Difference between revisions of "LaTeX:LaTeX on AoPS"
(→Equalities and Inequalities) |
|||
(35 intermediate revisions by the same user not shown) | |||
Line 11: | Line 11: | ||
LaTeX uses a special "math mode" to display mathematics. There are two types of this "math mode": | 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 <code>$</code> signs to enclose the math we want to display, and it displays in-line with our text. For example, typing <code>$\sqrt{x} = 5$</code> gives us <math>\sqrt{x} = 5.</math> |
− | < | + | |
+ | ==== 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, <code>$$\sqrt{x} = 5$$</code> gives us <cmath>\sqrt{x} = 5.</cmath> | ||
+ | |||
+ | ==== 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 <code>$\sum_{k=1}^n k^2$</code> gives us <math>\textstyle\sum_{k=1}^n k^2,</math> whereas <code>$$\sum_{k=1}^n k^2$$</code> gives us <cmath>\sum_{k=1}^n k^2.</cmath> | Besides displaying in-line vs. displaying centered and on a new line, the two modes render differently in other ways. Note that <code>$\sum_{k=1}^n k^2$</code> gives us <math>\textstyle\sum_{k=1}^n k^2,</math> whereas <code>$$\sum_{k=1}^n k^2$$</code> gives us <cmath>\sum_{k=1}^n k^2.</cmath> | ||
− | |||
− | |||
=== Basic Expressions === | === Basic Expressions === | ||
− | + | ==== Multiplication ==== | |
− | + | ||
− | + | Sometimes, when we're multiplying, we don't need a multiplication symbol. For instance, we can write <math>xy</math> instead of <math>x\cdot y</math> without ambiguity. However, when you're multiplying numbers, for instance, a multiplication symbol comes in handy. The standard symbol is given by <code>$\cdot$</code>. For example, <code>$12\cdot\frac{1}{2}$</code> gives us <math>\textstyle 12\cdot \frac 12.</math> | |
− | |||
− | + | ==== Fractions ==== | |
− | |||
− | |||
− | + | We can make fractions via <code>$\frac{...}{...}$</code>. For instance, <code>$\frac{x+y}{2}$</code> will give us <math>\textstyle\frac{x+y}{2}.</math> | |
− | |||
− | |||
− | + | ==== Roots ==== | |
− | |||
− | |||
− | < | + | Square roots in <math>\LaTeX</math> are pretty simple; we just type <code>$\sqrt{...}$</code>. For instance, <code>$\sqrt{2}$</code> gives us <math>\sqrt 2.</math> Cube roots, fourth roots, and so on are only slightly more difficult; we type <code>$\sqrt[n]{...}$</code>. For instance, <code>$\sqrt[4]{x-y}$</code> gives <math>\sqrt[4]{x-y}.</math> |
− | < | + | |
− | + | ==== Superscripts & Subscripts ==== | |
− | + | ||
+ | To get superscripts (or exponents), we use the caret symbol <code>^</code>. Typing <code>$x^2+y^2$</code> gives <math>x^2+y^2.</math> Subscripts are obtained via an underscore (holding shift and the minus sign on most keyboards). For instance, <code>$a_k$</code> yields <math>a_k.</math> | ||
+ | |||
+ | ==== Groups ==== | ||
+ | |||
+ | Most operations in <math>\LaTeX</math> (such as superscripts and subscripts) can only see the "group" of characters immediately following it. We use curly braces <code>{...}</code> to indicate groups longer than one character. For instance, if we wrote <code>$x^2015$</code>, we'd expect to get <math>x^{2015},</math> but we instead get <math>x^2015.</math> This is because each character in the string <code>2015</code> is in its own group until we tell <math>\LaTeX</math> that <code>2015</code> should be one whole group. To convey this information to <math>\LaTeX</math>, we write <code>$x^{2015}$</code> and we get <math>x^{2015}.</math> | ||
=== Beyond the Basic Expressions === | === Beyond the Basic Expressions === | ||
− | + | ==== Grouping Basic Expressions ==== | |
− | + | ||
− | + | Our ordinary parentheses <code>(...)</code> and brackets <code>[...]</code> work to group expressions in <math>\LaTeX</math>. For instance, <code>$(x+y)[z+w]$</code> gives us <math>(x+y)[z+w].</math> We can also group expressions using curly braces, but we can't just type <code>{...}</code>. Rather, we must type <code>\{...\}</code>. This is because <math>\LaTeX</math> uses plain curly braces for other things, such as fractions and superscripts and subscripts. | |
<br><br> | <br><br> | ||
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, <code>$$f(x) = \pi(\frac{\sqrt{x}}{x-1})$$</code> comes out as <cmath>f(x) = \pi(\frac{\sqrt{x}}{x-1}).</cmath> To automatically adjust the size of parentheses to fit the expression inside of them, we type <code>\left(...\right)</code>. If we do this for our <math>f</math> equation above, we get <cmath>f(x) = \pi\left(\frac{\sqrt{x}}{x-1}\right).</cmath> We can use <code>\left</code> and <code>\right</code> for all sorts of things... parentheses (as we saw), brackets <code>$\left[...\right]$</code>, braces <code>$\left\{...\right\}</code>, absolute values <code>$\left|...\right|$</code>, and much more (norms, floor and ceiling functions, inner products, etc.). | 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, <code>$$f(x) = \pi(\frac{\sqrt{x}}{x-1})$$</code> comes out as <cmath>f(x) = \pi(\frac{\sqrt{x}}{x-1}).</cmath> To automatically adjust the size of parentheses to fit the expression inside of them, we type <code>\left(...\right)</code>. If we do this for our <math>f</math> equation above, we get <cmath>f(x) = \pi\left(\frac{\sqrt{x}}{x-1}\right).</cmath> We can use <code>\left</code> and <code>\right</code> for all sorts of things... parentheses (as we saw), brackets <code>$\left[...\right]$</code>, braces <code>$\left\{...\right\}</code>, absolute values <code>$\left|...\right|$</code>, and much more (norms, floor and ceiling functions, inner products, etc.). | ||
− | |||
− | + | ==== Lists ==== | |
− | + | ||
− | + | To make a list, such as a sequence, we use <code>\dots</code>. For example, <code>$a_0,a_1,\dots,a_n$</code> will give us <math>a_0,a_1,\dots,a_n.</math> | |
+ | |||
+ | ==== Sums ==== | ||
− | + | There are two basic ways to write out sums. First, we can use <code>+</code> and <code>\cdots</code>. An example of this way would be <code>$a_1+a_2+\cdots+a_n$</code> This will give us <math>a_1+a_2+\cdots+a_n.</math> Second, we could use summation notation, or <code>\sum</code>. Such an example is <code>$\sum_{i=0}^n a_i$</code>, giving <math>\textstyle \sum_{i=0}^n a_i.</math> Note the use of superscripts and subscripts to obtain the summation index. | |
− | |||
− | |||
− | + | ==== Products ==== | |
− | |||
− | |||
− | </ | + | Again, there are two basic ways to display products. First, we can use <code>\cdot</code> and <code>\cdots</code>. An example is <code>$n! = n\cdot(n-1)\cdots 2\cdot 1$</code>, which of course gives <math>n! = n\cdot(n-1)\cdots 2 \cdot 1.</math> The alternative is to use product notation with <code>\prod</code>. For instance, <code>$n! = \prod_{k=1}^n k$</code>, giving <math>\textstyle n! = \prod_{k=1}^n k.</math> |
=== Equalities and Inequalities === | === Equalities and Inequalities === | ||
− | + | ==== Inequalities ==== | |
− | + | ||
− | + | the commands <code>>, <, \geq, \leq,</code> and <code>\neq</code> give us <math>>,</math> <math><,</math> <math>\geq,</math> <math>\leq,</math> and <math>\neq,</math> respectively. | |
− | |||
− | + | ==== Aligning Equations ==== | |
− | + | ||
+ | To align multiple equations, we use the <code>align*</code> environment. For example, we might type a system of equations as follows: | ||
<pre> | <pre> | ||
\begin{align*} | \begin{align*} | ||
Line 107: | Line 102: | ||
&= 6(2x+3)^2. | &= 6(2x+3)^2. | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
− | |||
− | + | ==== Numbering Equations ==== | |
− | + | ||
+ | To number equations, we use the <code>align</code> environment. This is the same environment as the <code>align*</code> environment, but we leave the <code>*</code> off. The <code>*</code> suppresses numbering. To number one equation, the code | ||
<pre> | <pre> | ||
\begin{align} | \begin{align} | ||
Line 133: | Line 128: | ||
gx + hy &= i. | gx + hy &= i. | ||
\end{align} </cmath> In general, <code>align</code> will auto-number your equations from first to last. | \end{align} </cmath> In general, <code>align</code> will auto-number your equations from first to last. | ||
− | |||
− | + | ==== Comments in Equations ==== | |
− | + | ||
+ | Again, we use the <code>align*</code> environment. The code | ||
<pre> | <pre> | ||
\begin{align*} | \begin{align*} | ||
Line 148: | Line 143: | ||
dx + ey &= f & \text{by such-and-such}. | dx + ey &= f & \text{by such-and-such}. | ||
\end{align*} </cmath> (You can use <code>align</code> to get numbering <i>and</i> comments!) | \end{align*} </cmath> (You can use <code>align</code> to get numbering <i>and</i> comments!) | ||
− | |||
− | + | ==== Definition by Cases ==== | |
− | + | ||
− | <pre> | + | To define, say, a function by cases, we use the <code>cases</code> environment. The code |
− | < | + | <pre><nowiki> |
− | </pre> | + | $$ </nowiki> \delta(i,j) = \begin{cases} 0 & \text{if } i \neq j \\ 1 &\text{if } i = j \end{cases} <nowiki> $$ |
+ | </nowiki></pre> | ||
gives us | gives us | ||
<cmath> | <cmath> | ||
\delta(i,j) = \begin{cases} 0 & \text{if } i \neq j \\ 1 &\text{if } i = j. \end{cases} | \delta(i,j) = \begin{cases} 0 & \text{if } i \neq j \\ 1 &\text{if } i = j. \end{cases} | ||
− | </cmath> | + | </cmath> As usual, the <code>&</code> is for aligning and the \\ is for line-breaking. |
− | < | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− |
Latest revision as of 19:09, 31 December 2015
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":
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.