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.

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Getting Started with LaTeX

The Very Basics

LaTeX uses a special "math mode" to display mathematics. There are two types of this "math mode":

In-line Math Mode

In in-line math mode, we use $ signs to enclose the math we want to display, and it displays in-line 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

In-line vs. Display

Besides displaying in-line 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, $\sqrt[4]{x-y}$ 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, $$f(x) = \pi(\frac{\sqrt{x}}{x-1})$$ 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 $n! = n\cdot(n-1)\cdots 2\cdot 1$, 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 auto-number 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 such-and-such}
\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 line-breaking.

Commonly Used Commands

Algebra

• Approximate: x\approx 1.5 gives us

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
  
  
• Inverse Trig Functions: \arcsin\theta, \arccos\theta, and \arctan\theta give and

Sets

• Denoting Membership: a\in A gives
  
  
• Complement of Set: A\setminus B yields
  
  
• Basic Set Notation: \{...\} gives
  
  
• Set-Builder Notation: \{x\in\mathbb{R}\mid P(x)\} gives
  
  
• Basic Union and Intersection: A\cup B and A\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
  
  
• And/Or: A\land B and A\lor B give and
  
  
• Negation: \neg A gives