Typing Mathematics

LaTeX is a typesetting language used heavily in mathematics. You must use it if

you are writing a document in Markdown, and
you want to insert a mathematics formula into your document.

You can type formulas on the keyboard using LaTeX. You must decide if you want an inline or display formula .

Create inline equation by placing the formula inside $ ... $
- The formula shows up in the middle of a paragraph of text.
- $a^2 + b^2 = c^2$ produces
  
  $a^2 + b^2 = c^2$
- You can mix math and non-math text in a single sentence
  Let $f(x) = 2x - 1$, then observe that $f'(x)=2$
Create display equation by putting the formula inside $$ ... $$
- The formula is displayed in a new paragraph, centered, and sometimes slightly larger.
- $$\sin^2 x + \cos^2 x = 1$$ produces the text \[\sin^2 x + \cos^2 x = 1\]
- Example $$x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ produces the text \[x=\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

Basic Formulas

Basic math formula syntax is given below, with examples.

Note: Only the LaTeX formulas are given below.
When used in a document the formulas must be enclosed inside $...$ or $$...$$.

Many symbols are exactly what you expect from the keyboard.
- + and - become $+$ and $-$
- numbers like 8 become $8$.
Normal letters are interpreted as variables. They are italicised, and spaces are ignored.
- x becomes $x$
- 3 x + 2 becomes $3 x + 2$
- x and y becomes $x and y$
Use \cdot to get a dot for multiplication.
- 2\cdot 3 = 6 becomes $2\cdot 3 = 6$
- x\cdot x^2 = x^3 becomes $x \cdot x^2 = x^3$
Use \frac{num}{denom} for fractions. Use \dfrac{num}{denom} to create the same fraction, but using more space
- \frac{a}{c} + \frac{b}{c} = \frac{a+b}{c} becomes $\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$
- \dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a+b}{c} becomes $\dfrac{a}{c} + \dfrac{b}{c} = \dfrac{a+b}{c}$
Use ^ to create an exponent. If you need to put more than one character in the exponent, enclose the exponent in {...}
- 2^a \cdot 2^b = 2^{a + b} becomes $2^a \cdot 2^b = 2^{a + b}$
- 2^{10} becomes $2^{10}$, whereas 2^10 becomes $2^10$
Use _ to create an subscript. If you need to put more than one character in the subscript, enclose it in {...}
- A_{i,j} becomes $A_{i,j}$
Use \sqrt{...} to create a square root. You can create a nth root using \sqrt[n]{...}
- \sqrt{9} = 3 becomes $\sqrt{9}=3$
- \sqrt[3]{8} = 2 becomes $\sqrt[3]{8} = 2$
You can get a nice version of most named functions by putting a \ before the function name. This includes \ln, \log, \sin, \cos, \tan, \sec, \csc, \cot
- \ln(e^3) = 3 becomes $\ln(e^3) = 3$
- \log_10(1000) = 3 becomes $\log_10(1000) = 3$
You can type many common symbols using their name, like \pi becomes $\pi$, \Delta becomes $\Delta$, and so on. A more complete table of common greek symbols is also given below.
You can combine any of the above
- \cos(3\pi) = -1 becomes $\cos(3\pi) = -1 $
- \tan^{-1}(-1) = -\frac{\pi}{4} becomes $\tan^{-1}(-1) = -\frac{\pi}{4}$
You can type strict inequalities using < and >.
- Use \leq for $\leq$ and \geq for $\geq$.
- Use |...| for absolute values.

Calculus Formulas

\lim inserts a limit. In inline mode, subscripts are typeset compactly. In display mode, they are written below the limit as normal.
- $\lim_{x \to 2} f(x)$ becomes $\lim_{x \to 2} f(x)$
- $$\lim_{x \to 2} f(x)$$ becomes $\displaystyle\lim_{x \to 2} f(x)$
Use the ' symbol when typesetting derivatives like $f'$, and use brackets [...] when writing $\dfrac{d}{dx}[2x]$. Otherwise, derivatives are typed using the same LaTeX tools from before.
- f'(x) = 2\cos(2x) becomes $f'(x) = 2\cos(2x)$
- \frac{dy}{dx} becomes $\frac{dy}{dx}$
- \frac{d}{dx}[2x] becomes $\frac{d}{dx}[2x]$
- \frac{d}{dx}[2x] becomes $\frac{d}{dx}[2x]$
- If you are taking the derivative of some functions in display mode, your brackets will look too short. You can use \left[ ... \right] instead, and they will scale appropriately
- $$\frac{d}{dx}\left[\frac{1}{x}\right]$$ becomes $\displaystyle\frac{d}{dx}\left[\frac{1}{x}\right]$
\int inserts an integral symbol. You must add the dx yourself.
- \int x dx becomes $\int x dx$, which is not correct, since there should be a space between x and the dx
- \, inserts a small space.
- \int x \,dx becomes $\int x \,dx$ which is much clearer.
- \int is formatted differently based on inline versus display mode.
  - $\int_1^2 x^2 \,dx$ becomes $\int_1^2 x^2 \,dx$
  - $$\int_1^2 x^2 \,dx$$ becomes $\displaystyle \int_1^2 x^2 \,dx$
\sum inserts a summation symbol. Note that this looks like a Greek \Sigma, but actually the \sum command is more intelligent in the placement of limits.
- $\sum_{n=1}^{\infty} \frac{1}{n}$ becomes $\sum_{n=1}^{\infty} \frac{1}{n}$
- $$\sum_{n=1}^{\infty} \frac{1}{n}$$ becomes $\displaystyle\sum_{n=1}^{\infty} \frac{1}{n}$
Multivariable Calculus uses the same main concepts, with a few extra symbols.
- Partial derivatives can be written using standard LaTeX, plus \partial for the partial derivative symbol $\partial$
- f_x(2,1) becomes $f_x(2,1)$
- \frac{\partial}{\partial x}[ 2xy ]= 2y becomes $\frac{\partial}{\partial x}[ 2xy ] = 2y$
- Double and triple integrals can be typeset using \iint for $\iint$ and \iiint for $\iiint$.
- \iint_D f(x,y)\,dA becomes $\iint_D f(x,y)\,dA$
- \iiint_V f(x,y,z)\,dV becomes $\iiint_V f(x,y,z)\,dV$
- For path integrals, you may want to make some letters bold, and write others in script.
- Use \mathbf{...} for math bold font. \mathbf{F} becomes $\mathbf{F}$
- Use \mathcal{...} for math calligraphy font. \mathcal{C} becomes $\mathcal{C}$
- \int_{\mathcal{C}} \mathbf{F}\cdot d\mathbf{r} becomes $\int_{\mathcal{C}} \mathbf{F}\cdot d\mathbf{r}$

Additional Symbols

Mathematicians frequently use symbols for definitions and shorthands. You can start with an in depth list of LaTeX math symbols, or consult the commonly used symbols below.

Common lowercase Greek letters in mathematics:

$ \alpha$	$ \beta$	$ \gamma$	$ \delta$	$ \theta$	$ \epsilon$	$ \eta$
`\alpha`	`\beta`	`\gamma`	`\delta`	`\theta`	`\epsilon`	`\eta`

$ \lambda$	$ \mu$	$ \nu$	$ \xi$	$ \rho$	$ \sigma$	$ \tau$
`\lambda`	`\mu`	`\nu`	`\xi`	`\rho`	`\sigma`	`\tau`

$ \phi$	$ \varphi$	$ \chi$	$ \psi$	$ \omega$
`\phi`	`\varphi`	`\chi`	`\psi`	`\omega`	$\phantom{\omega}$	$\phantom{\omega}$

Common uppercase greek letters in mathematics:
$ \Gamma $ \Gamma , $ \Delta $ \Delta , $ \Lambda $ \Lambda , $ \Phi $ \Phi , $ \Psi $ \Psi , $ \Omega $ \Omega
Natural language arguments
- $\implies$ \implies , $\iff$ \iff
Logic and Quantifiers
- $\exists$ \exists , $\forall$ \forall
- $\land$ \land, $\lor$ \lor, $\neg$ \neg, $\rightarrow$ \rightarrow
- $\equiv$ \equiv
Sets
- $\cup$ \cup, $\cap$ \cap, $\overline{A}$ \overline{A}
- $x\in A$ x \in A
- Use \{ and \} to type open and close set brackets ${\ }$
- $\subset$ \subset, $\subseteq$ \subseteq, $\subsetneq$ \subsetneq
- $3 \in\mathbb{N}$ 3\in\mathbb{N}
Other Comparisons
- $\prec$ \prec, $\preceq$ \preceq
- $\sqsubset$ \sqsubset, $\sqsubseteq$ \sqsubseteq
- $\neq$ \neq, $\gg$ \gg, $\ll$ \ll

See also Section 3.10 of the Not So Short Guide to LaTeX

Linear Algebra

Create matrices using the \begin{bmatrix} ... \end{bmatrix} environment - List out the entries from left to right, and top to bottom - Separate columns using & and separate rows using \\

For example, the code

\begin{bmatrix} 
1 & 2 & 3 \\ 
4 & 5 & 6 
\end{bmatrix}

creates the matrix $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$

Create a system of equations using the \begin{cases} ... \end{cases} environment.

List the equations inside the environment
Include a \\ after each equation
Place a & symbol next to the equal signs to make the equations line up

For example, the code

\begin{cases} 
2x - y  &= 5 \\
 x + y  &= 1 \\
 x      &=-2 \\
     y  &= 3 
\end{cases}

produces the output
$\begin{cases} 2x - y &= 5 \\ x + y &= 1 \\ x &=-2 \\ y &= 3 \end{cases}$

Create an aligned sequence of equations using the \begin{align} ... \end{align} environment. - This is like the 'cases' environment, but you can use multiple & symbols to specify multiple points to align.

For example

\begin{align}  
2&x - &y &= 5 \\ 
 &x + &y &= 1 \\ 
 &x   &  &=-2 \\ 
 &    &y &= 3 
\end{align}

produces the output
$\begin{align} 2&x - &y &= 5 \\ &x + &y &= 1 \\ &x & &=-2 \\ & &y &= 3 \end{align}$

Learning More

In these notes, I have emphasized Markdown due to its greater simplicity and accessibility. But LaTeX is in many ways a more powerful and flexible language.

You can quickly use and learn about LaTeX using Overleaf.

Overleaf is popular browser-based LaTeX editor.
Learn LaTeX in 30 Minutes is a brief introduction to writing whole documents in LaTeX.
Writing Mathematics in LaTeX describes in detail how to write mathematics in LaTeX. However, many examples are embedded in LaTex (rather than Markdown) documents.

You can also install a LaTeX compiler on your personal computer.

MiKTeX is a LaTeX distribution for Windows, Linux, and MacOS

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