The formula for kinetic energy is $E = \frac{1}{2}mv^2$.The Pythagorean theorem states $a^2 + b^2 = c^2$.Value of $\pi$ is approximately 3.14159.

The quadratic formula:$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$The Gaussian integral:$$\int_{-\infty}^{\infty} e^{-x^2} dx = \sqrt{\pi}$$

Addition: $a + b = c$Subtraction: $a - b = c$Multiplication: $a \times b = c$Division: $\frac{a}{b} = c$Exponentiation: $a^b = c$

Basic operations:- $2 + 3 = 5$- $10 - 4 = 6$- $5 \times 6 = 30$- $\frac{12}{4} = 3$- $2^3 = 8$

Variables: $x, y, z$Equation: $ax + by + c = 0$Solution: $x = \frac{-c}{a}$ (when $y = 0$)Inequality: $x > 0 \land y < 10$

Common Greek letters:- $\alpha, \beta, \gamma, \delta$- $\pi, \sigma, \theta, \lambda$- $\Omega, \Sigma, \Gamma, \Delta$Uppercase:- $\Alpha, \Beta, \Gamma, \Delta$- $\Pi, \Sigma, \Theta, \Lambda$

Simple fraction: $\frac{a}{b}$Complex fraction: $\frac{\frac{a}{b}}{\frac{c}{d}}$Mixed number: $2\frac{1}{2}$Partial fraction: $\frac{A}{B} + \frac{C}{D}$

Simple sum: $\sum_{i=1}^{n} x_i$Finite sum: $\sum_{i=1}^{10} i^2$Infinite sum: $\sum_{n=1}^{\infty} \frac{1}{n^2}$Double sum: $\sum_{i=1}^{n} \sum_{j=1}^{m} a_{ij}$

Definite integral: $\int_{a}^{b} f(x) dx$Indefinite integral: $\int f(x) dx$Multiple integral: $\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-(x^2+y^2)} dx dy$Differential: $df(x) = f'(x) dx$

Limit: $\lim_{x \to a} f(x) = L$One-sided limit: $\lim_{x \to a^+} f(x) = L$Infinite limit: $\lim_{x \to \infty} f(x) = L$Derivative limit: $\lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = f'(x)$

Column vector: $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}$Row vector: $\mathbf{u} = \begin{bmatrix} u_1 & u_2 & u_3 \end{bmatrix}$Matrix: $\mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}$Matrix multiplication: $\mathbf{A}\mathbf{B} = \mathbf{C}$

Mean: $\mu = \frac{1}{n}\sum_{i=1}^{n} x_i$Standard deviation: $\sigma = \sqrt{\frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2}$Variance: $\sigma^2 = \frac{1}{n}\sum_{i=1}^{n}(x_i - \mu)^2$Correlation: $\rho = \frac{\text{cov}(X,Y)}{\sigma_X \sigma_Y}$

Equality: $=$Inequality: $\neq, <, >, \leq, \geq$Approximation: $\approx, \sim, \simeq$Plus-minus: $\pm, \mp$Proportional: $\propto$Therefore: $\therefore$Because: $\because$

Element: $\in$Not element: $\notin$Union: $\cup$Intersection: $\cap$Empty set: $\emptyset$Subset: $\subset$Proper subset: \subsetneqUniversal set: $\forall$Existential: $\exists$

Logical AND: $\land$Logical OR: $\lor$Logical NOT: $\lnot$Implies: $\implies$If and only if: $\iff$For all: $\forall$There exists: $\exists$Unique: $\exists!$

Kinetic energy: $E = \frac{1}{2}mv^2$Potential energy: $PE = mgh$Force: $F = ma$Work: $W = Fd \cos \theta$Power: $P = \frac{W}{t} = Fv$

Big O notation: $O(n \log n)$Big Theta: $\Theta(n)$Logarithm: $\log_2 n$Exponential: $e^{nx}$Recurrence: $T(n) = 2T(n/2) + O(n)$

Chemical equation: $\ce{2H2 + O2 -> 2H2O}$Equilibrium: $H_2 + I_2 \rightleftharpoons 2HI$State symbols: $^{14}_{6}C$ (Carbon-14)Concentration: $[A] = \frac{n_A}{V}$

Parentheses: $(a + b)$Brackets: $[a, b, c]$Curly braces: $\{a, b, c\}$Absolute value: $|x|$Floor: $\lfloor x \rfloor$Ceiling: $\lceil x \rceil$Fraction: $\frac{a}{b+c}$

Add spacing: $a + b$Thin space: $a \; b$Medium space: $a \quad b$Large space: $a \qquad b$No space: $ab$Line break: $$a \\ b$$

Text in math: $\text{where } x > 0$Mixed text and math: $f(x) = x^2 \text{ for } x \in \mathbb{R}$Subscript text: $x_{\text{min}}$Superscript text: $x^{\text{max}}$

```math\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

# Good - Appropriate math typeInline for simple equations: $E = mc^2$Display for complex formulas:$$\int_{a}^{b} f(x) dx = F(b) - F(a)$$

# Good - Clear and simpleSolve: $x = \frac{-b}{2a}$# Bad - Overly complexSolve using quadratic formula:$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

## Quadratic FormulaThe quadratic formula solves $ax^2 + bx + c = 0$:$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$**Example:**For $x^2 - 5x + 6 = 0$ (where $a=1, b=-5, c=6$):$$x = \frac{5 \pm \sqrt{25 - 24}}{2} = \frac{5 \pm 1}{2}$$So $x = 3$ or $x = 2$.

# Good - Consistent notationLet $f(x)$ be a function, then $f'(x)$ is its derivative.# Bad - Inconsistent notationLet $f(x)$ be a function, then $g(x)$ is its derivative.

# Test on different platformsEquation: $E = mc^2$# Check for rendering issues- Test on GitHub, GitLab, your documentation platform- Verify accessibility with screen readers- Check mobile rendering

# Bad - Unclosed braces$\frac{a}{b$# Good - Proper syntax$\frac{a}{b}$

# Bad - Unclear grouping$\frac{a + b}{c + d}$# Good - Clear grouping$\frac{a+b}{c+d}$

# Bad - Not rendered as mathE = mc^2# Good - Proper dollar signs$E = mc^2$