Here is some inline math: $E = mc^2$. This is a classic.
Now, let's look at the Quadratic Formula as a display equation:
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
We can also write Euler's Identity:
$$e^{i\pi} + 1 = 0$$
Here is a summation (Sigma):
$$\sum_^{n} i^2 = \frac{n(n+1)(2n+1)}{6}$$
And an integral from calculus:
$$\int_^{b} f(x) , dx = F(b) - F(a)$$
This text includes various Greek letters like $\alpha$, $\beta$, $\gamma$, $\delta$, $\Omega$, and $\Sigma$. It also uses subscripts $x_1, x_2$ and superscripts $y^3, z^4$.
Here is a piecewise function using the cases environment:
$$
f(n) = \begin{cases}
n/2, & \text{if } n \text{ is even} \
3n+1, & \text{if } n \text{ is odd}
\end{cases}
$$
What about a matrix?
$$
A = \begin{pmatrix}
1 & 2 & 3 \
4 & 5 & 6 \
7 & 8 & 9
\end{pmatrix}
$$
Finally, here is a set of aligned equations using the align environment:
$$
\begin{align}
y &= (x+1)^2 \
&= x^2 + 2x + 1
\end{align}
$$