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LaTeX
LaTeX 
 Edited copy of http://en.wikibooks.org/wiki/LaTeX/Mathematics for testing of LaTeX commands in OEIS Wiki.
One of the greatest motivating forces for Donald Knuth^{[1]} when he began developing the original TeX system was to create something that allowed simple construction of mathematical formulas, whilst looking professional when printed. The fact that he succeeded was most probably why TeX (and later on, LaTeX) became so popular within the scientific community. Typesetting mathematics is one of LaTeX's greatest strengths. It is also a large topic due to the existence of so much mathematical notation.
If your document requires only a few simple mathematical formulas, plain LaTeX has most of the tools that you will need. If you are writing a scientific document that contains numerous complicated formulas, the amsmath package^{[2]} introduces several new commands that are more powerful and flexible than the ones provided by LaTeX. The mathtools package fixes some amsmath quirks and adds some useful settings, symbols, and environments to amsmath.^{[3]} To use either package, include
\usepackage{amsmath}
or
\usepackage{mathtools}
in the preamble of the document. The mathtools package loads the amsmath package and hence there is no need to \usepackage{amsmath} in the preamble if mathtools is used.
Contents
 1 Mathematics environments
 2 Symbols
 3 Greek letters
 4 Operators
 5 Powers and indices
 6 Fractions and binomials
 7 Roots
 8 Sums and integrals
 9 Brackets, braces and delimiters
 10 Matrices and arrays
 11 Adding text to equations
 12 Formatting mathematics symbols
 13 Color
 14 Plus and minus signs
 15 Controlling horizontal spacing
 16 Advanced Mathematics: AMS Math package
 17 List of mathematical symbols
 18 Summary
 19 Contentoriented LaTeX
 20 See also
 21 Notes
 22 Further reading
 23 External links
Mathematics environments
LaTeX needs to know beforehand that the subsequent text does indeed contain mathematical elements. This is because LaTeX typesets maths notation differently than normal text. Therefore, special environments have been declared for this purpose. They can be distinguished into two categories depending on how they are presented:
 text — text formulas are displayed inline, that is, within the body of text where it is declared, for example, I can say that within this sentence;
 displayed—displayed formulas are separate from the main text.
As maths require special environments, there are naturally the appropriate environment names you can use in the standard way. Unlike most other environments, however, there are some handy shorthands to declaring your formulas. The following table summarizes them:
Type  Inline (within text) formulas  Displayed equations  Displayed and automatically numbered equations 

Environment  math

displaymath

equation

Requires  amsmath  
LaTeX shorthand  \(...\)

\[...\]


TeX shorthand  $...$

$$...$$


Comment  equation* (starred version) suppresses numbering

Suggestion: Using the $$...$$
should be avoided, as it may cause problems, particularly with the AMSLaTeX macros. Furthermore, should a problem occur, the error messages may not be helpful.
The equation*
and displaymath
environments are functionally equivalent.
If you are typing text normally, you are said to be in text mode, but while you are typing within one of those mathematical environments, you are said to be in math mode, that has some differences compared to the text mode:
 Most spaces and line breaks do not have any significance, as all spaces are either derived logically from the mathematical expressions, or have to be specified with special commands such as
\quad
 Empty lines are not allowed. Only one paragraph per formula.
 Each letter is considered to be the name of a variable and will be typeset as such. If you want to typeset normal text within a formula (normal upright font and normal spacing) then you have to enter the text using dedicated commands.
Inserting "Displayed" maths inside blocks of text
In order for some operators, such as \lim
or \sum
to be displayed correctly inside some math environments (read $...$
), it might be convenient to write the \displaystyle
class inside the environment. Doing so might cause the line to be taller, but will cause exponents and indices to be displayed correctly for some math operators. For example, the $\textstyle \sum$
will print a smaller and $\displaystyle \sum$
will print a bigger one , like in equations (this only works with amsmath package).
Symbols
Mathematics has many symbols! One of the most difficult aspects of learning LaTeX is remembering how to produce symbols. There are of course a set of symbols that can be accessed directly from the keyboard:
+  = ! ? / ( ) [ ] < >  ' :
Some symbols that can be accessed directly from the keyboard need to be escaped:
\$ \% \_ \{ \}
Beyond those listed above, distinct commands must be issued in order to display the desired symbols. There are a great deal of examples such as Greek letters, set and relations symbols, arrows, binary operators, etc.
For example
\forall x \in X, \quad \exists y \leq \epsilon
Fortunately, there's a tool that can greatly simplify the search for the command for a specific symbol. Look for "Detexify" in the external links section below. Another option would be to look in the "The Comprehensive LaTeX Symbol List" in the external links section below.
Greek letters
Greek letters are commonly used in mathematics, and they are very easy to type in math mode. You just have to type the name of the letter after a backslash: if the first letter is lowercase, you will get a lowercase Greek letter, if the first letter is uppercase (and only the first letter), then you will get an uppercase letter. Note that some uppercase Greek letters look like Latin ones, so they are not provided by LaTeX (e.g. uppercase Alpha and Beta are just "A" and "B" respectively). Lowercase epsilon, theta, phi, pi, rho, and sigma are provided in two different versions. The alternate, or variant, version is created by adding "var" before the name of the letter:
\alpha, \Alpha, \beta, \Beta, \gamma, \Gamma, \pi, \varpi, \Pi, \phi, \varphi, \Phi
See List of LaTeX mathematical symbols for a complete list of Greek symbols.
Operators
An operator is a function that is written as a word: e.g. trigonometric functions (sin, cos, tan), logarithms and exponentials (log, exp). LaTeX has many of these defined as commands:
\cos (2\theta) = \cos^2 \theta  \sin^2 \theta
For certain operators such as limits, the subscript is placed underneath the operator:
\lim_{x \to \infty} \exp(x) = 0
For the modular operator there are two commands: \bmod
(b for binary operator form) and \pmod
(p for parenthesized form):
a \bmod b
or (for better spacing)
a \, \bmod \, b
x \equiv a \pmod b
To use operators which are not predefined, such as argmax, see custom operators.
Powers and indices
Powers and indices are equivalent to superscripts and subscripts in normal text mode. The caret (^
) character is used to raise something, and the underscore (_
) is for lowering. If more than one expression is raised or lowered, they should be grouped using curly braces ({
and }
).
k_{n+1} = n^2 + k_n^2  k_{n1}
An underscore (_
) can be used with a vertical bar (
) to denote evaluation using subscript notation in mathematics:
f(n) = n^5 + 4n^2 + 2 _{n=17}
or (for automatic sizing of the vertical bar, the \left.
standing for an invisible delimiter)
f(n) = \left. n^5 + 4n^2 + 2 \right_{n=17}
Fractions and binomials
A fraction is created using the \frac{numerator}{denominator}
command (for those who need their memories refreshed, that's the top and bottom respectively!). Likewise, the binomial coefficient (aka the choose function) may be written using the \binom
command:^{[4]}
\frac{n!}{k! \, (nk)!} = \binom{n}{k}
It is also possible to use the \choose
command without the amsmath package:
\frac{n!}{k! \, (nk)!} = {n \choose k}
You can embed fractions within fractions:
\frac{ \frac{1}{x} + \frac{1}{y} }{yz}
Note that when appearing inside another fraction, or in inline text , a fraction is noticeably smaller than in displayed mathematics. The \tfrac
and \dfrac
commands^{[4]} force the use of the respective styles, \textstyle
and \displaystyle
.
\frac{ \dfrac{1}{x} + \dfrac{1}{y} }{yz}
Similarly, the \tbinom
and \dbinom
commands typeset the binomial coefficient.
\tfrac{n!}{k! \, (nk)!} = \tbinom{n}{k}
Another way to write fractions is to use the \over
command without the amsmath package:
{n! \over k! \, (nk)!} = {n \choose k}
For relatively simple fractions, it may be more aesthetically pleasing to use powers and indices:
^3/_7 \quad ^{22}/_7
If you use them throughout the document, usage of xfrac package is recommended. This package provides \sfrac
command to create slanted fractions.
Usage:
Take $\sfrac{1}{2}$ cup of sugar, \dots 3\times\sfrac{1}{2}=1\sfrac{1}{2} Take ${}^1/_2$ cup of sugar, \dots 3\times{}^1/_2=1{}^1/_2
Alternatively, the nicefrac package provides the \nicefrac
command, whose usage is similar to \sfrac
.
Continued fractions
Continued fractions should be written using \cfrac
command^{[4]}:
x = a_0 + \cfrac{1}{a_1 + \cfrac{1}{a_2 + \cfrac{1}{a_3 + \cfrac{1}{a_4} } } }
Multiplication of two numbers
To make multiplication visually similar to a fraction, a nested array can be used, for example multiplication of numbers written one below the other.
\frac{ \begin{array}[b]{r} \left( x_1 x_2 \right) \\ \times \left( x'_1 x'_2 \right) \end{array} }{ \left( y_1 y_2 y_3 y_4 \right) }
Roots
The \sqrt
command creates a square root surrounding an expression. It accepts an optional argument specified in square brackets ([
and ]
) to change the root index:
\sqrt{\frac{a}{b}}
\sqrt[n]{1+x+x^2+x^3+\ldots}
Some people prefer writing the square root "closing" it over its content. This method arguably makes it more clear what is in the scope of the root sign. This habit is not normally used while writing with the computer, but if you still want to change the output of the square root, LaTeX gives you this possibility. Just add the following code in the preamble of your document:
% New definition of square root: % it renames \sqrt as \oldsqrt \let\oldsqrt\sqrt % it defines the new \sqrt in terms of the old one \def\sqrt{\mathpalette\DHLhksqrt} \def\DHLhksqrt#1#2{% \setbox0=\hbox{$#1\oldsqrt{#2\,}$}\dimen0=\ht0 \advance\dimen00.2\ht0 \setbox2=\hbox{\vrule height\ht0 depth \dimen0}% {\box0\lower0.4pt\box2}}
This TeX code first renames the \sqrt
command as \oldsqrt
, then redefines \sqrt
in terms of the old one, adding something more. The new square root can be seen in the picture on the left, compared to the old one on the right. Unfortunately this code won't work if you want to use multiple roots: if you try to write as \sqrt[b]{a}
after you used the code above, you'll just get a wrong output. In other words, you can redefine the square root this way only if you are not going to use multiple roots in the whole document.
An alternative piece of TeX code that does allow multiple roots is
\LetLtxMacro{\oldsqrt}{\sqrt} % makes all sqrts closed \renewcommand{\sqrt}[1][]{% \def\DHLindex{#1}\mathpalette\DHLhksqrt} \def\DHLhksqrt#1#2{% \setbox0=\hbox{$#1\oldsqrt[\DHLindex]{#2\,}$}\dimen0=\ht0 \advance\dimen00.2\ht0 \setbox2=\hbox{\vrule height\ht0 depth \dimen0}% {\box0\lower0.71pt\box2}} $\sqrt[a]{b} \quad \oldsqrt[a]{b}$
However this requires the \usepackage{letltxmacro}
package.
Sums and integrals
The \sum
and \int
commands insert the sum and integral symbols respectively, with limits specified using the caret (^
) and underscore (_
). The typical notation for sums is:
\sum_{i=1}^{10} t_i
The limits for the integrals follow the same notation. It's also important to represent the integration variables with an upright d, which in math mode is obtained through the \mathrm{}
command, and with a small space separating it from the integrand, which is attained with the \,
command.
\int_0^\infty \mathrm{e}^{x} \, \mathrm{d}x
There are many other "big" commands which operate in a similar manner:
LaTeX  Symbol  LaTeX  Symbol  LaTeX  Symbol 

\sum 
\prod 
\coprod 

\bigoplus 
\bigotimes 
\bigodot 

\bigcup 
\bigcap 
\biguplus 

\bigsqcup 
\bigvee 
\bigwedge 

\int 
\oint 
\iint ^{[4]} 

\iiint ^{[4]} 
\iiiint ^{[4]} 
\int \! \cdots \! \int 
For more integral symbols, including those not included by default in the Computer Modern font, try the esint package.
The \substack
command^{[4]} allows the use of \\
to write the limits over multiple lines:
\sum_{\substack{ 0<i<m \\ 0<j<n }} P(i,j)
If you want the limits of an integral to be specified above and below the symbol (like the sum), use the \limits
command:
\int\limits_a^b
However if you want this to apply to ALL integrals, it is preferable to specify the intlimits option when loading the amsmath package:
\usepackage[intlimits]{amsmath}

Subscripts and superscripts in other contexts as well as other parameters to amsmath package related to them are described in Advanced Mathematics chapter.
For bigger integrals, you may use personal declarations, or the bigints package ^{[5]}.
Brackets, braces and delimiters
 How to use braces in multiline equations is described in the Advanced Mathematics chapter.
The use of delimiters such as brackets soon becomes important when dealing with anything but the most trivial equations. Without them, formulas can become ambiguous. Also, special types of mathematical structures, such as matrices, typically rely on delimiters to enclose them.
There are a variety of delimiters available for use in LaTeX:
( a ), [ b ], \{ c \},  d , \ e \, \langle f \rangle, \lfloor g \rfloor, \lceil h \rceil, \ulcorner i \urcorner, \llcorner j \lrcorner
Automatic sizing
Very often mathematical features will differ in size, in which case the delimiters surrounding the expression should vary accordingly. This can be done automatically using the \left
, \right
, and \middle
commands. Any of the previous delimiters may be used in combination with these:
\left( \frac{x^2}{y^3} \right)
P \left( A = 2 \middle \frac{A^2}{B} > 4 \right)
P \left( A = 2 \middle \frac{A^2}{B} > 4 \right)
P \left( A = 2 \Bigg \frac{A^2}{B} > 4 \right)
Curly braces are defined differently by using \left\{
and \right\}
,
\left\{ \frac{x^2}{y^3} \right\}
If a delimiter on only one side of an expression is required, then an invisible delimiter on the other side may be denoted using a period (.
).
\left. \frac{x^3}{3} \right_0^1
Manual sizing
In certain cases, the sizing produced by the \left
and \right
commands may not be desirable, or you may simply want finer control over the delimiter sizes. In this case, the \big
, \Big
, \bigg
and \Bigg
modifier commands may be used:
( \big( \Big( \bigg( \Bigg(
These commands are primarily useful when dealing with nested delimiters. For example, when typesetting
\frac{\mathrm d}{\mathrm d x} \left( k g(x) \right)
we notice that the \left
and \right
commands produce the same size delimiters as those nested within it. This can be difficult to read. To fix this, we write
\frac{\mathrm d}{\mathrm d x} \big( k g(x) \big)
Manual sizing can also be useful when an equation is too large, trails off the end of the page, and must be separated into two lines using an align command. \left
and \right
will give errors if the left and right appear on different lines.
Matrices and arrays
A basic matrix may be created using the matrix environment^{[4]}: in common with other tablelike structures, entries are specified by row, with columns separated using an ampersand (&
) and a new rows separated with a double backslash (\\
)
\begin{matrix} a & b & c \\ d & e & f \\ g & h & i \end{matrix}
To specify alignment of columns in the table, use starred version^{[6]}:
\begin{matrix} 1 & 3 \\ 2 & 4 \end{matrix} = \begin{matrix} 1 & \,\;\;3 \\ \,\;\;2 & 4 \end{matrix}
The alignment by default is c but it can be any column type valid in array environment.
However matrices are usually enclosed in delimiters of some kind, and while it is possible to use the \left and \right commands, there are various other predefined environments which automatically include delimiters:
Environment name  Surrounding delimiter  Notes 

pmatrix^{[4]}  centers columns by default  
pmatrix*^{[6]}  allows to specify alignment of columns in optional parameter  
bmatrix^{[4]}  centers columns by default  
bmatrix*^{[6]}  allows to specify alignment of columns in optional parameter  
Bmatrix^{[4]}  centers columns by default  
Bmatrix*^{[6]}  allows to specify alignment of columns in optional parameter  
vmatrix^{[4]}  centers columns by default  
vmatrix*^{[6]}  allows to specify alignment of columns in optional parameter  
Vmatrix^{[4]}  centers columns by default  
Vmatrix*^{[6]}  allows to specify alignment of colums in optional parameter 
When writing down arbitrary sized matrices, it is common to use horizontal, vertical and diagonal triplets of dots (known as ellipses) to fill in certain columns and rows. These can be specified using the \cdots
, \vdots
and \ddots
respectively:
A_{m,n} = \begin{pmatrix} a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\ a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m,1} & a_{m,2} & \cdots & a_{m,n} \end{pmatrix}
In some cases you may want to have finer control of the alignment within each column, or want to insert lines between columns or rows. This can be achieved using the array environment, which is essentially a mathmode version of the tabular environment, which requires that the columns be prespecified:
\begin{array}{cc} 1 & 2 \\ \hline 3 & 4 \end{array}
You may see that the AMS matrix class of environments doesn't leave enough space when used together with fractions resulting in output similar to this:
M = \begin{bmatrix} \frac{5}{6} & \frac{1}{6} & 0 \\ \frac{5}{6} & 0 & \frac{1}{6} \\ 0 & \frac{5}{6} & \frac{1}{6} \end{bmatrix}
To counteract this problem, add additional leading space with the optional parameter to the \\
command:
M = \begin{bmatrix} \frac{5}{6} & \frac{1}{6} & 0 \\[0.3em] \frac{5}{6} & 0 & \frac{1}{6} \\[0.3em] 0 & \frac{5}{6} & \frac{1}{6} \end{bmatrix}
If you need "border" or "indexes" on your matrix, plain TeX provides the macro \bordermatrix
M = \bordermatrix{~ & x & y \cr A & 1 & 0 \cr B & 0 & 1 \cr}
Matrices in running text
To insert a small matrix, and not increase leading in the line containing it, use smallmatrix environment:
A matrix in text must be set smaller: $\bigl( \begin{smallmatrix} a&b \\ c&d \end{smallmatrix} \bigr)$ to not increase leading in a portion of text.
Adding text to equations
The math environment differs from the text environment in the representation of text. Here is an example of trying to represent text within the math environment:
50 apples \times 100 apples = lots of apples^2
There are two noticeable problems: there are no spaces between words or numbers, and the letters are italicized and more spaced out than normal. Both issues are simply artifacts of the maths mode, in that it treats it as a mathematical expression: spaces are ignored (LaTeX spaces mathematics according to its own rules), and each character is a separate element (so are not positioned as closely as normal text).
There are a number of ways that text can be added properly. The typical way is to wrap the text with the \text{...}
command ^{[4]} (a similar command is \mbox{...}
, though this causes problems with subscripts, and has a less descriptive name). Let's see what happens when the above equation code is adapted:
50 \text{apples} \times 100 \text{apples} = \text{lots of apples}^2
The text looks better. However, there are no gaps between the numbers and the words. Unfortunately, you are required to explicitly add these. There are many ways to add spaces between maths elements, but for the sake of simplicity you may literally add the space character in the affected \text
(s) itself (just before the text).
50 \text{ apples} \times 100 \text{ apples} = \text{lots of apples}^2
Formatted text
Using the \text
is fine and gets the basic result. Yet, there is an alternative that offers a little more flexibility. You may recall the introduction of font formatting commands, such as \textrm
, \textit
, \textbf
, etc. These commands format the argument accordingly, e.g., \textbf{bold text}
gives bold text. These commands are equally valid within a maths environment to include text. The added benefit here is that you can have better control over the font formatting, rather than the standard text achieved with \text
.
50 \textrm{ apples} \times 100 \textbf{ apples} = \textit{lots of apples}^2
Here we need to use ~ to get spaces.
50 \textrm{~apples} \times 100 \textbf{~apples} = \textit{lots~of~apples}^2
Formatting mathematics symbols
We can now format text; what about formatting mathematical expressions? There are a set of formatting commands very similar to the font formatting ones just used, except that they are specifically aimed at text in math mode (requires amsfonts)
LaTeX command  Sample  Description  Common use 

\mathnormal{…}

the default math font  most mathematical notation  
\mathrm{…}

this is the default or normal font, unitalicised  units of measurement, one word functions  
\mathit{…}

italicised font  
\mathbf{…}

bold font  vectors  
\mathsf{…}

Sansserif  
\mathtt{…}

ABCDEFabcdef123456  Monospace (fixedwidth) font  
\mathcal{…}

Calligraphy (uppercase only)  often used for sheaves/schemes and categories, used to denote cryptological concepts like an alphabet of definition (), message space (), ciphertext space () and key space (); Kleene's ; naming convention in description logic  
\mathfrak{…} ^{[7]}

Fraktur  Almost canonical font for Lie algebras, with superscript used to denote New Testament papyri, ideals in ring theory  
\mathbb{…} ^{[7]}

Blackboard bold  Used to denote special sets (e.g. real numbers)  
\mathscr{…} ^{[8]}

Script  An alternative font for categories and sheaves. 
The maths formatting commands can be wrapped around the entire equation, and not just on the textual elements: they only format letters, numbers, and uppercase Greek, and the rest of the maths syntax is ignored.
To bold lowercase Greek or other symbols use the \boldsymbol
command^{[4]}; this will only work if there exists a bold version of the symbol in the current font. As a last resort there is the \pmb
command^{[4]} (poor mans bold): this prints multiple versions of the character slightly offset against each other
\boldsymbol{\beta} = (\beta_1,\beta_2,\dotsc,\beta_n)
To change the size of the fonts in math mode, see [[../Advanced Mathematics#Changing font sizeChanging font size]].
Accents
So what to do when you run out of symbols and fonts? Well, the next step is to use accents.
Symbol  LaTeX  Comment  Symbol  LaTeX  Comment 

\acute{a} 
\grave{a} 

\hat{a} 
circumflex  \check{a} 

\breve{a} 
\tilde{a} 

\bar{a} 
\vec{a} 
vector  
\overline{a} 
\underline{a} 

\not\!{a} 

a' \, 
prime  a'' \, 
double prime  
a''' \, 
a'''' \, 

\dot{a} 
fluxion  \ddot{a} 
double fluxion  
\overrightarrow{AB} 
right ray  \overleftarrow{AB} 
left ray  
\overline{pbdq} 
\underline{pbdq} 

\widehat{pbdq} 

Color
The package xcolor, described in Colors, allows us to add color to our equations. For example
k = {\color{red}x}{\color{blue}}2
The only problem is that this disrupts the default LaTeX formatting around the 
operator. To fix this, we enclose it in a \mathbin
environment, since 
is a binary operator. This process is described here.
Plus and minus signs
Latex deals with the + and − signs in two possible ways. The most common is as a binary operator. When two maths elements appear either side of the sign, it is assumed to be a binary operator, and as such, allocates some space either side of the sign. The alternative way is a sign designation. This is when you state whether a mathematical quantity is either positive or negative. This is common for the latter, as in maths, such elements are assumed to be positive unless a − is prefixed to it. In this instance, you want the sign to appear close to the appropriate element to show their association. If you put a + or a − with nothing before it but you want it to be handled like a binary operator you can add an invisible character before the operator using {}
. This can be useful if you are writing multipleline formulas, and a new line could start with a = or a +, for example, then you can fix some strange alignments adding the invisible character where necessary.
A plusminus sign (or minusplus sign) used for uncertainty is written as
\pm (\mp)
Controlling horizontal spacing
LaTeX is obviously pretty good at typesetting maths—it was one of the chief aims of the core TeX system that LaTeX extends. However, it can't always be relied upon to accurately interpret formulas in the way you did. It has to make certain assumptions when there are ambiguous expressions. The result tends to be slightly incorrect horizontal spacing. In these events, the output is still satisfactory, yet any perfectionists will no doubt wish to finetune their formulas to ensure spacing is correct. These are generally very subtle adjustments.
There are other occasions where LaTeX has done its job correctly, but you just want to add some space, maybe to add a comment of some kind. For example, in the following equation, it is preferable to ensure there is a decent amount of space between the maths and the text.
f(n) = \begin{cases} n/2 & \quad \text{if } n \text{ is even}; \\ (n+1)/2 & \quad \text{if } n \text{ is odd}. \end{cases}
This code produces errors with Miktex 2.9 and does not yield the results seen on the right. Use \textrm instead of just \text.
(Note that this particular example can be expressed in more elegant code by the cases construct provided by the amsmath package described in Advanced Mathematics chapter.)
LaTeX has defined two commands that can be used anywhere in documents (not just maths) to insert some horizontal space. They are \quad
and \qquad
A \quad
is a space equal to the current font size. So, if you are using an 11pt font, then the space provided by \quad
will also be 11pt (horizontally, of course.) The \qquad
gives twice that amount. As you can see from the code from the above example, \quad
s were used to add some separation between the maths and the text.
OK, so back to the fine tuning as mentioned at the beginning of the document. A good example would be displaying the simple equation for the indefinite integral of y with respect to x:
If you were to try this, you may write:
\int y \mathrm{d}x
However, this doesn't give the correct result. LaTeX doesn't respect the whitespace left in the code to signify that the y and the dx are independent entities. Instead, it lumps them altogether. A \quad
would clearly be overkill in this situation—what is needed are some small spaces to be utilized in this type of instance, and that's what LaTeX provides:
LaTeX command  Description  Size 

\,

small space  3/18 of a quad 
\:

medium space  4/18 of a quad 
\;

large space  5/18 of a quad 
\!

negative space  3/18 of a quad 
N.B.: you can use more than one command in a sequence to achieve a greater space if necessary.
So, to rectify the current problem:
\int y \, \mathrm{d}x
\int y \: \mathrm{d}x
\int y { \;\;\!\! } \mathrm{d}x
\int y \; \mathrm{d}x
The negative space may seem like an odd thing to use, however, it wouldn't be there if it didn't have some use! Take the following example:
\left( \begin{array}{c} n \\ r \end{array} \right) = \frac{n!}{r!(nr)!}
The matrixlike expression for representing binomial coefficients is too padded. There is too much space between the brackets and the actual contents within. This can easily be corrected by adding a few negative spaces after the left bracket and before the right bracket.
\left(\! \begin{array}{c} n \\ r \end{array} \!\right) = \frac{n!}{r!(nr)!}
In any case, adding some spaces manually should be avoided whenever possible: it makes the source code more complex and it's against the basic principles of a What You See is What You Mean approach. The best thing to do is to define some commands using all the spaces you want and then, when you use your command, you don't have to add any other space. Later, if you change your mind about the length of the horizontal space, you can easily change it modifying only the command you defined before. Let us use an example: you want the d of a dx in an integral to be in roman font and a small space away from the rest. If you want to type an integral like \int x \; \mathrm{d} x
, you can define a command like this:
\newcommand{\dd}{ \; \mathrm{d} }

in the preamble of your document. We have chosen \dd
just because it reminds the "d" it replaces and it is fast to type. Doing so, the code for your integral becomes \int x \dd x
. Now, whenever you write an integral, you just have to use the \dd
instead of the "d", and all your integrals will have the same style. If you change your mind, you just have to change the definition in the preamble, and all your integrals will be changed accordingly.
Advanced Mathematics: AMS Math package
The AMS (American Mathematical Society) mathematics package is a powerful package that creates a higher layer of abstraction over the mathematical LaTeX language; if you use it it will make your life easier. Some commands amsmath introduces will make other plain LaTeX commands obsolete: in order to keep consistency in the final output you'd better use amsmath commands whenever possible. If you do so, you will get an elegant output without worrying about alignment and other details, keeping your source code readable. If you want to use it, you have to add this in the preamble:
\usepackage{amsmath}

Introducing text and dots in formulas
amsmath defines also the \dots
command, that is a generalization of the existing \ldots
. You can use \dots
in both text and math mode and LaTeX will replace it with three dots "…" but it will decide according to the context whether to put it on the bottom (like \ldots
) or centered (like \cdots
).
Dots
LaTeX gives you several commands to insert dots in your formulae. This can be particularly useful if you have to type big matrices omitting elements. First of all, here are the main dotsrelated commands LaTeX provides:
LaTeX code  Output  Comment 

a \dots b 
Generic dots, to be used in text (outside formulae as well). It automatically manages whitespaces before and after itself according to the context, it's a higher level command.  
a \ldots b 
The output is similar to the previous one, but there is no automatic whitespace management; it works at a lower level.  
a \cdots b 
These dots are centered relative to the height of a letter. There is also the binary multiplication operator, \cdot, mentioned below.  
\vdots 
vertical dots  
\ddots 
diagonal dots  
\iddots 
inverse diagonal dots (requires the mathdots package) 
Instead of using \ldots
and \cdots
, you should use the semantically oriented commands. It makes it possible to adapt your document to different conventions on the fly, in case (for example) you have to submit it to a publisher who insists on following house tradition in this respect. The default treatment for the various kinds follows American Mathematical Society conventions.
LaTeX code  Output  Comment 

A_1,A_2,\dotsc, 
File:LaTeX Dotsc.png  for "dots with commas" 
A_1+\dotsb+A_N 
File:LaTeX Dotsb.png  for "dots with binary operators/relations" 
A_1 \dotsm A_N 
File:LaTeX Dotsm.png  for "multiplication dots" 
\int_a^b \dotsi 
File:LaTeX Dotsi.png  for "dots with integrals" 
A_1\dotso A_N 
File:LaTeX Dotso.png  for "other dots" (none of the above) 
Write an equation with the align environment
How to write an equation with the align environment with the amsmath package is described in Advanced Mathematics.
List of mathematical symbols
See List of LaTeX mathematical symbols.
Summary
As you begin to see, typesetting math can be tricky at times. However, because Latex provides so much control, you can get professional quality mathematics typesetting with relatively little effort (once you've had a bit of practice, of course!). It would be possible to keep going and going with math topics because it seems potentially limitless. However, with this tutorial, you should be able to get along sufficiently.
TO DO. ^{(Tasks list)} ^{[9]}
Contentoriented LaTeX
Mathematical Knowledge Management (MKM) / Semantic Web:
 MKMIG.
 Christoph Lange and Michael Kohlhase, A Semantic Wiki for Mathematical Knowledge Management. (Semantic Web: Web of data for applications. Semantic MK Markup with OMDoc.)
 Massimo Marchiori, The Mathematical Semantic Web. (Semantic Web architecture based on RDF.)
 Christoph Lange, Integrating Mathematics into the Web of Data.
Semantic MK markup for LaTeX:
 N. Setzer, Content LaTeX2ε, 2006.
 Michael Kohlhase, Using LaTeX as a Semantic Markup Format.
 sTeX: Semantic Markup for LaTeX, ©2018 The KWARC Group.
 LaTeXML. (Support for representing and generating RDFa metadata.)
Other semantic MK markup formats:
 Content MathML: only covers mathematical formulae.
 OpenMath: only supports formulae and “content dictionaries” (with OpenMath Content Dictionary (CD) extensions) containing definitions of the symbols used in formulae.
 OMDoc (Open Mathematical Documents): covers the whole range of written mathematics.
See also
 TeX: {{TeX}} (TeX graphic template)
 LaTeX: {{LaTeX}} (LaTeX graphic template)
 LaTeX2ε: {{LaTeX2e}} (LaTeX2ε graphic template)
 Content LaTeX2ε
Notes
 ↑ Donald E. Knuth's Home Page.
 ↑ http://www.ams.org/publications/authors/tex/amslatex
 ↑ http://www.tex.ac.uk/ctan/macros/latex/contrib/mh/mathtools.pdf
 ↑ ^{4.00} ^{4.01} ^{4.02} ^{4.03} ^{4.04} ^{4.05} ^{4.06} ^{4.07} ^{4.08} ^{4.09} ^{4.10} ^{4.11} ^{4.12} ^{4.13} ^{4.14} ^{4.15} requires the amsmath package
 ↑ http://hdl.handle.net/2268/6219
 ↑ ^{6.0} ^{6.1} ^{6.2} ^{6.3} ^{6.4} ^{6.5} requires the mathtools package
 ↑ ^{7.0} ^{7.1} requires amsfonts or amssymb packages
 ↑ require mathrsfs package
 ↑ To do: Tasks list (
 introduce symbols from [1]
 add symbols from [2]
 consider adding symbols from [3]—the list of nearly all symbols available for LaTeX
 consider, instead of using the symbols from the above mentioned, using what has already been introduced in [4] instead of retyping the tables
 how to box an equation within an align environment
 color in equations
Further reading
 Help:Displaying a formula—Wikimedia MetaWiki.org. (Wikimedia uses a subset of LaTeX commands.)
External links
 LaTeX maths symbols
 detexify (applet for looking up LaTeX symbols by drawing them)
 User’s Guide for the amsmath Package (Version 2.0)—AMS.
 Michael Downes, Short Math Guide for LaTeX—AMS.
 LaTeX  The Student Room
 The Comprehensive LaTeX Symbol List
 LaTeX Mathematical Symbol
 The LaTeX Font Catalogue