A) General Form of an AP — Pg 337
B) Sum of an AP — Pg 337
C) Sum of an Infinite AP — Pg 337
D) Properties of AP — Pg 338
E) AP in Real Life — Pg 338
22.2 Geometric Progression (GP) — All FormulasPg 338
A) General Form of a GP — Pg 338
B) Sum of a GP — Pg 339
C) Properties of GP — Pg 339
D) GP in Real Life — Pg 339
22.3 Harmonic Progression (HP) & Special SeriesPg 340
A) General Form of an HP — Pg 340
B) Sum of an HP — Pg 341
C) Special Series — Pg 341
22.4 AM ≥ GM ≥ HM & All InequalitiesPg 342
A) Arithmetic Mean (AM) — Pg 342
B) Geometric Mean (GM) — Pg 342
C) Harmonic Mean (HM) — Pg 342
D) AM-GM Inequality — Pg 343
E) GM-HM Inequality — Pg 343
F) AM-HM Inequality — Pg 343
G) Combined AM-GM-HM Inequality — Pg 343
22.5 Infinite Series & Convergence TestsPg 344
A) Geometric Series — Pg 344
B) p-Series — Pg 344
C) Comparison Test — Pg 345
D) Ratio Test — Pg 345
E) Root Test — Pg 346
F) Integral Test — Pg 346
🧩23. Linear Programming & Mathematical Reasoning (Expanded)Pg 348▶
23.1 Linear Programming — Corner Point MethodPg 348
A) Formulating a Linear Programming Problem — Pg 348
B) Graphical Method (Corner Point Method) — Pg 349
C) Corner Point Method for Larger Problems — Pg 350
23.2 Mathematical Reasoning — Truth Tables & LogicPg 350
A) Propositions and Logical Connectives — Pg 350
B) Tautologies and Contradictions — Pg 351
C) Logical Equivalences — Pg 352
23.3 Boolean Algebra & Circuit LogicPg 352
A) Boolean Algebra Basics — Pg 353
B) Logic Gates — Pg 354
🌍24. World Mathematics — Tricks From Every CountryPg 355▶
24.1 Ancient Egyptian Mathematics — Multiplication & DivisionPg 355
A) Egyptian Multiplication — Pg 355
B) Egyptian Division — Pg 356
24.2 Babylonian Mathematics — Base 60 & AstronomyPg 356
A) Babylonian Numerals — Pg 356
B) Babylonian Multiplication — Pg 356
24.3 Chinese Mathematics — Counting Rods & Magic SquaresPg 357
A) Counting Rods — Pg 357
B) Magic Squares — Pg 357
24.4 Japanese Mathematics — Soroban & WasanPg 358
A) Soroban (Japanese Abacus) — Pg 358
B) Wasan (Traditional Japanese Mathematics) — Pg 358
24.5 Islamic Mathematics — Algebra & TrigonometryPg 359
A) Algebra — Pg 359
B) Trigonometry — Pg 360
24.6 Russian Mathematics — Trachtenberg SystemPg 360
A) Trachtenberg Multiplication — Pg 360
B) Trachtenberg Division — Pg 360
24.7 Mayan Mathematics — Base 20 & CalendarPg 361
A) Mayan Numerals — Pg 361
B) Mayan Calendar — Pg 361
24.8 African Math (Ethnomathematics)Pg 362
24.9 Native American Math (Inca Quipu)Pg 362
🔑25. Number Theory — All Theorems & ShortcutsPg 364▶
25.1 Divisibility & Congruence — All RulesPg 364
A) Divisibility by Powers of 2 and 5 — Pg 364
B) Divisibility by 3 and 9 — Pg 364
C) Divisibility by 11 — Pg 364
D) Congruence Modulo n — Pg 365
25.2 Fermat's Little Theorem & Wilson's TheoremPg 365
A) Fermat's Little Theorem — Pg 365
B) Wilson's Theorem — Pg 366
25.3 Chinese Remainder Theorem & Modular ArithmeticPg 366
A) Chinese Remainder Theorem — Pg 366
B) Modular Arithmetic — Pg 367
25.4 Diophantine Equations & Pell's EquationPg 368
A) Linear Diophantine Equations — Pg 368
B) Pell's Equation — Pg 368
25.5 Prime Numbers — Sieve, Tests & PropertiesPg 369
A) Sieve of Eratosthenes — Pg 369
B) Primality Tests — Pg 370
C) Properties of Prime Numbers — Pg 370
25.6 Number Theory in Cryptography — RSA & Diffie-HellmanPg 370
A) RSA Encryption — Pg 371
B) Diffie-Hellman Key Exchange — Pg 372
🕸️26. Discrete Mathematics — All Topics For JEE & BeyondPg 374▶
26.1 Graph Theory — All Basics & ShortcutsPg 374
A) Types of Graphs — Pg 374
B) Graph Terminology — Pg 374
C) Handshaking Lemma — Pg 375
D) Eulerian and Hamiltonian Paths — Pg 375
E) Trees — Pg 376
26.2 Combinatorics — All Identities & MethodsPg 376
A) Binomial Identities — Pg 376
B) Multinomial Identities — Pg 377
C) Inclusion-Exclusion Principle — Pg 377
26.3 Recurrence Relations & Generating FunctionsPg 377
A) Linear Recurrence Relations — Pg 377
B) Generating Functions — Pg 378
26.4 Set Theory — All Operations & CardinalityPg 378
A) Set Operations — Pg 378
B) Cardinality — Pg 379
26.5 Logic — Propositional & PredicatePg 379
A) Propositional Logic — Pg 379
B) Predicate Logic — Pg 379
🧮27. Linear Algebra — All Concepts SimplifiedPg 381▶
27.1 Vector Spaces & Subspaces — All PropertiesPg 381
A) Subspaces — Pg 382
B) Basis and Dimension — Pg 383
27.2 Linear Transformations & MatricesPg 383
A) Matrix Representation — Pg 383
B) Kernel and Image — Pg 384
27.3 Eigenvalues & Eigenvectors — All ShortcutsPg 384
A) Finding Eigenvalues — Pg 384
B) Finding Eigenvectors — Pg 385
C) Diagonalization — Pg 385
27.4 Inner Product Spaces & OrthogonalityPg 385
A) Dot Product in ℝⁿ — Pg 386
B) Orthogonality — Pg 386
C) Orthonormal Basis — Pg 386
27.5 Applications — Data Science, Physics & MorePg 387
A) Data Science — Pg 387
B) Physics — Pg 387
C) Computer Science — Pg 387
🌀28. Complex Analysis — All Theorems & ShortcutsPg 389▶
28.1 Analytic Functions & Cauchy-Riemann EquationsPg 389
A) Cauchy-Riemann Equations — Pg 389
28.2 Contour Integration & Residue TheoremPg 390
A) Contour Integration — Pg 390
B) Cauchy's Integral Theorem — Pg 390
C) Residue Theorem — Pg 390
28.3 Taylor & Laurent SeriesPg 391
A) Taylor Series — Pg 391
B) Laurent Series — Pg 391
28.4 Conformal Mappings & Riemann SurfacesPg 392
A) Conformal Mappings — Pg 392
B) Riemann Surfaces — Pg 392
🧵29. Differential Geometry & Topology BasicsPg 394▶
29.1 Curves in 2D & 3D — Parametric, Tangents & NormalsPg 394
A) Parametric Equations of a Curve — Pg 394
B) Tangent Vector — Pg 394
C) Normal Vector — Pg 395
D) Arc Length — Pg 395
E) Curvature — Pg 396
29.2 Surfaces in 3D — Parametric, Normals & CurvaturePg 396
A) Parametric Equations of a Surface — Pg 396
B) Tangent Plane — Pg 397
C) Normal Vector — Pg 397
D) First Fundamental Form — Pg 398
E) Second Fundamental Form — Pg 399
29.3 Manifolds & Differentiable StructuresPg 400
A) Examples of Manifolds — Pg 400
B) Differentiable Manifolds — Pg 400
C) Tangent Space — Pg 400
29.4 Topology Basics — Homeomorphisms & HomotopyPg 401
A) Topological Spaces — Pg 401
B) Homeomorphisms — Pg 401
C) Connectedness — Pg 402
D) Compactness — Pg 402
E) Homotopy — Pg 402
🎮30. Game Theory & Cryptography — Strategies & AlgorithmsPg 403▶
30.1 Game Theory Basics — Nash Equilibrium & MorePg 403
A) Types of Games — Pg 403
B) Payoff Matrix — Pg 403
C) Nash Equilibrium — Pg 404
D) Dominant Strategies — Pg 404
E) Zero-Sum Games — Pg 405
30.2 Cryptography — RSA, Diffie-Hellman & MorePg 405
A) Symmetric-Key Cryptography — Pg 406
B) Asymmetric-Key Cryptography (Public-Key Cryptography) — Pg 406
C) RSA Encryption (Recap) — Pg 406
D) Diffie-Hellman Key Exchange (Recap) — Pg 407
E) Elliptic Curve Cryptography (ECC) — Pg 407
30.3 Modular Arithmetic in Cryptography — All ShortcutsPg 408
1. Introduction to Modular Arithmetic — Pg 408
2. Applications in Cryptography — Pg 409
3. Shortcuts for Modular Arithmetic — Pg 409
4. Modular Arithmetic in RSA Encryption — Pg 410
5. Modular Arithmetic in Diffie-Hellman — Pg 411
6. Modular Arithmetic in Hash Functions — Pg 411
7. Modular Arithmetic in Digital Signatures — Pg 412
8. Common Modular Arithmetic Tricks for Competitive Exams — Pg 412
30.4 Hash Functions & Digital SignaturesPg 413
1. Introduction to Hash Functions — Pg 413
2. Types of Hash Functions — Pg 413
3. How Hash Functions Work — Pg 414
4. Applications of Hash Functions — Pg 414
5. Common Hash Function Algorithms — Pg 415
6. Hash Collisions and Security — Pg 416
7. Digital Signatures: A Deeper Dive — Pg 416
8. Elliptic Curve Digital Signature Algorithm (ECDSA) — Pg 417
9. Practical Examples of Hash Functions and Digital Signatures — Pg 418
10. Common Pitfalls in Hash Functions and Digital Signatures — Pg 418
💰31. Financial Mathematics — Compound Interest & AnnuitiesPg 420▶
From everyday interest calculations to options pricing — the complete shortcut toolkit for financial maths.
31.1 Simple Interest vs. Compound InterestPg 420
A) Simple Interest — Pg 420
B) Compound Interest — Pg 420
C) Continuous Compounding — Pg 421
D) Rule of 72 — Pg 421
31.2 Annuities & Sinking FundsPg 422
A) Types of Annuities — Pg 422
B) Future Value of an Ordinary Annuity — Pg 422
C) Present Value of an Ordinary Annuity — Pg 423
D) Annuity Due — Pg 423
E) Perpetuity — Pg 423
31.3 Amortization & Loan SchedulesPg 424
A) Amortization Formula — Pg 424
B) Amortization Schedule — Pg 424
31.4 Stocks, Bonds & Investment StrategiesPg 426
A) Stocks — Pg 426
B) Bonds — Pg 426
C) Investment Strategies — Pg 427
31.5 Options & Futures – Black-Scholes & Binomial ModelPg 427
1. Introduction to Derivatives — Pg 427
2. Options Basics — Pg 428
3. Binomial Option Pricing Model — Pg 428
4. Multi-Step Binomial Model — Pg 430
5. Black-Scholes Model — Pg 431
6. Calculating d₁ and d₂ — Pg 432
7. Put-Call Parity — Pg 432
8. Greeks in Options Trading — Pg 433
9. Applications of Options — Pg 434
10. Common Pitfalls in Options Trading — Pg 434
💻32. Computer Science Mathematics — Algorithms & ComplexityPg 436▶
32.1 Algorithms — Sorting, Searching & Big-OPg 436
A) Sorting Algorithms — Pg 436
B) Searching Algorithms — Pg 437
C) Big-O Notation — Pg 438
32.2 Data Structures — Arrays, Trees, GraphsPg 439
A) Arrays — Pg 439
B) Linked Lists — Pg 439
C) Stacks — Pg 440
D) Queues — Pg 440
E) Trees — Pg 440
F) Graphs — Pg 441
32.3 Computational Complexity — P, NP, NP-CompletePg 442
A) Complexity Classes — Pg 442
B) P vs. NP Problem — Pg 443
C) Examples of Problems in P and NP — Pg 443
32.4 Automata Theory & Regular LanguagesPg 443
A) Finite Automata (FA) — Pg 444
B) Regular Languages — Pg 444
C) Pushdown Automata (PDA) — Pg 445
D) Turing Machines — Pg 445
🪐33. Physics Mathematics — Mechanics, E&M & QuantumPg 447▶
33.1 Classical Mechanics — Kinematics & DynamicsPg 447
A) Kinematics — Pg 447
B) Dynamics — Pg 448
C) Work, Energy, and Power — Pg 448
33.2 Electromagnetism — Maxwell's Equations & MorePg 449
A) Coulomb's Law — Pg 450
B) Electric Field — Pg 450
C) Magnetic Field — Pg 450
D) Maxwell's Equations — Pg 451
33.3 Quantum Mechanics — Wave Functions & Schrödinger's EquationPg 451
A) Wave Function — Pg 452
B) Schrödinger's Equation — Pg 452
C) Heisenberg's Uncertainty Principle — Pg 453
D) Particle in a Box — Pg 453
33.4 Relativity — Lorentz Transformations & MorePg 454
1. Introduction to Special Relativity — Pg 454
2. Lorentz Transformations — Pg 454
3. Time Dilation — Pg 455
4. Length Contraction — Pg 455
5. Relativity of Simultaneity — Pg 456
6. Velocity Addition Formula — Pg 456
7. Energy-Momentum Relation — Pg 457
8. Mass-Energy Equivalence — Pg 458
9. Twin Paradox — Pg 458
10. General Relativity Basics — Pg 459
11. Applications of Relativity — Pg 459
12. Common Pitfalls in Relativity — Pg 460
