\documentclass[12pt, a4paper]{article}
\usepackage{amsmath, amssymb, amsfonts}
\usepackage{geometry}
\geometry{margin=1in}

\title{Mathematics in LaTeX}
\author{Pulakesh Pradhan}
\date{\today}

\begin{document}

\maketitle

\section{Introduction}
LaTeX is the de facto standard for writing mathematical documents.

\section{Inline vs Display Math}
The equation for a line is $y = mx + b$.

For more complex equations, use display mode:
$$ E = mc^2 $$

\section{Fractions and Roots}
Simple Fraction: $\frac{1}{2}$
Nested Fraction: $\frac{1}{1 + \frac{1}{x}}$
Square Root: $\sqrt{2}$
Cube Root: $\sqrt[3]{27}$

\section{Common Notation}
\subsection{Greek Letters}
$\alpha, \beta, \gamma, \delta, \epsilon, \zeta, \eta, \theta, \iota, \kappa, \lambda, \mu, \nu, \xi, \pi, \rho, \sigma, \tau, \upsilon, \phi, \chi, \psi, \omega$

$\Gamma, \Delta, \Theta, \Lambda, \Xi, \Pi, \Sigma, \Upsilon, \Phi, \Psi, \Omega$

\subsection{Operators}
Summation: $\sum_{i=1}^n i^2$
Integral: $\int_a^b f(x) dx$
Limit: $\lim_{x \to \infty} \frac{1}{x} = 0$

\section{Matrices}
Matrix using \texttt{pmatrix}:
$$
\begin{pmatrix}
1 & 2 \\
3 & 4
\end{pmatrix}
$$

Matrix using \texttt{bmatrix}:
$$
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i
\end{bmatrix}
$$

\section{Aligned Equations}
\begin{align}
(a+b)^2 &= (a+b)(a+b) \\
          &= a^2 + ab + ba + b^2 \\
          &= a^2 + 2ab + b^2
\end{align}

\end{document}
