LaTeX:LaTeX on AoPS
LaTeX 
About  Getting Started  Diagrams  Symbols  Downloads  Basics  Math  Examples  Pictures  Layout  Commands  Packages  Help 
This article explains how to use LaTeX in the AoPSWiki, the AoPS Community, and the AoPS Classroom. See Packages to know which packages are prebuilt into the AoPS site.
Contents
Getting Started with LaTeX
The Very Basics
LaTeX uses a special "math mode" to display mathematics. There are two types of this "math mode":
Inline Math Mode
In inline math mode, we use $
signs to enclose the math we want to display, and it displays inline with our text. For example, typing $\sqrt{x} = 5$
gives us
Display Math Mode
In display math mode, we enclose our code in double dollar signs, and it displays the math centered and on its own line. For example, $$\sqrt{x} = 5$$
gives us
Inline vs. Display
Besides displaying inline vs. displaying centered and on a new line, the two modes render differently in other ways. Note that $\sum_{k=1}^n k^2$
gives us whereas $$\sum_{k=1}^n k^2$$
gives us
Basic Expressions
Multiplication
Sometimes, when we're multiplying, we don't need a multiplication symbol. For instance, we can write instead of without ambiguity. However, when you're multiplying numbers, for instance, a multiplication symbol comes in handy. The standard symbol is given by $\cdot$
. For example, $12\cdot\frac{1}{2}$
gives us
Fractions
We can make fractions via $\frac{...}{...}$
. For instance, $\frac{x+y}{2}$
will give us
Roots
Square roots in are pretty simple; we just type $\sqrt{...}$
. For instance, $\sqrt{2}$
gives us Cube roots, fourth roots, and so on are only slightly more difficult; we type $\sqrt[n]{...}$
. For instance, $\sqrt[4]{xy}$
gives
Superscripts & Subscripts
To get superscripts (or exponents), we use the caret symbol ^
. Typing $x^2+y^2$
gives Subscripts are obtained via an underscore (holding shift and the minus sign on most keyboards). For instance, $a_k$
yields
Groups
Most operations in (such as superscripts and subscripts) can only see the "group" of characters immediately following it. We use curly braces {...}
to indicate groups longer than one character. For instance, if we wrote $x^2015$
, we'd expect to get but we instead get This is because each character in the string 2015
is in its own group until we tell that 2015
should be one whole group. To convey this information to , we write $x^{2015}$
and we get
Beyond the Basic Expressions
Grouping Basic Expressions
Our ordinary parentheses (...)
and brackets [...]
work to group expressions in . For instance, $(x+y)[z+w]$
gives us We can also group expressions using curly braces, but we can't just type {...}
. Rather, we must type \{...\}
. This is because uses plain curly braces for other things, such as fractions and superscripts and subscripts.
When we put (vertically) large expressions inside of parentheses (or brackets, or curly braces, etc.), the parentheses don't resize to fit the expression and instead remain relatively small. For instance, $$f(x) = \pi(\frac{\sqrt{x}}{x1})$$
comes out as To automatically adjust the size of parentheses to fit the expression inside of them, we type \left(...\right)
. If we do this for our equation above, we get We can use \left
and \right
for all sorts of things... parentheses (as we saw), brackets $\left[...\right]$
, braces $\left\{...\right\}
, absolute values $\left...\right$
, and much more (norms, floor and ceiling functions, inner products, etc.).
Lists
To make a list, such as a sequence, we use \dots
. For example, $a_0,a_1,\dots,a_n$
will give us
Sums
There are two basic ways to write out sums. First, we can use +
and \cdots
. An example of this way would be $a_1+a_2+\cdots+a_n$
This will give us Second, we could use summation notation, or \sum
. Such an example is $\sum_{i=0}^n a_i$
, giving Note the use of superscripts and subscripts to obtain the summation index.
Products
Again, there are two basic ways to display products. First, we can use \cdot
and \cdots
. An example is $n! = n\cdot(n1)\cdots 2\cdot 1$
, which of course gives The alternative is to use product notation with \prod
. For instance, $n! = \prod_{k=1}^n k$
, giving
Equalities and Inequalities
Inequalities
the commands >, <, \geq, \leq,
and \neq
give us and respectively.
Aligning Equations
To align multiple equations, we use the align*
environment. For example, we might type a system of equations as follows:
\begin{align*} ax + by &= 1 \\ cx + dy &= 2 \\ ex + fy &= 3. \end{align*}
(You do not need dollar signs.) The &
symbol tells where to align to and the \\ symbols break to the next line. This code will output
An example of a string of equations is:
\begin{align*} ((2x+3)^3)' &= 3(2x+3)^2 \cdot (2x+3)' \\ &= 3(2x+3)^2 \cdot 2 \\ &= 6(2x+3)^2. \end{align*}
Again, the &
symbol tells where to align to, and the \\ symbols break to the next line. This code outputs
Numbering Equations
To number equations, we use the align
environment. This is the same environment as the align*
environment, but we leave the *
off. The *
suppresses numbering. To number one equation, the code
\begin{align} ax + by = c \end{align}
will produce
We don't have to use &
or \\ since there is nothing to align and no lines to break. To number several equations, such as a system, the code
\begin{align} ax + by &= c \\ dx + ey &= f \\ gx + hy &= i \end{align}
will produce
In general, align
will autonumber your equations from first to last.
Comments in Equations
Again, we use the align*
environment. The code
\begin{align*} ax + by &= c & \text{because blah} \\ dx + ey &= f & \text{by suchandsuch} \end{align*}
will produce
(You can use align
to get numbering and comments!)
Definition by Cases
To define, say, a function by cases, we use the cases
environment. The code
$$ \delta(i,j) = \begin{cases} 0 & \text{if } i \neq j \\ 1 &\text{if } i = j \end{cases} $$
gives us
As usual, the &
is for aligning and the \\ is for linebreaking.
Commonly Used Commands
Algebra
Geometry/Trig

Degrees Symbol:
90^\circ
gives 
Triangle Symbol:
\triangle ABC
gives 
Parallel Symbol:
AB\parallel XY
gives 
Perpendicular Symbol:
AB\perp CD
gives 
Trig Functions:
\sin\theta
,\cos\theta
,\tan\theta
,\sec\theta
, and\csc\theta
give and
Counting & Probability
Number Theory
Linear Algebra
Calculus
Sets

Denoting Membership:
a\in A
gives 
Complement of Set:
A\setminus B
yields 
Basic Set Notation:
\{...\}
gives 
SetBuilder Notation:
\{x\in\mathbb{R}\mid P(x)\}
gives 
Basic Union and Intersection:
A\cup B
andA\cap B
give and 
Indexed Union and Intersection:
\bigcup_{i=1}^n A_i
and\bigcap_{i=1}^n A_i
give and Mostly used in display math mode. 
Subset:
A \subseteq B
gives 
Cartesian Product:
A \times B
gives 
Common Sets:
\mathbb{N}
,\mathbb{Z}
,\mathbb{Q}
,\mathbb{R}
, and\mathbb{C}
give us and 
Cardinality of a Set:
A
gives
Logic

Implication:
A\implies B
gives 
Biconditional:
A\iff B
yields 
Connectives:
A\land B
andA\lor B
give and 
Negation:
\neg A
gives