LaTeX Math and Equations - Complete Mathematical Typesetting Guide

% Inline math with dollar signs
The equation $E = mc^2$ was discovered by Einstein.

% Variables in text
Let $x$ and $y$ be real numbers such that $x < y$.

% Multiple inline expressions
For $a = 2$ and $b = 3$, we have $a + b = 5$.

% Display math with brackets
\[
    E = mc^2
\]

% Longer equation
\[
    x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

% With preceding text
The quadratic formula is:
\[
    x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

% Addition and subtraction (use symbols directly)
$a + b - c$

% Multiplication
$a \times b$   % Cross symbol
$a \cdot b$    % Dot symbol
$ab$           % Implicit (most common)

% Division
$a \div b$     % Division symbol
$a / b$        % Forward slash
$\frac{a}{b}$  % Fraction (best for display)

% Plus-minus
$x = 5 \pm 2$

% Equals and inequalities
$a = b$        % Equal
$a \neq b$     % Not equal
$a < b$        % Less than
$a > b$        % Greater than
$a \leq b$     % Less than or equal
$a \geq b$     % Greater than or equal

% Approximation
$\pi \approx 3.14$     % Approximately equal
$x \sim y$             % Similar to

% Much less/greater
$a \ll b$      % Much less than
$a \gg b$      % Much greater than

% Simple fraction
\[
    \frac{1}{2}
\]

% Fraction with expressions
\[
    \frac{a + b}{c + d}
\]

% Nested fractions
\[
    \frac{1}{1 + \frac{1}{2}}
\]

% Complex fraction
\[
    \frac{x^2 + 2x + 1}{x^2 - 1} = \frac{(x+1)^2}{(x-1)(x+1)}
\]

% Regular frac in inline (may look cramped)
The ratio is $\frac{1}{2}$ which equals 0.5.

% Better for inline: use slash
The ratio is $1/2$ which equals 0.5.

% Or use \tfrac for smaller fractions
The ratio is $\tfrac{1}{2}$ which equals 0.5.

% Binomial coefficient
\[
    \binom{n}{k} = \frac{n!}{k!(n-k)!}
\]

% Pascal's triangle identity
\[
    \binom{n}{k} = \binom{n-1}{k-1} + \binom{n-1}{k}
\]

% Simple exponent
$x^2$

% Multiple character exponent (use braces)
$x^{10}$
$e^{-x^2}$

% Nested exponents
$x^{y^z}$
$(x^y)^z = x^{yz}$

% In equations
\[
    a^2 + b^2 = c^2
\]

% Simple subscript
$x_1$

% Multiple character subscript
$x_{10}$
$x_{max}$

% Multiple subscripts
$x_1, x_2, x_3, \ldots, x_n$

% Subscript and superscript together
$x_i^2$
$\sum_{i=1}^{n} x_i^2$

% Common Greek letter usage
$\alpha, \beta, \gamma, \delta$
$\pi \approx 3.14159$
$\theta = 45^\circ$
$\mu = 100, \sigma = 15$ % mean and std deviation
$\lambda = 550$ nm        % wavelength
$\Omega = 100$ $\Omega$   % resistance

% Plain matrix (no brackets) - requires amsmath
\[
\begin{matrix}
a & b \\
c & d
\end{matrix}
\]

% Parentheses matrix
\[
\begin{pmatrix}
a & b \\
c & d
\end{pmatrix}
\]

% Square brackets matrix
\[
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
\]

% Curly braces matrix
\[
\begin{Bmatrix}
a & b \\
c & d
\end{Bmatrix}
\]

% Determinant (vertical bars)
\[
\begin{vmatrix}
a & b \\
c & d
\end{vmatrix}
\]

% 3×3 matrix
\[
A = \begin{bmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{bmatrix}
\]

% Identity matrix
\[
I = \begin{pmatrix}
1 & 0 & 0 \\
0 & 1 & 0 \\
0 & 0 & 1
\end{pmatrix}
\]

% Matrix with ellipsis
\[
\begin{bmatrix}
a_1 & a_2 & \cdots & a_n \\
b_1 & b_2 & \cdots & b_n \\
\vdots & \vdots & \ddots & \vdots \\
z_1 & z_2 & \cdots & z_n
\end{bmatrix}
\]

\usepackage{amsmath}  % Required in preamble

\begin{align}
x + y &= 5 \\
2x - y &= 1
\end{align}

% Alignment with explanations
\begin{align}
f(x) &= x^2 + 2x + 1 \\
     &= (x + 1)^2 \\
     &= (x + 1)(x + 1)
\end{align}

% Two alignment points
\begin{align}
x &= a + b  &  y &= c + d \\
  &= e + f  &    &= g + h
\end{align}

% Long derivation
\begin{align}
(a + b)^2 &= (a + b)(a + b) \\
          &= a(a + b) + b(a + b) \\
          &= a^2 + ab + ba + b^2 \\
          &= a^2 + 2ab + b^2
\end{align}

% gather: centered equations without alignment
\begin{gather}
x + y = 5 \\
x^2 + y^2 = 25
\end{gather}

% multline: first line left, last line right, middle centered
\begin{multline}
a + b + c + d + e + f \\
+ g + h + i + j + k
\end{multline}

% split: for breaking single equation (inside equation)
\begin{equation}
\begin{split}
f(x) = a_0 + a_1 x + a_2 x^2 \\
       + a_3 x^3 + a_4 x^4
\end{split}
\end{equation}

% equation environment automatically numbers
\begin{equation}
E = mc^2
\end{equation}

% align numbers each line
\begin{align}
x + y &= 5 \\
2x - y &= 1
\end{align}

% Suppress numbering with *
\begin{equation*}
E = mc^2  % No number
\end{equation*}

% Or use \nonumber on specific lines
\begin{align}
x + y &= 5 \\
2x - y &= 1 \nonumber
\end{align}

% Label an equation
\begin{equation} \label{eq:einstein}
E = mc^2
\end{equation}

% Reference it later
As shown in Equation~\ref{eq:einstein}, energy and mass are equivalent.

% With eqref (adds parentheses automatically)
As shown in Equation~\eqref{eq:einstein}...
% Produces: "Equation (1)"

% Multiple labels
\begin{align}
x + y &= 5   \label{eq:first} \\
2x - y &= 1  \label{eq:second}
\end{align}

From Equations~\eqref{eq:first} and~\eqref{eq:second}, we can solve for $x$ and $y$.

% Use \sin, \cos, etc. (not sin, cos)
$\sin(x), \cos(x), \tan(x)$
$\arcsin(x), \arccos(x), \arctan(x)$
$\sinh(x), \cosh(x), \tanh(x)$  % Hyperbolic

% In equations
\[
\sin^2(x) + \cos^2(x) = 1
\]

% Logarithms
$\log(x), \ln(x), \lg(x)$
$\log_2(x), \log_{10}(x)$

% Exponential
$e^x, \exp(x)$

% Limits
\[
\lim_{x \to \infty} \frac{1}{x} = 0
\]

% Summation
\[
\sum_{i=1}^{n} i = \frac{n(n+1)}{2}
\]

% Product
\[
\prod_{i=1}^{n} i = n!
\]

% Integral
\[
\int_{0}^{1} x^2 \, dx = \frac{1}{3}
\]

% Multiple integral
\[
\iint_D f(x,y) \, dx \, dy
\]

% Limits of integration
\[
\int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}
\]

% Derivative notation
$f'(x), f''(x), f'''(x)$
$\frac{df}{dx}, \frac{d^2f}{dx^2}$

% Partial derivatives
$\frac{\partial f}{\partial x}$
$\frac{\partial^2 f}{\partial x \partial y}$

% Nabla (del) operator
$\nabla f$
$\nabla \cdot \mathbf{F}$  % Divergence
$\nabla \times \mathbf{F}$  % Curl

% WRONG - variables in text mode look wrong
Let x = 5 and y = 10.

% CORRECT - use math mode for variables
Let $x = 5$ and $y = 10$.

% WRONG - * shows as superscript placeholder
$a * b$

% CORRECT - use \times, \cdot, or implicit
$a \times b$  % Cross
$a \cdot b$   % Dot
$ab$          % Implicit (best for variables)

% WRONG - only first character becomes superscript
$x^10$ produces x¹0 instead of x¹⁰

% CORRECT - use braces
$x^{10}$ produces x¹⁰