Mathematics Formula Cheat Sheet

A compact but broad reference of the most-used mathematical formulas.

This sheet is designed for quick lookup rather than derivations. It covers school mathematics, higher-secondary mathematics, undergraduate calculus, coordinate geometry, vectors, matrices, complex numbers, sequences and series, probability, statistics, number theory, discrete mathematics, numerical methods, and common applied formulas.

Important: No finite document can literally contain every formula in mathematics—mathematics is an open-ended discipline with infinitely many identities and specialized results. This version aims to cover the standard formulas most useful for students, competitive programming, engineering, computer science, and general mathematical work.

Arithmetic & Foundations

1. Basic Arithmetic

Order of Operations & Arithmetic Properties

No.	Property / Rule	Formula	Conditions / Notes
1	Order of Operations	$\text{PEMDAS / BODMAS}$	Parentheses/Brackets $\to$ Exponents/Orders $\to$ Multiplication/Division $\to$ Addition/Subtraction (left-to-right for equal rank)
2	Commutative (Addition)	$a + b = b + a$	Order of terms does not affect sum
3	Commutative (Multiplication)	$ab = ba$	Order of factors does not affect product
4	Associative (Addition)	$(a + b) + c = a + (b + c)$	Grouping does not affect sum
5	Associative (Multiplication)	$(ab)c = a(bc)$	Grouping does not affect product
6	Distributive (Over Addition)	$a(b + c) = ab + ac$	Factor distributes across sum
7	Distributive (Over Subtraction)	$a(b - c) = ab - ac$	Factor distributes across difference
8	Additive Identity	$a + 0 = a$	$0$ is the additive identity
9	Multiplicative Identity	$a \cdot 1 = a$	$1$ is the multiplicative identity
10	Additive Inverse	$a + (-a) = 0$	Opposite value yields zero
11	Multiplicative Inverse	$a \cdot \frac{1}{a} = 1$	Reciprocal yields unity ( $a \ne 0$ )

Absolute Value

No.	Property	Formula	Conditions / Notes
12	Piecewise Definition	$\vert x\vert = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}$	Geometric distance from $0$ on number line
13	Absolute Product	$\vert ab\vert = \vert a\vert \vert b\vert$	Magnitude of product equals product of magnitudes
14	Absolute Quotient	$\left\vert\frac{a}{b}\right\vert = \frac{\vert a\vert}{\vert b\vert}$	Magnitude of quotient ( $b \ne 0$ )
15	Triangle Inequality	$\vert a + b\vert \le \vert a\vert + \vert b\vert$	Upper bound on magnitude of sum
16	Reverse Triangle Inequality	$\big\vert\vert a\vert - \vert b\vert\big\vert \le \vert a - b\vert$	Lower bound on magnitude of difference

2. Fractions, Ratios, Proportions and Percentages

Fraction Operations

No.	Operation	Formula	Conditions / Notes
17	Fraction Addition	$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$	$b, d \ne 0$
18	Fraction Subtraction	$\frac{a}{b} - \frac{c}{d} = \frac{ad - bc}{bd}$	$b, d \ne 0$
19	Fraction Multiplication	$\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$	$b, d \ne 0$
20	Fraction Division	$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} = \frac{ad}{bc}$	$b, c, d \ne 0$

Ratios & Proportions

No.	Concept	Formula	Conditions / Notes
21	Ratio Notation	$a : b = \frac{a}{b}$	Comparison of two relative quantities ( $b \ne 0$ )
22	Proportion Cross-Product	$a : b = c : d \iff ad = bc$	Equality of two ratios ( $b, d \ne 0$ )
23	Direct Proportion	$y \propto x \implies y = kx$	$k$ is constant of proportionality
24	Inverse Proportion	$y \propto \frac{1}{x} \implies xy = k$	$k$ is constant of variation

Percentages

No.	Metric	Formula	Conditions / Notes
25	Percentage Calculation	$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%$	Ratio expressed per hundred
26	Part from Percentage	$\text{Part} = \frac{\text{Percentage}}{100} \times \text{Whole}$	Amount corresponding to percentage rate
27	Percentage Change	$\%\text{ Change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%$	Relative change over baseline
28	Successive Changes	$\text{Net Change} = \left(a + b + \frac{ab}{100}\right)\%$	For consecutive percentage changes $a\%$ and $b\%$
29	Percentage Error	$\%\text{ Error} = \frac{\vert\text{Experimental} - \text{True}\vert}{\vert\text{True}\vert} \times 100\%$	Accuracy relative to standard/true value

3. Powers, Exponents and Radicals

Laws of Exponents

No.	Law	Formula	Conditions / Notes
30	Product of Powers	$a^m a^n = a^{m+n}$	Add exponents when multiplying same base
31	Quotient of Powers	$\frac{a^m}{a^n} = a^{m-n}$	Subtract exponents ( $a \ne 0$ )
32	Power of a Power	$(a^m)^n = a^{mn}$	Multiply inner and outer exponents
33	Power of a Product	$(ab)^n = a^n b^n$	Distribute exponent across factors
34	Power of a Quotient	$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$	Distribute exponent across terms ( $b \ne 0$ )
35	Zero Exponent	$a^0 = 1$	Nonzero base to the zero power ( $a \ne 0$ )
36	Negative Exponent	$a^{-n} = \frac{1}{a^n}$	Reciprocal with positive power ( $a \ne 0$ )
37	Unit Fractional Exponent	$a^{1/n} = \sqrt[n]{a}$	Principal $n$ -th root ( $a \ge 0$ for even $n$ )
38	General Rational Exponent	$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$	$m, n \in \mathbb{Z}, n > 0$

Radical Rules & Rationalization

No.	Property	Formula	Conditions / Notes
39	Radical Product	$\sqrt{ab} = \sqrt{a}\sqrt{b}$	$a, b \ge 0$
40	Radical Quotient	$\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$	$a \ge 0, b > 0$
41	Root of a Power	$\sqrt[n]{a^m} = a^{m/n}$	Radical to fractional power equivalence
42	Nested Radicals	$\sqrt[n]{\sqrt[m]{a}} = \sqrt[mn]{a}$	Product of root indices
43	Radical Power Inverse	$(\sqrt[n]{a})^n = a$	$a \ge 0$ if $n$ is even
44	Monomial Rationalization	$\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a}$	Multiply top & bottom by $\sqrt{a}$
45	Binomial Rationalization	$\frac{1}{a + \sqrt{b}} = \frac{a - \sqrt{b}}{a^2 - b}$	Multiply by conjugate $(a - \sqrt{b})$
46	Conjugate Product	$(a + \sqrt{b})(a - \sqrt{b}) = a^2 - b$	Difference of squares eliminates radical

Algebra

4. Algebraic Identities

Squares & Quadratics

No.	Identity	Formula	Conditions / Notes
47	Square of Sum	$(a + b)^2 = a^2 + 2ab + b^2$	Trinomial expansion
48	Square of Difference	$(a - b)^2 = a^2 - 2ab + b^2$	Middle term negative
49	Difference of Squares	$a^2 - b^2 = (a + b)(a - b)$	Factoring conjugate binomials
50	Sum of Squares (via Sum)	$a^2 + b^2 = (a + b)^2 - 2ab$	Expressed using sum and product
51	Sum of Squares (via Diff)	$a^2 + b^2 = (a - b)^2 + 2ab$	Expressed using difference and product

Cubes & Higher Powers

No.	Identity	Formula	Conditions / Notes
52	Cube of Sum	$(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3$	$= a^3 + b^3 + 3ab(a + b)$
53	Cube of Difference	$(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$	$= a^3 - b^3 - 3ab(a - b)$
54	Sum of Cubes	$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$	Linear factor with alternating quadratic
55	Difference of Cubes	$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$	Linear factor with positive quadratic
56	Difference of Fourth Powers	$a^4 - b^4 = (a - b)(a + b)(a^2 + b^2)$	Iterated difference of squares
57	Fourth Power of Sum	$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$	Binomial coefficients: 1, 4, 6, 4, 1
58	Fourth Power of Difference	$(a - b)^4 = a^4 - 4a^3b + 6a^2b^2 - 4ab^3 + b^4$	Alternating sign expansion

Multi-Variable & General Powers

No.	Identity	Formula	Conditions / Notes
59	Square of Trinomial	$(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$	Sum of squares + all pairwise products
60	Alternating Trinomial Square	$(a - b - c)^2 = a^2 + b^2 + c^2 - 2ab - 2ac + 2bc$	Sign distribution rule
61	Sum of Three Cubes	$a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$	Euler’s factorization identity
62	Zero-Sum Cubic Special Case	$a + b + c = 0 \implies a^3 + b^3 + c^3 = 3abc$	Direct consequence of Euler’s identity
63	General Difference of $n$ -th Powers	$a^n - b^n = (a - b)(a^{n-1} + a^{n-2}b + \cdots + b^{n-1})$	Valid for all positive integers $n$
64	General Sum of $n$ -th Powers	$a^n + b^n = (a + b)(a^{n-1} - a^{n-2}b + \cdots + b^{n-1})$	Valid for odd integers $n$

5. Polynomials

Theorems & Quadratics

No.	Concept / Form	Formula	Conditions / Notes
65	Polynomial Form	$P(x) = a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x + a_0$	Degree $n$ with leading coefficient $a_n \ne 0$
66	Remainder Theorem	$\text{Remainder} = P(a)$	Remainder when $P(x)$ is divided by $(x - a)$
67	Factor Theorem	$P(a) = 0 \iff (x - a) \text{ is a factor of } P(x)$	Root-factor equivalence
68	Quadratic Formula	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$	Roots of $ax^2 + bx + c = 0$ ( $a \ne 0$ )
69	Quadratic Discriminant	$\Delta = b^2 - 4ac$	Determines nature and multiplicity of roots
70	Vieta’s Sum of Roots (Quadratic)	$\alpha + \beta = -\frac{b}{a}$	Sum of roots of $ax^2 + bx + c = 0$
71	Vieta’s Product of Roots (Quadratic)	$\alpha\beta = \frac{c}{a}$	Product of roots of $ax^2 + bx + c = 0$
72	Factored Quadratic Form	$ax^2 + bx + c = a(x - \alpha)(x - \beta)$	Factorization via roots $\alpha, \beta$

Cubic Root Relations (Vieta’s Formulas)

No.	Relation	Formula	Conditions / Notes
73	Sum of Roots	$\alpha + \beta + \gamma = -\frac{b}{a}$	For $ax^3 + bx^2 + cx + d = 0$
74	Sum of Pairwise Products	$\alpha\beta + \beta\gamma + \gamma\alpha = \frac{c}{a}$	Pairwise root products
75	Product of Roots	$\alpha\beta\gamma = -\frac{d}{a}$	Product of all three roots

6. Equations and Inequalities

Linear & Systems of Equations

No.	Form / Concept	Formula	Conditions / Notes
76	Linear Equation	$ax + b = 0 \implies x = -\frac{b}{a}$	$a \ne 0$
77	System Determinant $D$	$D = a_1 b_2 - a_2 b_1$	Determinant of $2\times 2$ coefficient matrix
78	Cramer’s Rule for $x$	$x = \frac{c_1 b_2 - c_2 b_1}{D}$	Unique solution when $D \ne 0$
79	Cramer’s Rule for $y$	$y = \frac{a_1 c_2 - a_2 c_1}{D}$	Unique solution when $D \ne 0$

Quadratic Nature & Parabola Vertex

No.	Feature	Formula	Conditions / Notes
80	Two Distinct Real Roots	$\Delta = b^2 - 4ac > 0$	Graph intersects $x$ -axis twice
81	One Repeated Real Root	$\Delta = b^2 - 4ac = 0$	Graph tangent to $x$ -axis at double root
82	Complex Conjugate Roots	$\Delta = b^2 - 4ac < 0$	No real roots; graph does not touch $x$ -axis
83	Parabola Vertex $x$ -Coordinate	$x_v = -\frac{b}{2a}$	Axis of symmetry: $x = -\frac{b}{2a}$
84	Parabola Vertex $y$ -Coordinate	$y_v = f\left(-\frac{b}{2a}\right) = \frac{4ac - b^2}{4a}$	Extremum value of quadratic function
85	Parabola Vertex Form	$y = a(x - h)^2 + k$	Vertex located at $(h, k)$

Absolute-Value Equations & Inequalities

No.	Expression	Solution / Equivalent Form	Conditions / Notes
86	Absolute Value Equality	$\vert x\vert = a \implies x = \pm a$	Valid for $a \ge 0$
87	Absolute Value Less-Than	$\vert x\vert < a \iff -a < x < a$	Bounded interval ( $a > 0$ )
88	Absolute Value Greater-Than	$\vert x\vert > a \iff x < -a \text{ or } x > a$	Disjoint exterior rays ( $a > 0$ )

7. Logarithms

Definitions & Core Properties

No.	Property / Rule	Formula	Conditions / Notes
89	Logarithm Definition	$\log_a b = c \iff a^c = b$	$a > 0, a \ne 1, b > 0$
90	Logarithm of 1	$\log_a 1 = 0$	$a^0 = 1$
91	Logarithm of Base	$\log_a a = 1$	$a^1 = a$
92	Log of Base Power	$\log_a(a^x) = x$	Inverse cancellation law
93	Base to Log Power	$a^{\log_a x} = x$	Inverse cancellation law ( $x > 0$ )

Operational Laws & Base Conversion

No.	Law	Formula	Conditions / Notes
94	Product Rule	$\log_a(xy) = \log_a x + \log_a y$	$x, y > 0$
95	Quotient Rule	$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$	$x, y > 0$
96	Power Rule	$\log_a(x^r) = r\log_a x$	$x > 0$
97	Change of Base Formula	$\log_a x = \frac{\log_b x}{\log_b a}$	Convert to any convenient base $b > 0, b \ne 1$
98	Change to Natural Log	$\log_a x = \frac{\ln x}{\ln a}$	Base $e$ conversion
99	Change to Common Log	$\log_a x = \frac{\log x}{\log a}$	Base $10$ conversion
100	Reciprocal Identity	$\log_a b = \frac{1}{\log_b a}$	Inverts base and argument ( $b \ne 1$ )
101	Natural Log Definition	$\ln x = \log_e x$	Base $e \approx 2.71828$
102	Exponential-Log Identity	$e^{\ln x} = x \text{ and } \ln(e^x) = x$	$x > 0$ for $e^{\ln x}$

8. Sequences and Series

Arithmetic Progression (AP)

No.	Form / Concept	Formula	Conditions / Notes
103	$n$ -th Term of AP	$a_n = a + (n - 1)d$	First term $a$ , common difference $d$
104	Sum of First $n$ Terms (with $d$ )	$S_n = \frac{n}{2}[2a + (n - 1)d]$	Total sum of $n$ terms
105	Sum of First $n$ Terms (with $l$ )	$S_n = \frac{n}{2}(a + l)$	$l = a_n$ is the last term
106	Arithmetic Mean	$\text{AM} = \frac{a + b}{2}$	Midpoint between $a$ and $b$

Geometric Progression (GP)

No.	Form / Concept	Formula	Conditions / Notes
107	$n$ -th Term of GP	$a_n = a r^{n-1}$	First term $a$ , common ratio $r$
108	Finite GP Sum ( $r \ne 1$ )	$S_n = a\frac{1 - r^n}{1 - r} = a\frac{r^n - 1}{r - 1}$	$r \ne 1$
109	Finite GP Sum ( $r = 1$ )	$S_n = na$	Identical repeated terms
110	Infinite Geometric Series	$S_\infty = \frac{a}{1 - r}$	Converges iff $\vert r\vert < 1$
111	Geometric Mean	$\text{GM} = \sqrt{ab}$	$a, b > 0$

Harmonic Progression & Standard Series Sums

No.	Formula / Series	Expression	Conditions / Notes
112	Harmonic Progression Definition	$a, b, c \in \text{HP} \iff \frac{1}{a}, \frac{1}{b}, \frac{1}{c} \in \text{AP}$	Reciprocal sequence form
113	Harmonic Mean (Two Numbers)	$\text{HM} = \frac{2ab}{a + b}$	$a, b > 0$
114	Harmonic Mean ( $n$ Numbers)	$\text{HM} = \frac{n}{\sum_{i=1}^n \frac{1}{x_i}}$	Reciprocal of average of reciprocals
115	AM-GM-HM Inequality	$\text{AM} \ge \text{GM} \ge \text{HM}$	Equality holds iff all terms are equal
116	Sum of First $n$ Integers	$\sum_{k=1}^n k = \frac{n(n + 1)}{2}$	Triangular numbers
117	Sum of First $n$ Squares	$\sum_{k=1}^n k^2 = \frac{n(n + 1)(2n + 1)}{6}$	Sum of squares formula
118	Sum of First $n$ Cubes	$\sum_{k=1}^n k^3 = \left[\frac{n(n + 1)}{2}\right]^2$	Square of the sum of integers

9. Binomial Theorem

No.	Formula / Identity	Expression	Conditions / Notes
119	Binomial Expansion	$(a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$	Nonnegative integer $n$
120	Binomial Coefficient	$\binom{n}{k} = \frac{n!}{k!(n - k)!}$	Combinations formula
121	General Term $T_{k+1}$	$T_{k+1} = \binom{n}{k} a^{n-k} b^k$	$(k+1)$ -th term of expansion
122	Boundary Values	$\binom{n}{0} = \binom{n}{n} = 1$	Endpoints of row in Pascal’s triangle
123	Symmetry Identity	$\binom{n}{k} = \binom{n}{n - k}$	Symmetry of Pascal’s triangle
124	Pascal’s Recurrence	$\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$	Additive rule of Pascal’s triangle
125	Sum of Coefficients	$\sum_{k=0}^n \binom{n}{k} = 2^n$	Total number of subsets of $n$ elements
126	Alternating Coefficient Sum	$\sum_{k=0}^n (-1)^k \binom{n}{k} = 0$	Valid for $n > 0$

10. Permutations and Combinations

No.	Concept / Rule	Formula	Conditions / Notes
127	Factorial Definition	$n! = n(n - 1)(n - 2)\cdots 2\cdot 1$	Product of first $n$ positive integers
128	Factorial of Zero	$0! = 1$	By definition / empty product
129	Permutations without Repetition	${}^n P_r = \frac{n!}{(n - r)!}$	Ordered arrangements of $r$ from $n$ distinct
130	Combinations without Repetition	${}^n C_r = \binom{n}{r} = \frac{n!}{r!(n - r)!}$	Unordered selections of $r$ from $n$ distinct
131	Permutation-Combination Relation	${}^n P_r = {}^n C_r \cdot r!$	Accounts for $r!$ internal orderings
132	Permutations with Repetition	$\frac{n!}{n_1! n_2! \cdots n_k!}$	Groups of identical objects of sizes $n_i$
133	Circular Permutations (Distinct)	$(n - 1)!$	Distinct clockwise vs counterclockwise
134	Circular Permutations (Unoriented)	$\frac{(n - 1)!}{2}$	Flipping allowed (e.g. bead necklace)
135	Combinations with Repetition	$\binom{n + r - 1}{r}$	Stars and bars selection of $r$ items from $n$ types

11. Set Theory

No.	Law / Identity	Formula	Conditions / Notes
136	Inclusion-Exclusion (2 Sets)	$\vert A \cup B\vert = \vert A\vert + \vert B\vert - \vert A \cap B\vert$	Prevents double-counting intersection
137	Inclusion-Exclusion (3 Sets)	$\vert A \cup B \cup C\vert = \vert A\vert + \vert B\vert + \vert C\vert - \sum \vert A \cap B\vert + \vert A \cap B \cap C\vert$	Alternating inclusion-exclusion
138	Complement Cardinality	$\vert A^c\vert = \vert U\vert - \vert A\vert$	Elements outside $A$ in universe $U$
139	Set Difference	$A - B = A \cap B^c$	Elements in $A$ but not in $B$
140	De Morgan’s Law (Union)	$(A \cup B)^c = A^c \cap B^c$	Complement of union is intersection of complements
141	De Morgan’s Law (Intersection)	$(A \cap B)^c = A^c \cup B^c$	Complement of intersection is union of complements
142	Cartesian Product Cardinality	$\vert A \times B\vert = \vert A\vert \vert B\vert$	Ordered pairs $(a, b)$
143	Power Set Cardinality	$\vert\mathcal{P}(A)\vert = 2^n$	Total distinct subsets of set of size $n$

12. Functions

No.	Concept	Formula	Conditions / Notes
144	Composite Function	$(f \circ g)(x) = f(g(x))$	Output of $g$ becomes input to $f$
145	Inverse Function Identity	$f(f^{-1}(x)) = x \text{ and } f^{-1}(f(x)) = x$	Bidirectional cancellation
146	Even Function	$f(-x) = f(x)$	Symmetric about the $y$ -axis
147	Odd Function	$f(-x) = -f(x)$	Symmetric about the origin (rotational)
148	Average Rate of Change	$\frac{f(b) - f(a)}{b - a}$	Secant line slope over $[a, b]$
149	Linear Function	$f(x) = mx + c$	Constant slope $m$ , $y$ -intercept $c$

Geometry

13. Coordinate Geometry

No.	Formula Name	Formula	Conditions / Notes
150	Distance Formula	$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$	Euclidean distance between $P_1(x_1, y_1)$ and $P_2(x_2, y_2)$
151	Midpoint Formula	$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$	Halfway point of line segment
152	Internal Section Formula	$P = \left(\frac{mx_2 + nx_1}{m + n}, \frac{my_2 + ny_1}{m + n}\right)$	Divides segment internally in ratio $m:n$
153	External Section Formula	$P = \left(\frac{mx_2 - nx_1}{m - n}, \frac{my_2 - ny_1}{m - n}\right)$	Divides segment externally in ratio $m:n$ ( $m \ne n$ )
154	Triangle Centroid	$G = \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right)$	Intersection of three medians
155	Area from Coordinates	$A = \frac{1}{2}\vert x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)\vert$	Shoelace formula for vertices $(x_i, y_i)$
156	Collinearity Condition	$x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0$	Area of triangle is zero iff points are collinear

14. Straight Lines

No.	Form / Concept	Equation / Formula	Conditions / Notes
157	Slope of Line	$m = \frac{y_2 - y_1}{x_2 - x_1}$	Line through $(x_1, y_1)$ and $(x_2, y_2)$ ( $x_1 \ne x_2$ )
158	Point-Slope Form	$y - y_1 = m(x - x_1)$	Line through $(x_1, y_1)$ with slope $m$
159	Slope-Intercept Form	$y = mx + c$	Slope $m$ , $y$ -intercept $(0, c)$
160	Two-Point Form	$y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$	Line passing through two designated points
161	Intercept Form	$\frac{x}{a} + \frac{y}{b} = 1$	$x$ -intercept $a$ , $y$ -intercept $b$ ( $a, b \ne 0$ )
162	General Form Slope	$Ax + By + C = 0 \implies m = -\frac{A}{B}$	Slope of general linear form ( $B \ne 0$ )
163	Angle Between Lines	$\tan\theta = \left\vert\frac{m_2 - m_1}{1 + m_1 m_2}\right\vert$	Acute angle between slopes $m_1, m_2$
164	Perpendicular Lines	$m_1 m_2 = -1$	Slopes are negative reciprocals
165	Parallel Lines	$m_1 = m_2$	Slopes are equal
166	Point to Line Distance	$d = \frac{\vert Ax_1 + By_1 + C\vert}{\sqrt{A^2 + B^2}}$	Perpendicular distance from $(x_1, y_1)$ to $Ax + By + C = 0$
167	Distance Between Parallel Lines	$d = \frac{\vert C_1 - C_2\vert}{\sqrt{A^2 + B^2}}$	Between $Ax + By + C_1 = 0$ and $Ax + By + C_2 = 0$

15. Circles

No.	Property / Form	Formula	Conditions / Notes
168	Standard Equation	$(x - h)^2 + (y - k)^2 = r^2$	Center $(h, k)$ , radius $r$
169	Center at Origin	$x^2 + y^2 = r^2$	Center $(0, 0)$ , radius $r$
170	General Equation Center	$\left(-\frac{D}{2}, -\frac{E}{2}\right)$	For $x^2 + y^2 + Dx + Ey + F = 0$
171	General Equation Radius	$r = \frac{1}{2}\sqrt{D^2 + E^2 - 4F}$	Real circle requires $D^2 + E^2 - 4F > 0$
172	Circumference	$C = 2\pi r = \pi d$	Perimeter of circle
173	Area	$A = \pi r^2 = \frac{\pi d^2}{4}$	Total enclosed surface
174	Arc Length (Radians)	$s = r\theta$	Angle $\theta$ in radians
175	Arc Length (Degrees)	$s = \frac{\theta}{360^\circ} 2\pi r$	Angle $\theta$ in degrees
176	Sector Area (Radians)	$A = \frac{1}{2}r^2\theta$	Central angle $\theta$ in radians
177	Sector Area (Degrees)	$A = \frac{\theta}{360^\circ}\pi r^2$	Central angle $\theta$ in degrees
178	Chord Length	$c = 2r\sin\left(\frac{\theta}{2}\right)$	Central angle $\theta$

16. Conic Sections

Parabola

No.	Form / Property	Formula	Conditions / Notes
179	Horizontal Standard Parabola	$y^2 = 4ax$	Opens right ( $a > 0$ ) or left ( $a < 0$ )
180	Vertical Standard Parabola	$x^2 = 4ay$	Opens up ( $a > 0$ ) or down ( $a < 0$ )
181	Parabola Focus ( $y^2=4ax$ )	Focus: $(a, 0)$	Directrix: $x = -a$
182	Parabola Vertex & Latus Rectum	Vertex: $(0, 0)$ , Latus Rectum: $4a$	Focal chord length perpendicular to axis

Ellipse

No.	Form / Property	Formula	Conditions / Notes
183	Horizontal Ellipse Equation	$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$	Major axis horizontal ( $a > b > 0$ )
184	Vertical Ellipse Equation	$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$	Major axis vertical ( $a > b > 0$ )
185	Focal Distance Relation	$c^2 = a^2 - b^2$	$c$ is distance from center to foci
186	Eccentricity	$e = \frac{c}{a} = \sqrt{1 - \frac{b^2}{a^2}}$	$0 < e < 1$ for an ellipse
187	Foci Coordinates	(\pm c, 0)	Foci located on major axis
188	Ellipse Area	$A = \pi ab$	Total enclosed area

Hyperbola

No.	Form / Property	Formula	Conditions / Notes
189	Horizontal Hyperbola Equation	$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$	Transverse axis on $x$ -axis
190	Vertical Hyperbola Equation	$\frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$	Transverse axis on $y$ -axis
191	Focal Distance Relation	$c^2 = a^2 + b^2$	$c$ is distance from center to foci
192	Eccentricity	$e = \frac{c}{a} = \sqrt{1 + \frac{b^2}{a^2}}$	$e > 1$ for a hyperbola
193	Asymptotes (Horizontal Form)	$y = \pm \frac{b}{a}x$	Slant asymptotes through origin
194	Asymptotes (Vertical Form)	$y = \pm \frac{a}{b}x$	Slant asymptotes through origin

17. Plane Geometry

No.	Concept / Theorem	Formula	Conditions / Notes
195	Straight Angle	$180^\circ = \pi \text{ rad}$	Angle on a straight line
196	Full Revolution	$360^\circ = 2\pi \text{ rad}$	Complete circular angle around a point
197	Complementary Angles	$A + B = 90^\circ$	Two angles sum to right angle
198	Supplementary Angles	$A + B = 180^\circ$	Two angles sum to straight angle
199	Interior Angle Sum ( $n$ -gon)	$S = (n - 2)180^\circ$	Total interior angles of any $n$ -sided polygon
200	Regular $n$ -gon Interior Angle	$\theta_{\text{int}} = \frac{(n - 2)180^\circ}{n}$	Each interior angle of regular polygon
201	Sum of Exterior Angles	$S_{\text{ext}} = 360^\circ$	Exterior angles sum to $360^\circ$ for any convex polygon
202	Regular $n$ -gon Exterior Angle	$\theta_{\text{ext}} = \frac{360^\circ}{n}$	Each exterior angle of regular polygon
203	Number of Diagonals	$D = \frac{n(n - 3)}{2}$	Total diagonals connecting non-adjacent vertices
204	Similar Figures Area Ratio	$\frac{A_2}{A_1} = k^2$	For linear scale factor $k = \frac{L_2}{L_1}$
205	Similar Figures Volume Ratio	$\frac{V_2}{V_1} = k^3$	For linear scale factor $k = \frac{L_2}{L_1}$

18. Triangles

No.	Formula Name	Formula	Conditions / Notes
206	Basic Area	$A = \frac{1}{2}bh$	Base $b$ , perpendicular height $h$
207	Semiperimeter	$s = \frac{a + b + c}{2}$	Half of triangle perimeter
208	Heron’s Formula	$A = \sqrt{s(s - a)(s - b)(s - c)}$	Area from three side lengths $a, b, c$
209	Equilateral Triangle Height	$h = \frac{\sqrt{3}}{2}a$	For side length $a$
210	Equilateral Triangle Area	$A = \frac{\sqrt{3}}{4}a^2$	Area of equilateral triangle
211	Pythagorean Theorem	$a^2 + b^2 = c^2$	Right triangle with hypotenuse $c$
212	Law of Sines	$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R$	$R$ is circumradius
213	Law of Cosines	$a^2 = b^2 + c^2 - 2bc\cos A$	Solves SAS and SSS configurations
214	Area via Sine (SAS)	$A = \frac{1}{2}bc\sin A = \frac{1}{2}ca\sin B = \frac{1}{2}ab\sin C$	Area from two sides and included angle
215	Circumradius Formula	$R = \frac{abc}{4A}$	Radius of circumscribed circle
216	Inradius Formula	$r = \frac{A}{s} \implies A = rs$	Radius of inscribed circle
217	Euler’s Triangle Relation	$OI^2 = R(R - 2r)$	Distance between circumcenter $O$ and incenter $I$
218	Median Length (to side $a$ )	$m_a = \frac{1}{2}\sqrt{2b^2 + 2c^2 - a^2}$	Apollonius theorem for median length
219	Angle Bisector (to side $a$ )	$t_a = \frac{2bc\cos(A/2)}{b + c}$	Internal bisector of angle $A$

19. Quadrilaterals and Polygons

No.	Shape	Property / Formula	Conditions / Notes
220	Square	$P = 4a,\; A = a^2,\; d = a\sqrt{2}$	Side length $a$ , diagonal $d$
221	Rectangle	$P = 2(l + w),\; A = lw,\; d = \sqrt{l^2 + w^2}$	Length $l$ , width $w$ , diagonal $d$
222	Parallelogram	$A = bh,\; P = 2(a + b)$	Base $b$ , height $h$ , adjacent side $a$
223	Rhombus	$A = \frac{1}{2}d_1 d_2,\; P = 4a$	Diagonals $d_1, d_2$ are perpendicular
224	Trapezoid	$A = \frac{1}{2}(a + b)h$	Parallel bases $a, b$ with height $h$
225	Kite	$A = \frac{1}{2}d_1 d_2$	Diagonals $d_1, d_2$ are perpendicular

20. 3D Geometry and Mensuration

No.	Solid	Formulas	Conditions / Notes
226	Cube	$V = a^3,\; S = 6a^2,\; d = a\sqrt{3}$	Edge $a$ , total surface area $S$ , space diagonal $d$
227	Rectangular Prism (Cuboid)	$V = lwh,\; S = 2(lw + lh + wh),\; d = \sqrt{l^2 + w^2 + h^2}$	Length $l$ , width $w$ , height $h$
228	Right Circular Cylinder	$V = \pi r^2 h,\; \text{CSA} = 2\pi rh,\; \text{TSA} = 2\pi r(h + r)$	Base radius $r$ , height $h$
229	Right Circular Cone	$l = \sqrt{r^2 + h^2},\; V = \frac{1}{3}\pi r^2 h,\; \text{CSA} = \pi rl,\; \text{TSA} = \pi r(l + r)$	Slant height $l$ , radius $r$ , vertical height $h$
230	Sphere	$S = 4\pi r^2,\; V = \frac{4}{3}\pi r^3$	Surface area $S$ , volume $V$
231	Hemisphere	$\text{CSA} = 2\pi r^2,\; \text{TSA} = 3\pi r^2,\; V = \frac{2}{3}\pi r^3$	Curved and total surface area
232	Right Prism	$V = Bh,\; \text{LSA} = P_{\text{base}} h$	Base area $B$ , base perimeter $P_{\text{base}}$
233	Regular Pyramid	$V = \frac{1}{3}Bh$	Base area $B$ , vertical height $h$

Trigonometry

21. Trigonometry

Right-Triangle Ratios & Reciprocals

No.	Ratio / Function	Formula	Conditions / Notes
234	Sine Ratio	$\sin\theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$	Right triangle definition
235	Cosine Ratio	$\cos\theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$	Right triangle definition
236	Tangent Ratio	$\tan\theta = \frac{\text{Opposite}}{\text{Adjacent}}$	Right triangle definition
237	Cosecant Ratio	$\csc\theta = \frac{1}{\sin\theta}$	Reciprocal of sine
238	Secant Ratio	$\sec\theta = \frac{1}{\cos\theta}$	Reciprocal of cosine
239	Cotangent Ratio	$\cot\theta = \frac{1}{\tan\theta}$	Reciprocal of tangent
240	Tangent Quotient	$\tan\theta = \frac{\sin\theta}{\cos\theta}$	Ratio of sine to cosine
241	Cotangent Quotient	$\cot\theta = \frac{\cos\theta}{\sin\theta}$	Ratio of cosine to sine

Pythagorean, Parity & Cofunction Identities

No.	Identity	Formula	Conditions / Notes
242	Pythagorean Identity I	$\sin^2\theta + \cos^2\theta = 1$	Fundamental identity on unit circle
243	Pythagorean Identity II	$1 + \tan^2\theta = \sec^2\theta$	Divide by $\cos^2\theta$
244	Pythagorean Identity III	$1 + \cot^2\theta = \csc^2\theta$	Divide by $\sin^2\theta$
245	Sine Parity (Odd)	$\sin(-x) = -\sin x$	Odd function
246	Cosine Parity (Even)	$\cos(-x) = \cos x$	Even function
247	Tangent Parity (Odd)	$\tan(-x) = -\tan x$	Odd function
248	Cotangent Parity (Odd)	$\cot(-x) = -\cot x$	Odd function
249	Secant Parity (Even)	$\sec(-x) = \sec x$	Even function
250	Cosecant Parity (Odd)	$\csc(-x) = -\csc x$	Odd function
251	Sine Cofunction	$\sin\left(\frac{\pi}{2} - \theta\right) = \cos\theta$	Complementary angle relation
252	Cosine Cofunction	$\cos\left(\frac{\pi}{2} - \theta\right) = \sin\theta$	Complementary angle relation
253	Tangent Cofunction	$\tan\left(\frac{\pi}{2} - \theta\right) = \cot\theta$	Complementary angle relation

Compound, Multiple & Half-Angle Formulas

No.	Identity Name	Formula	Conditions / Notes
254	Sine of Sum	$\sin(A + B) = \sin A\cos B + \cos A\sin B$	Compound angle formula
255	Sine of Difference	$\sin(A - B) = \sin A\cos B - \cos A\sin B$	Compound angle formula
256	Cosine of Sum	$\cos(A + B) = \cos A\cos B - \sin A\sin B$	Compound angle formula
257	Cosine of Difference	$\cos(A - B) = \cos A\cos B + \sin A\sin B$	Compound angle formula
258	Tangent of Sum	$\tan(A + B) = \frac{\tan A + \tan B}{1 - \tan A\tan B}$	Compound angle formula
259	Tangent of Difference	$\tan(A - B) = \frac{\tan A - \tan B}{1 + \tan A\tan B}$	Compound angle formula
260	Sine Double Angle	$\sin 2\theta = 2\sin\theta\cos\theta$	Double angle formula
261	Cosine Double Angle (I)	$\cos 2\theta = \cos^2\theta - \sin^2\theta$	Double angle formula
262	Cosine Double Angle (II)	$\cos 2\theta = 2\cos^2\theta - 1$	Cosine-only form
263	Cosine Double Angle (III)	$\cos 2\theta = 1 - 2\sin^2\theta$	Sine-only form
264	Tangent Double Angle	$\tan 2\theta = \frac{2\tan\theta}{1 - \tan^2\theta}$	Double angle formula
265	Sine Triple Angle	$\sin 3\theta = 3\sin\theta - 4\sin^3\theta$	Triple angle expansion
266	Cosine Triple Angle	$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$	Triple angle expansion
267	Tangent Triple Angle	$\tan 3\theta = \frac{3\tan\theta - \tan^3\theta}{1 - 3\tan^2\theta}$	Triple angle expansion
268	Sine Half Angle	$\sin\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 - \cos\theta}{2}}$	Sign determined by quadrant of $\theta/2$
269	Cosine Half Angle	$\cos\left(\frac{\theta}{2}\right) = \pm\sqrt{\frac{1 + \cos\theta}{2}}$	Sign determined by quadrant of $\theta/2$
270	Tangent Half Angle	$\tan\left(\frac{\theta}{2}\right) = \frac{1 - \cos\theta}{\sin\theta} = \frac{\sin\theta}{1 + \cos\theta}$	Rational half-angle forms

Product-Sum & Power Reduction Transformations

No.	Transformation	Formula	Conditions / Notes
271	Product to Sum: $\sin\sin$	$\sin A\sin B = \frac{1}{2}[\cos(A - B) - \cos(A + B)]$	Converts products to sums
272	Product to Sum: $\cos\cos$	$\cos A\cos B = \frac{1}{2}[\cos(A - B) + \cos(A + B)]$	Converts products to sums
273	Product to Sum: $\sin\cos$	$\sin A\cos B = \frac{1}{2}[\sin(A + B) + \sin(A - B)]$	Converts products to sums
274	Sum to Product: $\sin + \sin$	$\sin A + \sin B = 2\sin\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)$	Converts sums to products
275	Sum to Product: $\sin - \sin$	$\sin A - \sin B = 2\cos\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)$	Converts differences to products
276	Sum to Product: $\cos + \cos$	$\cos A + \cos B = 2\cos\left(\frac{A + B}{2}\right)\cos\left(\frac{A - B}{2}\right)$	Converts sums to products
277	Sum to Product: $\cos - \cos$	$\cos A - \cos B = -2\sin\left(\frac{A + B}{2}\right)\sin\left(\frac{A - B}{2}\right)$	Converts differences to products
278	Sine Power Reduction	$\sin^2\theta = \frac{1 - \cos 2\theta}{2}$	Useful in calculus integrations
279	Cosine Power Reduction	$\cos^2\theta = \frac{1 + \cos 2\theta}{2}$	Useful in calculus integrations
280	Tangent Power Reduction	$\tan^2\theta = \frac{1 - \cos 2\theta}{1 + \cos 2\theta}$	Ratio of power reduction forms

22. Inverse Trigonometry

No.	Property / Identity	Expression	Conditions / Notes
281	$\arcsin x$ Principal Range	$\arcsin x \in \left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$	Domain: $x \in [-1, 1]$
282	$\arccos x$ Principal Range	$\arccos x \in [0, \pi]$	Domain: $x \in [-1, 1]$
283	$\arctan x$ Principal Range	$\arctan x \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$	Domain: $x \in \mathbb{R}$
284	Complementary Arc Sum	$\arcsin x + \arccos x = \frac{\pi}{2}$	Valid for $x \in [-1, 1]$
285	Reciprocal Argument $\arctan$	$\arctan x + \arctan\left(\frac{1}{x}\right) = \begin{cases} \frac{\pi}{2}, & x > 0 \\ -\frac{\pi}{2}, & x < 0 \end{cases}$	Reciprocal argument identity
286	Addition of $\arctan$	$\arctan x + \arctan y = \arctan\left(\frac{x + y}{1 - xy}\right)$	For $xy < 1$ (quadrant adjustments if $xy > 1$ )

Complex Numbers

23. Complex Numbers

No.	Concept / Property	Formula	Conditions / Notes
287	Standard Algebraic Form	$z = a + bi$	$a = \operatorname{Re}(z), b = \operatorname{Im}(z), i^2 = -1$
288	Powers of $i$ Cycle	$i^0=1,\; i^1=i,\; i^2=-1,\; i^3=-i,\; i^4=1$	Powers repeat with period 4
289	Complex Conjugate	$\bar{z} = a - bi$	Reflection across real axis
290	Modulus (Absolute Value)	$\vert z\vert = \sqrt{a^2 + b^2}$	Euclidean distance from origin in complex plane
291	Modulus-Conjugate Relation	$z\bar{z} = \vert z\vert^2$	Crucial for division / rationalization
292	Complex Addition	$(a + bi) + (c + di) = (a + c) + (b + d)i$	Add real and imaginary parts
293	Complex Multiplication	$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$	FOIL expansion with $i^2 = -1$
294	Complex Division	$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$	Multiply numerator & denominator by $\bar{w}$
295	Polar Trigonometric Form	$z = r(\cos\theta + i\sin\theta)$	$r = \vert z\vert, \theta = \operatorname{arg}(z)$
296	Euler’s Formula	$e^{i\theta} = \cos\theta + i\sin\theta$	Bridges complex exponential and trigonometry
297	Exponential Complex Form	$z = r e^{i\theta}$	Compact polar representation
298	De Moivre’s Theorem	$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$	$(e^{i\theta})^n = e^{in\theta}$ for $n \in \mathbb{Z}$
299	$n$ -th Roots of Complex Number	$z_k = r^{1/n} e^{i\frac{\theta + 2k\pi}{n}}$	$k = 0, 1, \ldots, n-1$ (equally spaced roots)
300	Euler’s Identity	$e^{i\pi} + 1 = 0$	Fundamental identity linking 5 core constants

Differentiation

24. Limits

No.	Limit Law / Standard Limit	Formula	Conditions / Notes
301	Derivative Definition Limit	$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$	Instantaneous rate of change
302	Fundamental Trig Limit I	$\lim_{x \to 0} \frac{\sin x}{x} = 1$	Angle $x$ measured in radians
303	Fundamental Trig Limit II	$\lim_{x \to 0} \frac{\tan x}{x} = 1$	Angle $x$ measured in radians
304	Cosine Limit	$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$	Derived via half-angle identity
305	Exponential Limit	$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$	Derivative of $e^x$ at $x=0$
306	Logarithmic Limit	$\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$	Derivative of $\ln x$ at $x=1$
307	Euler’s Number Limit	$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$	Continuous compounding base
308	Generalized Euler Limit	$\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^x = e^a$	Exponential power limit
309	Sum & Product Laws	$\lim(f \pm g) = \lim f \pm \lim g,\; \lim(fg) = (\lim f)(\lim g)$	Provided limits exist independently
310	Quotient Law	$\lim \frac{f}{g} = \frac{\lim f}{\lim g}$	Valid when $\lim g \ne 0$

25. Differentiation

Basic & Core Differentiation Rules

No.	Rule	Formula	Conditions / Notes
311	Constant Rule	$\frac{d}{dx}(c) = 0$	Derivative of constant is zero
312	Power Rule	$\frac{d}{dx}(x^n) = n x^{n-1}$	Valid for all real powers $n$
313	Constant Multiple Rule	$\frac{d}{dx}[c f(x)] = c f'(x)$	Scalar factors out of derivative
314	Sum and Difference Rule	$\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$	Linearity of derivative operator
315	Product Rule	$\frac{d}{dx}[f g] = f' g + f g'$	Leibniz product rule
316	Quotient Rule	$\frac{d}{dx}\left[\frac{f}{g}\right] = \frac{f' g - f g'}{g^2}$	Valid where $g(x) \ne 0$
317	Chain Rule (Function Form)	$\frac{d}{dx}[f(g(x))] = f'(g(x)) g'(x)$	Derivative of composite functions
318	Chain Rule (Leibniz Form)	$\frac{dy}{dx} = \frac{dy}{du}\frac{du}{dx}$	Chain rule with intermediate variable $u$

Derivatives of Elementary Functions

No.	Function	Derivative	Conditions / Notes
319	Natural Exponential	$\frac{d}{dx}(e^x) = e^x$	Eigenfunction of differentiation
320	Composite Exponential	$\frac{d}{dx}(e^u) = e^u u'$	Chain rule form ( $u = u(x)$ )
321	General Exponential	$\frac{d}{dx}(a^x) = a^x \ln a$	$a > 0, a \ne 1$
322	Natural Logarithm	$\frac{d}{dx}(\ln x) = \frac{1}{x}$	$x > 0$
323	General Logarithm	$\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}$	$a > 0, a \ne 1, x > 0$
324	Sine Derivative	$\frac{d}{dx}(\sin x) = \cos x$	Radians required
325	Cosine Derivative	$\frac{d}{dx}(\cos x) = -\sin x$	Radians required
326	Tangent Derivative	$\frac{d}{dx}(\tan x) = \sec^2 x$	Radians required
327	Cotangent Derivative	$\frac{d}{dx}(\cot x) = -\csc^2 x$	Radians required
328	Secant Derivative	$\frac{d}{dx}(\sec x) = \sec x\tan x$	Radians required
329	Cosecant Derivative	$\frac{d}{dx}(\csc x) = -\csc x\cot x$	Radians required
330	Inverse Sine Derivative	$\frac{d}{dx}(\arcsin x) = \frac{1}{\sqrt{1 - x^2}}$	$\vert x\vert < 1$
331	Inverse Cosine Derivative	$\frac{d}{dx}(\arccos x) = -\frac{1}{\sqrt{1 - x^2}}$	$\vert x\vert < 1$
332	Inverse Tangent Derivative	$\frac{d}{dx}(\arctan x) = \frac{1}{1 + x^2}$	Valid for all $x \in \mathbb{R}$
333	Hyperbolic Sine	$\frac{d}{dx}(\sinh x) = \cosh x$	$\sinh x = \frac{e^x - e^{-x}}{2}$
334	Hyperbolic Cosine	$\frac{d}{dx}(\cosh x) = \sinh x$	$\cosh x = \frac{e^x + e^{-x}}{2}$
335	Hyperbolic Tangent	$\frac{d}{dx}(\tanh x) = \operatorname{sech}^2 x$	$\tanh x = \frac{\sinh x}{\cosh x}$

26. Applications of Differentiation

No.	Application / Concept	Formula / Rule	Conditions / Notes
336	Tangent Line Equation	$y - f(a) = f'(a)(x - a)$	Point-slope line with derivative as slope
337	Normal Line Equation	$y - f(a) = -\frac{1}{f'(a)}(x - a)$	Perpendicular to tangent ( $f'(a) \ne 0$ )
338	Critical Point Condition	$f'(x) = 0 \text{ or } f'(x) \text{ undefined}$	Candidates for local extrema
339	Second Derivative Test (Min)	$f'(c) = 0 \text{ and } f''(c) > 0 \implies \text{Local Min}$	Graph concave up at $c$
340	Second Derivative Test (Max)	$f'(c) = 0 \text{ and } f''(c) < 0 \implies \text{Local Max}$	Graph concave down at $c$
341	Monotonicity Test	$f'(x) > 0 \implies \text{Increasing},\; f'(x) < 0 \implies \text{Decreasing}$	Over open interval $I$
342	Concavity Test	$f''(x) > 0 \implies \text{Concave Up},\; f''(x) < 0 \implies \text{Concave Down}$	Over open interval $I$
343	Linear Approximation	$f(x) \approx f(a) + f'(a)(x - a)$	First-order Taylor polynomial near $a$
344	Mean Value Theorem (MVT)	$f'(c) = \frac{f(b) - f(a)}{b - a}$	Continuous on $[a, b]$ , differentiable on $(a, b)$
345	Rolle’s Theorem	$f(a) = f(b) \implies \exists c \in (a, b): f'(c) = 0$	Special case of MVT with equal endpoints

Integration

27. Integration

Indefinite Integrals & Core Techniques

No.	Rule / Form	Formula	Conditions / Notes
346	Constant Rule	$\int c\,dx = cx + C$	$c$ is constant, $C$ is integration constant
347	Power Rule of Integration	$\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$	Valid for all $n \ne -1$
348	Reciprocal Integral	$\int \frac{1}{x}\,dx = \ln\vert x\vert + C$	$x \ne 0$
349	Exponential Integral	$\int e^x\,dx = e^x + C$	Natural base exponential
350	General Exponential Integral	$\int a^x\,dx = \frac{a^x}{\ln a} + C$	$a > 0, a \ne 1$
351	Linearity of Integration	$\int [a f(x) + b g(x)]dx = a\int f(x)dx + b\int g(x)dx$	Linear operator property
352	Integration by Substitution	$\int f(g(x))g'(x)\,dx = \int f(u)\,du$	$u = g(x), du = g'(x)dx$
353	Integration by Parts	$\int u\,dv = uv - \int v\,du$	Derived from product rule

Trigonometric & Special Integrals

No.	Integrand	Antiderivative	Conditions / Notes
354	Sine Integral	$\int \sin x\,dx = -\cos x + C$	Antiderivative of sine
355	Cosine Integral	$\int \cos x\,dx = \sin x + C$	Antiderivative of cosine
356	Tangent Integral	$\int \tan x\,dx = -\ln\vert\cos x\vert + C = \ln\vert\sec x\vert + C$	Integrable via substitution
357	Cotangent Integral	$\int \cot x\,dx = \ln\vert\sin x\vert + C$	Integrable via substitution
358	Secant Squared Integral	$\int \sec^2 x\,dx = \tan x + C$	Standard derivative inverse
359	Cosecant Squared Integral	$\int \csc^2 x\,dx = -\cot x + C$	Standard derivative inverse
360	Secant-Tangent Integral	$\int \sec x\tan x\,dx = \sec x + C$	Standard derivative inverse
361	Cosecant-Cotangent Integral	$\int \csc x\cot x\,dx = -\csc x + C$	Standard derivative inverse
362	Inverse Tangent Form	$\int \frac{1}{a^2 + x^2}\,dx = \frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$	$a > 0$
363	Inverse Sine Form	$\int \frac{1}{\sqrt{a^2 - x^2}}\,dx = \arcsin\left(\frac{x}{a}\right) + C$	$a > 0, \vert x\vert < a$
364	Fundamental Theorem of Calculus (I)	$\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$	Differentiation of integral function
365	Fundamental Theorem of Calculus (II)	$\int_a^b f(x)\,dx = F(b) - F(a)$	$F'(x) = f(x)$ on $[a, b]$

28. Applications of Integration

No.	Application	Formula	Conditions / Notes
366	Area Under Curve	$A = \int_a^b f(x)\,dx$	For $f(x) \ge 0$ on $[a, b]$
367	Area Between Curves ( $x$ -slices)	$A = \int_a^b [f(x) - g(x)]\,dx$	$f(x) \ge g(x)$ (upper minus lower)
368	Area Between Curves ( $y$ -slices)	$A = \int_c^d [f(y) - g(y)]\,dy$	$f(y) \ge g(y)$ (right minus left)
369	Solid of Revolution: Disk Method	$V = \pi\int_a^b [f(x)]^2\,dx$	Rotation around the $x$ -axis
370	Solid of Revolution: Washer Method	$V = \pi\int_a^b [R(x)^2 - r(x)^2]\,dx$	Outer radius $R(x)$ , inner radius $r(x)$
371	Solid of Revolution: Shell Method	$V = 2\pi\int_a^b x f(x)\,dx$	Rotation around the $y$ -axis ( $x \ge 0$ )
372	Arc Length ( $y = f(x)$ )	$L = \int_a^b \sqrt{1 + [f'(x)]^2}\,dx$	Curve length along $x$ -interval $[a, b]$
373	Arc Length ( $x = g(y)$ )	$L = \int_c^d \sqrt{1 + [g'(y)]^2}\,dy$	Curve length along $y$ -interval $[c, d]$
374	Surface Area of Revolution	$S = 2\pi\int_a^b f(x)\sqrt{1 + [f'(x)]^2}\,dx$	Rotation of $y=f(x)$ around $x$ -axis
375	Average Value of Function	$f_{\text{avg}} = \frac{1}{b - a}\int_a^b f(x)\,dx$	Mean height of function over $[a, b]$

29. Differential Equations

No.	ODE Type	Form & Solution	Conditions / Notes
376	Separable ODE	$\frac{dy}{dx} = g(x)h(y) \implies \int \frac{dy}{h(y)} = \int g(x)\,dx$	Separate variables and integrate both sides
377	First-Order Linear ODE	$\frac{dy}{dx} + P(x)y = Q(x)$	Standard linear differential equation
378	Integrating Factor	$IF = e^{\int P(x)\,dx}$	Factor converting LHS into product derivative
379	First-Order Linear Solution	$y \cdot IF = \int Q(x)IF\,dx + C$	General solution via integrating factor
380	Exponential Growth / Decay	$\frac{dy}{dt} = ky \implies y(t) = y_0 e^{kt}$	$k > 0$ growth, $k < 0$ decay
381	Radioactive Half-Life	$t_{1/2} = \frac{\ln 2}{\vert k\vert}$	Time required for half of substance to decay
382	Logistic Differential Equation	$\frac{dP}{dt} = rP\left(1 - \frac{P}{K}\right)$	Carrying capacity $K$ , intrinsic growth rate $r$
383	Logistic Analytical Solution	$P(t) = \frac{K}{1 + A e^{-rt}},\; A = \frac{K - P_0}{P_0}$	S-shaped sigmoidal population curve
384	Second-Order Homogeneous ODE	$a y'' + b y' + c y = 0$	Constant coefficients $a, b, c$
385	Characteristic Equation	$a r^2 + b r + c = 0$	Quadratic auxiliary equation in $r$

Vectors & Matrices

30. Vectors

No.	Vector Concept / Operation	Formula	Conditions / Notes
386	3D Component Representation	$\vec{v} = \langle v_1, v_2, v_3\rangle = v_1\mathbf{i} + v_2\mathbf{j} + v_3\mathbf{k}$	Cartesian vector components
387	Vector Magnitude	$\vert\vec{v}\vert = \sqrt{v_1^2 + v_2^2 + v_3^2}$	Euclidean length / norm
388	Unit Vector	$\hat{v} = \frac{\vec{v}}{\vert\vec{v}\vert}$	Normalization of nonzero vector $\vec{v}$
389	Vector Addition	$\vec{a} + \vec{b} = \langle a_1 + b_1, a_2 + b_2, a_3 + b_3\rangle$	Component-wise addition
390	Dot Product (Algebraic)	$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$	Scalar result
391	Dot Product (Geometric)	$\vec{a} \cdot \vec{b} = \vert\vec{a}\vert \vert\vec{b}\vert\cos\theta$	$\theta$ is angle between vectors
392	Angle Between Vectors	$\cos\theta = \frac{\vec{a} \cdot \vec{b}}{\vert\vec{a}\vert \vert\vec{b}\vert}$	Orthogonal iff $\vec{a} \cdot \vec{b} = 0$
393	Cross Product Formula	$\vec{a} \times \vec{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}$	Vector perpendicular to both $\vec{a}$ and $\vec{b}$
394	Cross Product Magnitude	$\vert\vec{a} \times \vec{b}\vert = \vert\vec{a}\vert \vert\vec{b}\vert\sin\theta$	Parallel iff $\vec{a} \times \vec{b} = \vec{0}$
395	Scalar Projection	$\operatorname{comp}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{\vert\vec{b}\vert}$	Component of $\vec{a}$ along $\vec{b}$
396	Vector Projection	$\operatorname{proj}_{\vec{b}}\vec{a} = \left(\frac{\vec{a} \cdot \vec{b}}{\vert\vec{b}\vert^2}\right)\vec{b}$	Vector projected onto direction of $\vec{b}$
397	Parallelogram Area	$A = \vert\vec{a} \times \vec{b}\vert$	Area spanned by vectors $\vec{a}$ and $\vec{b}$
398	Triangle Area from Vectors	$A = \frac{1}{2}\vert\vec{a} \times \vec{b}\vert$	Half of parallelogram area
399	Scalar Triple Product	$\vec{a} \cdot (\vec{b} \times \vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$	Volume of parallelepiped $= \vert\vec{a} \cdot (\vec{b} \times \vec{c})\vert$

31. Matrices

No.	Operation / Property	Formula	Conditions / Notes
400	Matrix Addition	$(A + B)_{ij} = a_{ij} + b_{ij}$	Matrices must share identical dimensions
401	Scalar Multiplication	$(c A)_{ij} = c a_{ij}$	Multiplies every entry by scalar $c$
402	Matrix Multiplication	$(A B)_{ij} = \sum_{k=1}^n a_{ik} b_{kj}$	$A$ is $m\times n$ , $B$ is $n\times p$ ; generally $AB \ne BA$
403	Matrix Transpose	$(A^T)_{ij} = a_{ji}$	Swaps rows and columns
404	Transpose of Product	$(AB)^T = B^T A^T$	Reverses order of matrix product
405	Inverse of $2\times 2$ Matrix	$\begin{pmatrix} a & b \\ c & d \end{pmatrix}^{-1} = \frac{1}{ad - bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$	Invertible iff $\det A = ad - bc \ne 0$
406	General Matrix Inverse	$A^{-1} = \frac{\operatorname{adj}(A)}{\det A}$	Invertible iff $\det A \ne 0$
407	Matrix Trace	$\operatorname{tr}(A) = \sum_{i=1}^n a_{ii}$	Sum of main diagonal elements
408	Eigenvalue Equation	$A\vec{v} = \lambda\vec{v} \iff \det(A - \lambda I) = 0$	Characteristic equation determines eigenvalues $\lambda$
409	Characteristic Polynomial ( $2\times 2$ )	$\lambda^2 - \operatorname{tr}(A)\lambda + \det(A) = 0$	Characteristic equation for $2\times 2$ matrices

32. Determinants

No.	Concept / Property	Formula	Conditions / Notes
410	$2\times 2$ Determinant	$\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$	Product of main diag minus anti-diag
411	$3\times 3$ Determinant	$\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = a(ei - fh) - b(di - fg) + c(dh - eg)$	Cofactor expansion along first row
412	Multiplicative Property	$\det(AB) = \det(A)\det(B)$	Determinant of matrix product
413	Transpose Property	$\det(A^T) = \det(A)$	Transpose preserves determinant
414	Inverse Property	$\det(A^{-1}) = \frac{1}{\det(A)}$	Reciprocal of determinant
415	Row Multiplication by Scalar	$\det(A') = k\det(A)$	Multiplying a single row by $k$
416	Triangular Matrix Determinant	$\det A = \prod_{i=1}^n a_{ii}$	Product of diagonal entries for triangular matrices

Probability & Statistics

33. Probability

No.	Rule / Theorem	Formula	Conditions / Notes
417	Classical Probability	$P(A) = \frac{\text{Favorable Outcomes}}{\text{Total Possible Outcomes}}$	Equally likely outcome space
418	Complement Rule	$P(A^c) = 1 - P(A)$	Probability of event not occurring
419	Addition Rule (General)	$P(A \cup B) = P(A) + P(B) - P(A \cap B)$	Union of any two events
420	Addition Rule (Mutually Exclusive)	$P(A \cup B) = P(A) + P(B)$	When $A \cap B = \emptyset$
421	Conditional Probability	$P(A\mid B) = \frac{P(A \cap B)}{P(B)}$	Probability of $A$ given $B$ occurred ( $P(B) > 0$ )
422	Multiplication Rule	$P(A \cap B) = P(A\mid B)P(B) = P(B\mid A)P(A)$	Joint probability calculation
423	Independent Events	$P(A \cap B) = P(A)P(B) \iff P(A\mid B) = P(A)$	Events exert no mutual influence
424	Bayes’ Theorem	$P(A\mid B) = \frac{P(B\mid A)P(A)}{P(B)}$	Posterior probability from likelihood and prior
425	Law of Total Probability	$P(B) = \sum_{i} P(B\mid A_i)P(A_i)$	For partition $\{A_1, A_2, \ldots, A_n\}$

34. Statistics

No.	Statistical Measure	Formula	Conditions / Notes
426	Arithmetic Mean (Sample)	$\bar{x} = \frac{1}{n}\sum_{i=1}^n x_i$	Sum divided by sample size $n$
427	Weighted Mean	$\bar{x}_w = \frac{\sum w_i x_i}{\sum w_i}$	Accounts for varying observations weights $w_i$
428	Range	$\text{Range} = x_{\max} - x_{\min}$	Difference between extreme values
429	Population Variance	$\sigma^2 = \frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2$	For entire population of size $N$
430	Population Standard Deviation	$\sigma = \sqrt{\frac{1}{N}\sum_{i=1}^N (x_i - \mu)^2}$	Square root of population variance
431	Sample Variance	$s^2 = \frac{1}{n - 1}\sum_{i=1}^n (x_i - \bar{x})^2$	Bessel’s correction with $n-1$ degrees of freedom
432	Sample Standard Deviation	$s = \sqrt{\frac{1}{n - 1}\sum_{i=1}^n (x_i - \bar{x})^2}$	Sample variability metric
433	Variance Computational Shortcut	$s^2 = \frac{1}{n - 1}\left[\sum x_i^2 - \frac{(\sum x_i)^2}{n}\right]$	Single-pass calculation formula
434	Coefficient of Variation	$CV = \frac{\sigma}{\mu} \times 100\%$	Relative dispersion metric
435	Standard Score ( $Z$ -Score)	$Z = \frac{x - \mu}{\sigma}$	Number of standard deviations from mean
436	Sample Covariance	$s_{xy} = \frac{1}{n - 1}\sum (x_i - \bar{x})(y_i - \bar{y})$	Joint variability of two variables
437	Pearson Correlation ( $r$ )	$r = \frac{s_{xy}}{s_x s_y} = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum (x_i - \bar{x})^2 \sum (y_i - \bar{y})^2}}$	Linear association ( $-1 \le r \le 1$ )
438	Regression Line Slope	$m = \frac{n\sum xy - \sum x\sum y}{n\sum x^2 - (\sum x)^2} = r\frac{s_y}{s_x}$	Slope of least-squares line $y = mx + c$
439	Regression Line Intercept	$c = \bar{y} - m\bar{x}$	$y$ -intercept of least-squares regression line

Discrete Mathematics & Number Theory

35. Discrete Mathematics

No.	Logical Law / Principle	Equivalence / Formula	Conditions / Notes
440	Conditional (Implication)	$p \to q \equiv \neg p \lor q$	Fundamental implication identity
441	Contrapositive Law	$p \to q \equiv \neg q \to \neg p$	Logically identical to conditional
442	Biconditional Law	$p \leftrightarrow q \equiv (p \to q) \land (q \to p)$	Mutual implication equivalence
443	De Morgan’s Law (AND)	$\neg(p \land q) \equiv \neg p \lor \neg q$	Negation of conjunction
444	De Morgan’s Law (OR)	$\neg(p \lor q) \equiv \neg p \land \neg q$	Negation of disjunction
445	Double Negation	$\neg(\neg p) \equiv p$	Involution property
446	Idempotent Laws	$p \lor p \equiv p \text{ and } p \land p \equiv p$	Self-combination yields same statement
447	Identity Laws	$p \lor F \equiv p \text{ and } p \land T \equiv p$	Neutral truth elements
448	Domination Laws	$p \lor T \equiv T \text{ and } p \land F \equiv F$	Annihilating truth elements
449	Complement Laws	$p \lor \neg p \equiv T \text{ and } p \land \neg p \equiv F$	Law of excluded middle and non-contradiction

36. Number Theory

No.	Concept / Theorem	Formula / Property	Conditions / Notes
450	Divisibility Definition	$a \mid b \iff b = ak$	For some integer $k$
451	Euclidean Algorithm	$\gcd(a, b) = \gcd(b, a \bmod b)$	Iterated reduction until remainder is zero
452	LCM-GCD Product Relation	$\operatorname{lcm}(a, b) \cdot \gcd(a, b) = a b$	Positive integers $a, b$
453	Bézout’s Identity	$a x + b y = \gcd(a, b)$	Always solvable for integer coefficients $x, y$
454	Congruence Addition	$a \equiv b \pmod m \implies a + c \equiv b + c \pmod m$	Preserves arithmetic under addition
455	Congruence Multiplication	$a \equiv b \pmod m \implies ac \equiv bc \pmod m$	Preserves arithmetic under multiplication
456	Modular Multiplicative Inverse	$a x \equiv 1 \pmod m$	Unique solution exists iff $\gcd(a, m) = 1$
457	Fermat’s Little Theorem	$a^{p-1} \equiv 1 \pmod p$	Prime $p$ and $\gcd(a, p) = 1$
458	Euler’s Totient Product	$\phi(n) = n\prod_{p \mid n}\left(1 - \frac{1}{p}\right)$	Count of integers $1 \le k \le n$ coprime to $n$
459	Euler’s Theorem	$a^{\phi(n)} \equiv 1 \pmod n$	Generalization of FLT for $\gcd(a, n) = 1$
460	Prime Factorization	$n = p_1^{a_1} p_2^{a_2} \cdots p_k^{a_k}$	Unique prime factorization of $n$
461	Number of Divisors $d(n)$	$d(n) = (a_1 + 1)(a_2 + 1)\cdots(a_k + 1)$	Total positive factors
462	Sum of Divisors $\sigma(n)$	$\sigma(n) = \prod_{i=1}^k \frac{p_i^{a_i + 1} - 1}{p_i - 1}$	Total sum of all positive divisors

37. Graph Theory

No.	Theorem / Property	Formula	Conditions / Notes
463	Handshaking Lemma	$\sum_{v \in V} \deg(v) = 2\vert E\vert$	Every edge contributes to two endpoint degrees
464	Odd Degree Vertex Count	$\vert\{v : \deg(v) \text{ is odd}\}\vert \text{ is even}$	Consequence of Handshaking Lemma
465	Complete Graph Edges	$\vert E(K_n)\vert = \frac{n(n - 1)}{2}$	Graph where every pair of vertices is connected
466	Complete Bipartite Edges	$\vert E(K_{m, n})\vert = mn$	Partitions of sizes $m$ and $n$
467	Tree Edge Count	$\vert E\vert = n - 1$	Connected acyclic graph with $n$ vertices
468	Forest Edge Count	$\vert E\vert = n - c$	Acyclic graph with $c$ connected components
469	Euler’s Planar Formula	$V - E + F = 2$	Connected planar graph with faces $F$
470	Planar Edge Upper Bound	$E \le 3V - 6$	For simple planar graphs with $V \ge 3$

38. Set/Counting/Combinatorial Identities

No.	Identity Name	Formula	Conditions / Notes
471	Binomial Sum Identity	$\sum_{k=0}^n \binom{n}{k} = 2^n$	Total subsets of an $n$ -element set
472	Alternating Binomial Sum	$\sum_{k=0}^n (-1)^k \binom{n}{k} = 0$	Valid for $n > 0$
473	Weighted Binomial Sum	$\sum_{k=0}^n k \binom{n}{k} = n 2^{n-1}$	Evaluated via derivative of $(1 + x)^n$
474	Vandermonde’s Identity	$\sum_k \binom{r}{k} \binom{s}{n - k} = \binom{r + s}{n}$	Convolution of binomial coefficients
475	Hockey-Stick Identity	$\sum_{k=r}^n \binom{k}{r} = \binom{n + 1}{r + 1}$	Sum of diagonal entries in Pascal’s triangle
476	Catalan Number Formula	$C_n = \frac{1}{n + 1}\binom{2n}{n}$	Sequence: 1, 1, 2, 5, 14, 42, 132…
477	Catalan Recurrence	$C_{n+1} = \frac{2(2n + 1)}{n + 2}C_n$	Multiplicative growth recurrence

39. Financial Mathematics

No.	Financial Metric	Formula	Conditions / Notes
478	Simple Interest	$SI = \frac{P R T}{100}$	Principal $P$ , rate $R\%$ , time $T$ years
479	Simple Total Amount	$A = P + SI = P\left(1 + \frac{RT}{100}\right)$	Total accumulated balance
480	Compound Interest Amount	$A = P\left(1 + \frac{r}{n}\right)^{nt}$	Compounded $n$ times/year at decimal rate $r$
481	Continuous Compounding	$A = P e^{rt}$	Limit as compounding frequency $n \to \infty$
482	Exponential Depreciation	$V(t) = P(1 - r)^t$	Value with annual depreciation rate $r$
483	Present Value	$PV = \frac{FV}{(1 + r)^n}$	Discounted current worth of future cash flow
484	Future Value	$FV = PV(1 + r)^n$	Compounded future worth of present balance
485	Annuity Future Value	$FV = PMT \cdot \frac{(1 + r)^n - 1}{r}$	Ordinary annuity with periodic deposit $PMT$
486	Annuity Present Value	$PV = PMT \cdot \frac{1 - (1 + r)^{-n}}{r}$	Capitalized value of future income stream
487	Loan Periodic Payment	$PMT = PV \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1}$	Amortization periodic payment formula

Applied Mathematics & Numerical Methods

40. Numerical Methods

No.	Method	Iteration / Quadrature Formula	Conditions / Notes
488	Bisection Method Midpoint	$c = \frac{a + b}{2}$	Requires $f(a)f(b) < 0$
489	Newton-Raphson Iteration	$x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}$	Quadratic convergence near simple roots
490	Secant Method Iteration	$x_{n+1} = x_n - f(x_n)\frac{x_n - x_{n-1}}{f(x_n) - f(x_{n-1})}$	Derivative-free approximation
491	Single Trapezoidal Rule	$\int_a^b f(x)\,dx \approx \frac{b - a}{2}[f(a) + f(b)]$	Linear approximation of area
492	Composite Trapezoidal Rule	$\int_a^b f(x)\,dx \approx \frac{h}{2}\left[f(x_0) + f(x_n) + 2\sum_{i=1}^{n-1} f(x_i)\right]$	$h = \frac{b - a}{n}$
493	Simpson’s $1/3$ Rule	$\int_a^b f(x)\,dx \approx \frac{h}{3}\left[f(x_0) + 4\sum_{i \text{ odd}} f(x_i) + 2\sum_{j \text{ even}} f(x_j) + f(x_n)\right]$	Parabolic quadrature (requires even $n$ )

41. Common Mathematical Constants

No.	Constant	Symbol	Approximate Value	Significance
494	Pi	$\pi$	$\approx 3.141592653589793$	Ratio of circumference to diameter of circle
495	Euler’s Number	$e$	$\approx 2.718281828459045$	Base of natural logarithm
496	Pythagoras’ Constant	$\sqrt{2}$	$\approx 1.414213562373095$	Diagonal of unit square / irrational root
497	Theodorus’ Constant	$\sqrt{3}$	$\approx 1.732050807568877$	Diagonal of unit cube
498	Golden Ratio	$\varphi = \frac{1 + \sqrt{5}}{2}$	$\approx 1.618033988749895$	Positive root of $x^2 - x - 1 = 0$
499	Euler-Mascheroni Constant	$\gamma$	$\approx 0.577215664901532$	Limiting difference between harmonic series and natural log
500	Imaginary Unit	$i$	$\sqrt{-1}$	Fundamental unit of complex numbers ( $i^2 = -1$ )

42. Useful Unit Conversions

No.	Dimension	Conversion Identity	Scale Factor
501	Length	$1\text{ km} = 1000\text{ m}$	Metric length
502	Length	$1\text{ m} = 100\text{ cm} = 1000\text{ mm}$	Metric length
503	Length	$1\text{ in} = 2.54\text{ cm}$	Imperial to metric exact definition
504	Length	$1\text{ ft} = 12\text{ in} = 0.3048\text{ m}$	Imperial foot
505	Length	$1\text{ yd} = 3\text{ ft} = 0.9144\text{ m}$	Imperial yard
506	Length	$1\text{ mi} = 5280\text{ ft} \approx 1.60934\text{ km}$	Statute mile
507	Area	$1\text{ m}^2 = 10^4\text{ cm}^2 = 10^6\text{ mm}^2$	Metric area
508	Area	$1\text{ km}^2 = 10^6\text{ m}^2 = 100\text{ ha}$	1 hectare = $10^4\text{ m}^2$
509	Volume	$1\text{ m}^3 = 10^6\text{ cm}^3 = 1000\text{ L}$	1 cubic meter
510	Volume	$1\text{ L} = 1000\text{ mL} = 1000\text{ cm}^3$	Metric liquid volume
511	Angle	$180^\circ = \pi\text{ rad}$	Straight angle equivalence
512	Angle	$1\text{ rad} = \frac{180^\circ}{\pi} \approx 57.2958^\circ$	Radian to degree conversion

43. Quick Reference Tables

Common Derivatives Reference

No.	Function $f(x)$	Derivative $f'(x)$	Domain / Notes
513	$c$	$0$	Constant function
514	$x^n$	$n x^{n-1}$	Power rule ( $n \in \mathbb{R}$ )
515	$e^x$	$e^x$	Natural exponential
516	$a^x$	$a^x \ln a$	$a > 0, a \ne 1$
517	$\ln x$	$\frac{1}{x}$	$x > 0$
518	$\sin x$	$\cos x$	$x \in \mathbb{R}$
519	$\cos x$	$-\sin x$	$x \in \mathbb{R}$
520	$\tan x$	$\sec^2 x$	$x \ne \frac{\pi}{2} + k\pi$
521	$\cot x$	$-\csc^2 x$	$x \ne k\pi$
522	$\sec x$	$\sec x\tan x$	$x \ne \frac{\pi}{2} + k\pi$
523	$\csc x$	$-\csc x\cot x$	$x \ne k\pi$
524	$\arcsin x$	$\frac{1}{\sqrt{1 - x^2}}$	$\vert x\vert < 1$
525	$\arccos x$	$-\frac{1}{\sqrt{1 - x^2}}$	$\vert x\vert < 1$
526	$\arctan x$	$\frac{1}{1 + x^2}$	$x \in \mathbb{R}$
527	$\sinh x$	$\cosh x$	$x \in \mathbb{R}$
528	$\cosh x$	$\sinh x$	$x \in \mathbb{R}$
529	$\tanh x$	$\operatorname{sech}^2 x$	$x \in \mathbb{R}$

Common Integrals Reference

No.	Function $f(x)$	Indefinite Integral $\int f(x)\,dx$	Domain / Notes
530	$c$	$cx + C$	Constant
531	$x^n$	$\frac{x^{n+1}}{n+1} + C$	$n \ne -1$
532	$\frac{1}{x}$	$\ln\vert x\vert + C$	$x \ne 0$
533	$e^x$	$e^x + C$	Natural exponential
534	$a^x$	$\frac{a^x}{\ln a} + C$	$a > 0, a \ne 1$
535	$\sin x$	$-\cos x + C$	Trig antiderivative
536	$\cos x$	$\sin x + C$	Trig antiderivative
537	$\sec^2 x$	$\tan x + C$	$x \ne \frac{\pi}{2} + k\pi$
538	$\csc^2 x$	$-\cot x + C$	$x \ne k\pi$
539	$\sec x\tan x$	$\sec x + C$	$x \ne \frac{\pi}{2} + k\pi$
540	$\csc x\cot x$	$-\csc x + C$	$x \ne k\pi$
541	$\frac{1}{1 + x^2}$	$\arctan x + C$	$x \in \mathbb{R}$
542	$\frac{1}{\sqrt{1 - x^2}}$	$\arcsin x + C$	$\vert x\vert < 1$

Common Trigonometric Values

$\theta$	$0^\circ$ ( $0$ )	$30^\circ$ ( $\pi/6$ )	$45^\circ$ ( $\pi/4$ )	$60^\circ$ ( $\pi/3$ )	$90^\circ$ ( $\pi/2$ )
$\sin\theta$	$0$	$\frac{1}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{3}}{2}$	$1$
$\cos\theta$	$1$	$\frac{\sqrt{3}}{2}$	$\frac{\sqrt{2}}{2}$	$\frac{1}{2}$	$0$
$\tan\theta$	$0$	$\frac{1}{\sqrt{3}}$	$1$	$\sqrt{3}$	undefined

Reference & Appendices

Formula Selection Guide

Problem Type	Target Section / Formulas	Key Method / Identity
Expand or factor algebraic expressions	Algebraic Identities (Sec. 4)	Binomial, difference of squares, cubes
Find roots of polynomial equations	Equations and Inequalities (Sec. 5 & 6)	Quadratic formula, Vieta’s relations, factoring
Solve exponential / logarithmic equations	Logarithms (Sec. 7)	Change of base, product and power rules
Summing series with constant difference	Sequences and Series (Sec. 8)	Arithmetic Progression ( $S_n$ )
Summing geometric series / decay processes	Sequences and Series (Sec. 8)	Geometric Progression ( $S_n, S_\infty$ )
Combinations & permutations of objects	Counting & Combinatorics (Sec. 10)	${}^n P_r,$ {}^n C_r, stars and bars
Finding angles, lengths, or sides in triangles	Trigonometry & Triangles (Sec. 18 & 21)	SOHCAHTOA, Law of Sines, Law of Cosines
Distance, slope, or line equations in 2D	Coordinate Geometry & Lines (Sec. 13 & 14)	Distance formula, slope-intercept, point-to-line
Circles, arcs, and sector areas	Circles (Sec. 15)	Standard circle form, arc length $s = r\theta$
Conic curves (parabolas, ellipses, hyperbolas)	Conic Sections (Sec. 16)	Standard forms, focal distance $c^2$ , eccentricity $e$
3D volumes, surface areas, and slant heights	3D Mensuration (Sec. 20)	Cylinder, cone, sphere, prism formulas
Rates of change, tangents, optimization	Differentiation (Sec. 25 & 26)	Product/chain rules, critical points $f'(x)=0$
Accumulation, curve areas, solid volumes	Integration (Sec. 27 & 28)	Fundamental theorem, substitution, disk/shell
Dynamic rate processes, growth, half-life	Differential Equations (Sec. 29)	Separation of variables, integrating factor
Spatial vectors, angles, planes, projections	Vectors (Sec. 30)	Dot product (angles), cross product (normal/area)
Linear systems, transformations, determinants	Matrices & Determinants (Sec. 31 & 32)	Matrix multiplication, $A^{-1}$ , $\det A$
Chance events, conditional odds, Bayesian inference	Probability (Sec. 33)	Addition/product rules, Bayes’ theorem
Dataset spread, variance, linear regression	Statistics (Sec. 34)	Mean, sample variance $s^2$ , correlation $r$
Logic propositions, set intersections	Discrete Mathematics (Sec. 35)	De Morgan’s laws, inclusion-exclusion
Divisibility, modular equations, primes	Number Theory (Sec. 36)	Euclidean algorithm, modular inverse, totient
Networks, trees, graph connectivity	Graph Theory (Sec. 37)	Handshaking lemma, Euler formula $V - E + F = 2$
Interest, depreciation, loan payments	Financial Mathematics (Sec. 39)	Compound interest, amortization payment $PMT$
Numerical root finding and numerical integration	Numerical Methods (Sec. 40)	Newton-Raphson, composite trapezoidal rule

Important Notation

Symbol	Mathematical Meaning	Standard Context
$\pi$	Circle ratio constant ( $3.14159\dots$ )	Geometry, trigonometry, calculus
$e$	Euler’s natural logarithm base ( $2.71828\dots$ )	Growth models, calculus, complex analysis
$i$	Imaginary unit ( $i^2 = -1$ )	Complex numbers, algebra
$\sum$	Summation operator	Series, statistics, discrete math
$\prod$	Product operator	Factorials, number theory, linear algebra
$\int$	Integral operator	Calculus, continuous distributions
$\partial$	Partial derivative	Multivariable calculus, PDEs
$\nabla$	Del / gradient vector differential operator	Vector calculus
$\Delta$	Difference / change; discriminant	Algebra, calculus
$\infty$	Infinity	Limits, unbounded domains
$\forall$	Universal quantifier (“for all”)	Logic, proofs, set theory
$\exists$	Existential quantifier (“there exists”)	Logic, proofs
$\in$	Element of	Set membership
$\subseteq$	Subset of	Set theory
$\approx$	Approximately equal to	Numerical methods, approximations
$\equiv$	Identically equal / congruent modulo $m$	Algebra, number theory, logic
$\propto$	Proportional to	Ratios, variation
$\perp$	Perpendicular to	Geometry, orthogonal vectors
$\parallel$	Parallel to	Geometry, lines

Final Notes

Angle Units: Trigonometric calculus formulas (derivatives and integrals) strictly assume angles are measured in radians.
Logarithmic Notation: $\log x$ without an explicit base may refer to base $10$ in applied contexts or the natural logarithm $\ln x$ (base $e$ ) in advanced mathematics; $\ln x$ unambiguously denotes base $e$ .
Principal Values: For square roots over real numbers, $\sqrt{x}$ denotes the non-negative principal root.
Domain Restrictions: Always verify domain validity (e.g. non-zero denominators, strictly positive logarithmic arguments, non-negative radical radicands) before applying identities.

Explore practical applications of these mathematical principles in software engineering:

Binary Number Conversions — Positional number systems, radix mathematics, and conversion algorithms between binary, octal, decimal, and hexadecimal.
How I Cut My Palindrome Algorithm’s Execution Time in Half — Algorithmic number theory, integer digit extraction, and eliminating memory overhead.
Python Fundamentals — Foundations of Python covering architecture, CPython execution model, syntax rules, variables, memory mechanics, and operators.
Mathematics Topic Hub — The complete mathematical pillar page and structured learning paths.

Last updated: 2026-09-30