Help:Editing
You should feel welcome to contribute, even if you think what you write doesn't match the 'house style'.
In that case, this page is more to explain the meaning behind any minor changes that may be applied to your work after it has been submitted.
Over the course of prolonged contribution, it is considered both polite and very desirable that contributors put effort in mastering and adhering to house style, which is located at Help:House Style.
Should there be any questions, Help:Questions is the place to raise issues.
Page Editing
This reference provides the general outline on creating pages and instructions for specific sections; for the rest, see Help:House Style. Furthermore, this page does not dwell on details pertaining to the MediaWiki architecture; the reference for such things is Help:Wiki Editing.
Creating a Page
After searching, you may conclude that the page you're looking for does not exist yet.
There are several ways to create it, as follows:
By Searching
Search for the page name which you would like to create.
As you have found out by now, the page has not already been created.
You will see something that looks like this:
 Create the page "Proof name" on this wiki!
Clicking on Proof name will open the page editing window, and you can start writing.
By following a red link
All you have to do is click the red link and start writing.
By entering a direct URL
If you know for sure that the proof is not on the site, then simply type that name into the URL.
For example, if you wanted to prove the Riemann Hypothesis, you would type:
http://www.proofwiki.org/wiki/Riemann Hypothesis
If this page does not exist then, you will get a single line saying:
 There is currently no text in this page. You can search for this page title in other pages, search the related logs, or create this page.
Clicking create this page will open the editing page where you can edit the page.
Deleting a page
Don't do this. Only trusted users can delete pages in any event.
If you feel a page needs to be deleted, add {{Delete}}
to the top of the page in question.
Page Naming
Choosing a good page title is important for multiple reasons:
 It makes the page easier to search for.
 The title gives a first impression of what the page is about. A good title adds to the understanding.
 The title is what is used in links. It's always nice to know what a page is about before clicking the link.
 A descriptive title avoids name conflicts.
Some pages are subject to specific naming conventions. See the corresponding help pages for instructions:
Page titles that do not meet the standards can be flagged for renaming.
Tips to choose a good name
Be specific
There's no such thing as being too specific!
In particular, page names consisting of a single word should generally be avoided: you never know if there are other concepts in mathematics with the same name.
That is:
 not Definition:Prime Ideal but Definition:Prime Ideal of Ring
 not Definition:Supremum or Definition:Supremum of Set but Definition:Supremum of Subset of Real Numbers
The word "of" comes in handy here.
In the same spirit, it is a good practice to always include at least one noun in the title.
That is:
 not Definition:Differentiable but Definition:Differentiable Mapping
 not Definition:Simple but Definition:Simple Group
Note how the shorter page names are always disambiguation pages.
For theorems, the same philosophy applies:
Redundant words
Page names should not be started with articles such as "A" or "The", as this makes it significantly harder to find pages alphabetically in their categories.
For example: Pythagorean Theorem, not The Pythagorean Theorem.
Similarly, is not necessary to begin the name with "Proof of ...", and this should be avoided.
Since $\mathsf{Pr} \infty \mathsf{fWiki}$ is a collection of proofs, it is assumed that each page is going to be a proof of something.
This also makes searching for articles much easier.
In general, the words "the", "a" or "an" are strongly discouraged in page names, as they make the name longer than necessary and rarely add clarity to the concept being defined.
The main exception to this rule is for concepts split into subtypes named along the lines "of the $n$th kind", for example: Definition:Elliptic Integral of the First Kind.
Descriptiveness
A good page title describes the content accurately.
Because theorems are linked to using their exact page title, when reading a proof it is useful when you can guess what a theorem is about without having to visit the page.
In particular, if a theorem contains an equivalence proof, the title should makes this clear, by using "iff".
Don't blindly trust literature
Books, thanks to their limited scope, can afford to use simplified terminology without running into ambiguity problems.
At $\mathsf{Pr} \infty \mathsf{fWiki}$ we cannot.
Thus it is a good idea to check if there is a danger for name conflicts by doing some research.
Alternatively, make the name overly specific.
As for theorems, books may call a theorem "Fundamental Property of Homomorphisms" or "Continuity Property".
This does not mean that the theorem is everywhere known by that name.
While at $\mathsf{Pr} \infty \mathsf{fWiki}$ we do prefer to use as a title the name of a theorem rather than a description of the result, we do so only if there is no ambiguity.
Naming conventions
Namespaces
When you would like to create a page for a definition, all you have to do different from naming a proof is to add Definition:
in front of the name.
So for example if you wanted to create a page for the definition of calculus you would name the page:
Definition:Calculus
Also, after you create the page, be sure to add the definition to the appropriate "Definitions" categories (see Category:Definitions).
The same method that is used for Definitions is also used to name and categorize axiom and symbols pages.
Simply substitute Symbol
or Axiom
for Definition:
in the page name and, mutatis mutandis, in the category name.
See Help:Categories.
Capitalizing
Page titles are casesensitive.
For all types of pages, major words in the title of the page should be capitalized.
For example:
So as to promote consistency, be informed that in particular, the following words are not considered to be "major" and ought to be used in their lowercase form when naming a page:
 Prepositions, pronouns and conjunctions:
 and, around, as, between, by, for, from, if, iff, in, its, minus, no, not, of, on, or, over, plus, such that, that, the, then, to, under, with, with respect to, which, whose
 Short verbs, such as:
 are, can, cannot, does, equals, form, has, have, implies, is
These lists are not exhaustive.
Variables in formulas tend not to be capitalized.
See Names with Formulas.
Names with Formulas
If the name of a page contains mathematical statements, the following formatting practices should be adhered to:
 Put no spaces between numbers/elements that are added, subtracted, divided, or multiplied together.
 Example:
(1+2)x3
 Example:
 Do, however, put spaces between elements that are put into equality or inequality with one another. Use
!=
to signify inequality. Example:
1 = 3/3 != 3/4
 Example:
 Variables appearing in the formulas need not be capitalized.
 Example:
Primitive of x squared over a x + b
 Example:
 A page name is no place for $\LaTeX$ commands.
Things named after a mathematician
A theorem that is named after a Mathematician gets a call of the {{Namedfor}}
template.
When done properly, the page is then automatically placed in a corresponding category in Named Theorems, which itself has to be created manually.
Similarly, for named definitions there is the {{NamedforDef}}
template.
Disambiguation pages are treated in the same way.
Note that theorem disambiguations do not otherwise get categories.
See Help:Disambiguation.
A page whose name contains a word named after a Mathematician does not fall under this category.
For example, not every theorem about Krull dimension needs a call of {{Namedfor}}
.
Multiple Names
Some mathematical concepts have several names, according to the sources you consult.
Which of these names is used in $\mathsf{Pr} \infty \mathsf{fWiki}$ is largely a matter of happenstance.
However, if one of the names of a concept is for a particular mathematician, that name is in general to be used in preference, unless that usage is obscure.
An example of this is Definition:Chebyshev Distance, which is otherwise known as the Definition:Maximum Metric or the Definition:Chessboard Metric.
An example where we do not do this is for Definition:Spherical Triangle: although this is also known as an Definition:Euler Triangle, we use the former in preference.
See also Help:Also known as.
Special Characters
Disallowed Characters
The following characters should not be used in page names:
# < > [ ]  { } * & $ @
Accented Characters
A theorem named after someone gets this exact name, including accented characters.
For search convenience, a redirect can be set up.
See Help:Redirects#Accented characters
Page Structure
The purpose of this page is to describe the general structure that the various sections most used on $\mathsf{Pr} \infty \mathsf{fWiki}$ are to adhere to. While this naturally intersects with House Style at some points, an attempt is made to separate the global editing instructions and sectionspecific instructions.
Introduction
On all pages (except for talk and user pages), the House Style applies.
This page gives the general structure that applies to all pages. Click on the links below for the more precise expected format, which depends on the type of page:
General Format
Generally, pages follow this format:
== Theorem == State the theorem here. == Proof == State the proof here. == Also see == * List of (internal) links to closely related material. == Sources == Add citations here. [[Category:The Category]]
Sections
Below, various recurring sections on $\mathsf{Pr} \infty \mathsf{fWiki}$ pages and their particular rules are explained. All of these should have a type 2 heading.
The sections should be placed in the following order, with this exact capitalization:
 Definition / Theorem
 Proof(s)
 Also known as
 Also defined as
 Also see
 Named for
 Historical Note
 Linguistic Note
 Technical Note
 Sources
Other optional sections include:
and more, such as remarks, comments, notations, for which there are no written guidelines yet.
Definitions and Theorems
These are in practice split into two parts (which is made visual by extra blank lines separating them).
Namely, first there is a series of lines, typically starting with "Let", introducing all names and concepts needed for stating the actual definition or theorem.
Then, separated by two blank lines, the definition or theorem itself is stated. Thus, we obtain the following structure (analogous for Theorems):
== Definition == Let ... ... Let ... Then '''what is to be defined''' is defined as ...
The concept that is to be defined is to be displayed in bold (i.e., with three apostrophes, '
, on either side) throughout the page to make it stand out.
This article is complete as far as it goes, but it could do with expansion. In particular: Discuss corollaries and other subpages You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. 
Proofs
Besides adhering to house style, it is a good idea to separate different stages of the proof by subsections or whitespace. Other than that, rigour is the only real prerequisite for proofs.
If you would like other contributors to check your proof, please use the proofread template.
Notes (avoid)
Try to avoid having a general "Notes" section on a page you create or edit. Instead, try to put any remarks you want to add into one of the sections listed next.
Explanation
This section can be used to explain concepts, in particular on pages that give definitions or describe types of proofs. To give context, consider using an "Also see" section.
Also known as
Use this section when a concept or result is referred to in multiple ways; this is most commonly used for definitions.
All names should appear in bold. Should an alternative name coincide with the $\mathsf{Pr} \infty \mathsf{fWiki}$ name of something else, it is good to draw the reader's attention to this by including a link and a comment.
Also defined as
Use this section when a single name is used in the literature for multiple definitions. Typically, it is to be used mainly when these definitions are in the same field of mathematics. In other cases, a disambiguation is usually more appropriate. See Help:Disambiguation for instructions on that.
It is advisable to create a synthesis of this and the "Also known as" section. That is, to place "Also known as" sections on pages that are referred to in this section.
Also see
The "Also see" section is intended to contain references to closely related concepts and/or results. These are entered as a bulleted list:
== Also see == * [[Check This Out 1]] * [[Check This Out 2]] * Etc.
It is understood that definitions should be referenced in this section directly, without providing a reader view. This is to make it easy to see which entries are definitions and which are proofs.
Thus:
* [[Definition:Increasing Sequence of Sets]]
is a correct entry.
In addition to the above, when a definition has an associated category, this category is to be referenced as well.
For example, Definition:Set Union refers to Category:Set Union. This is accomplished by the {{Linktocategory}}
template, entered as:
{{LinktocategorySet Unionset union}}
More documentation for this template can be found on its page: Template:Linktocategory.
Source of Name
This section is exclusively created by the {{Namedfor}}
template.
It is entered as:
{{NamedforName of mathematiciancat = Surname of mathematician}}
where Surname of mathematician
is actually the name of the mathematician's subcategory of Category:Named Theorems  multiple notable mathematicians with identical surnames exist.
If some page is named for multiple mathematicians (e.g. CayleyHamilton Theorem) they should all be listed, via:
{{NamedforName 1cat = Surname 1name2 = Name 2cat2 = Surname 2}} {{NamedforName 1cat = Surname 1name2 = Name 2cat2 = Surname 2name3 = Name 3cat3 = Surname 3}}
A similar technique is used for definitions.
If a definition is named for a particular mathematician, then the {{NamedforDef}}
template is used:
{{NamedforDefName of mathematiciancat = Surname of mathematician}}
and again for axioms:
{{NamedforAxiomName of mathematiciancat = Surname of mathematician}}
The same extensions apply for multiple mathematicians.
Historical Note
The Historical Note section is intended as a relatively freeform section in which any interesting information about the concept can be elaborated on.
If there is already a "Source of Name" section, then if what you want to say consists of a sentence or two, it may be better just to add it to directly after your invocation of the namedfor template. See Fermat's Little Theorem for a simple example. On the other hand, see Fermat's Two Squares Theorem for an example of where the author has considered it appropriate to create a separate section.
If you have a great deal to say about the subject in question, then it is worth considering whether to write it as a separate transcluded page.
Indeed, if you have a strong interest and expertise in the history of mathematics and wish to impart that knowledge on this website, then it may be a worthwhile future task setting up a properly structured category for the history of mathematics, into which we may find it worthwhile to migrate, for example, our Mathematicians space into.
This is one area of $\mathsf{Pr} \infty \mathsf{fWiki}$ whose evolution is in progress.
Linguistic Note
If a term being defined is not a common word in natural language, then it may be appropriate to give an indication of various linguistic characteristics of that word.
Examples of this are:
 Its pronunciation (for example: see Definition:Integer)
 Its plural form (for example: see Definition:Continuum (Topology))
 Its etymology (for example: see Definition:Summand)
Boldface is used for all words which directly relate to the term being defined.
The pronunciation is given in simple, phonetic English, with syllables separated by hyphens.
Stressed syllables are indicated in italics, hence the rendering: syllable.
Note that the Linguistic Note section is not mandatory for any page; it is created only when there is a need.
It needs to be remembered that $\mathsf{Pr} \infty \mathsf{fWiki}$ is accessed by users worldwide, to whom English is not their first language, and may not be familiar with many aspects of mathematical language which may be taken for granted by a native English speaker.
Also note that if there are differences between UK and US English forms of the spelling, the pronunciation or the plural form of any term, then this is the section to document it.
Technical Note
On definitions pages, typically some notation is introduced.
When rendering this notation using $\LaTeX$ requires some involved trickery, the code for achieving this may be explained in a section named "Technical Note".
See Definition:Convergence in Measure for an example.
Sources
This section serves to list the sources backing up a certain page. Because this section is of paramount importance for the reliability of $\mathsf{Pr} \infty \mathsf{fWiki}$, it and its constituents are discussed in detail on a dedicated page, Help:Sources.
Categories
At the very bottom of the page, categories have to be added. See Help:Categories for documentation.
Wiki Editing
The definitive reference on wiki editing is of course MediaWiki's own site, in particular the MediaWiki Help Pages.
The below gives an overview of the wiki functionality that is most frequently used on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Text styles
Text you would like to be bold should be enclosed in three single quotes. For example:
'''Bold text'''
creates: Bold text.
Italic font is produced with two single quotes, for example:
''Italic text''
creates: Italic text
To produce bold, italicised writing, use five consecutive single quotes:
'''''Bold and italic text'''''
creates: Bold and italic text.
Fixed width text:
<code> Fixed width text</code>
creates: Fixed width text
Note that while:
<tt> Fixed width text </tt>
produces the same result, the tt
tag is known to be badly behaved in combination with MediaWiki, creating rendering issues.
Therefore, the code
tag is to be used universally on $\mathsf{Pr} \infty \mathsf{fWiki}$ to generate fixed width text.
Line breaking
Breaking a line in the editor does not break a line in the displayed text. For example:
''I will continue this'' ''on the next line''
displays as:
I will continue this on the next line.
To break a line, space the text with a single blank line:
''I will continue this'' ''on the next line''
Sections and subsections
A section heading is produced by enclosing the desired title between two equals (=
) signs. For example:
== Definition ==
will produce a section entitled "Definition".
A subsection should be enclosed between three equals signs, for example:
=== Subsection ===
will produce a subsection within the section that it is located.
One can continue making subsubsections etc. with four or more equals signs.
For documentation of the accepted house style regarding sections, see Help:House Style.
Inclined to move the below to referenced Help:House Style.
As a general rule on spacing, each section or subsection heading should be preceded by two blank lines and followed by a single blank line.
Each sentence should appear on a separate line, with a single blank line above and below. For example:
This line precedes the section. == Section heading == Line one of the section. Line two of the section.
Indenting and bullet points
To indent a sentence or equation, add a colon (:
) at the start of the line. For example:
:Indented material
will produce:
 Indented material
For a larger indent use two or more colons.
For example, it is required by house style that each displayed equation be preceded with a single colon.
To produce a list of bullet points, write each point on a line beginning with an asterisk (*
). For example:
* Bullet point 1 * Bullet point 2
produces:
 Bullet point 1
 Bullet point 2
As a notable discrepancy in house style, the empty lines guideline does not apply to successive items of lists.
This is because doing so results in a slight spacing issue, as seen in:
* Bullet point 1 * Bullet point 2 * Bullet point 3
which renders as:
 Bullet point 1
 Bullet point 2
 Bullet point 3
On $\mathsf{Pr} \infty \mathsf{fWiki}$, these lists are mainly used in the Also see and Sources sections.
Link to internal page
To link to another page on $\mathsf{Pr} \infty \mathsf{fWiki}$, you must enclose the page name in double square brackets.
For example, to link to Pythagoras's Theorem, you would type:
[[Pythagoras's Theorem]]
To change the text to say something different and still link to that page you need to use a pipe character (
).
For example to link here you would type:
[[Pythagoras's Theoremhere]]
To link to a particular section on a page, append a octothorpe (#
) to the page name followed by the subsection.
For example, to link here you would type:
[[Pythagoras's Theorem#The Classic Proofhere]]
This means of reference is generally going out of fashion on $\mathsf{Pr} \infty \mathsf{fWiki}$ due to the advent of the transclusion method.
In the above example, it would be more correct to refer to the subpage containing the "Classic Proof", viz here:
[[Pythagoras's Theorem/Classic Proofhere]]
There may be instances though where its use will be continued, for example to link to particular sections of the main wiki talk page in discussions.
For more information on the house style stipulations regarding internal reference, see Help:House Style.
Redirecting a page
See also Help:Redirects
If a theorem or definition is commonly known by more than one name, rather than create two separate pages just one page should be created, and the second redirected to the first.
This should be done with the command:
#Redirect [[Page Name]]
For example, to redirect a page to Pythagoras's Theorem enter:
#Redirect [[Pythagoras's Theorem]]
See Help:Redirects for more information on redirecting pages and the associated house style.
Link to external page
Linking to an external page is almost the same as linking to an internal page.
You need to enclose the page name in square brackets, []
.
To add text the link, you just have to add what you want to name the link inside the brackets.
For example:
[http://www.google.com Google]
would create a link that looked like this: Google
References
See also Help:Citations
References can easily be created by enclose the reference inside a <ref></ref> tag in the area you would like referenced. Then near the end of your article include this: <references/> . This will be where the references are actually listed. For an example see the below:
According to scientists, the Sun is pretty big.^{[1]} The Moon, however, is not so big.^{[2]}
Notes
The code for this:
According to scientists, the Sun is pretty big.<ref>E. Miller, The Sun, (New York: Academic Press, 2005), 235.</ref> The Moon, however, is not so big.<ref>R. Smith, "Size of the Moon", Scientific American, 46 (April 1978): 446.</ref> '''Notes''' <references/>
Typically, it is desirable to collect the references under a designated 'type2' header named 'References', whose code is:
== References ==
This section is best placed directly above the "Sources" section. For more information on page structuring, see Help:Page Editing.
Transclusion
Transclusion is a MediaWiki feature that allows (part of) a page to be displayed verbatim on another.
Since the construct is rather involved, it is explained on its own page, Help:Transclusion.
The $\mathsf{Pr} \infty \mathsf{fWiki}$ extension
The $\mathsf{Pr} \infty \mathsf{fWiki}$ extension is an amendment of MediaWiki's code that enables certain sitespecific constructs.
See Help:ProofWiki Extension for documentation on this feature.
$\LaTeX$ Editing
In general, contributors are assumed to be up to speed with some form of $\LaTeX$; a web search should be sufficient to find ample reference on how to get started with it, should you still need to.
The "External references" section below may also be consulted.
The $\LaTeX$ interpreter used on this site is brought to you by MathJax.
This produces an experience different from that produced by the MediaWiki interpreter which is (at time of writing) the one used by Wikipedia and other places.
It also has a subtly different syntax in places. Specific instances will be detailed where relevant.
$\LaTeX$ delimiters
To display an equation in line with some text, the equation should be enclosed in single dollar signs: $ ... $
Note that \( ... \)
also works, but takes more effort to type and so is less recommended.
There may (but we hope not) still be some pages with <math> ... </math>
in them. This is a holdover from when MediaWiki was the interpreter used for $\LaTeX$ commands. It still works in MathJax after a fashion but on transcluded pages, such enclosed $\LaTeX$ will not be converted to mathematical symbols.
If you see any, then feel free to change them to $
signs, as they should not be there.
No longer supported
The following $\LaTeX$ commands are not supported in MathJax, but may still be present in some pages. When found they need to be replaced.
 For $\lor$:
\or
to be replaced by\lor
 For $\land$:
\and
to be replaced by\land
 For $\R$:
\reals
to be replaced by\R
 For $\exists$:
\exist
to be replaced by\exists
 For producing fixed width text in math mode:
\texttt
needs to be replaced by\mathtt
.
If you find any more examples, add them here.
New commands
New commands can be requested and discussed at Symbols:LaTeX Commands/ProofWiki Specific, transcluded here:
\(\AA\)  $\quad:\quad$\AA

$\qquad$that is: \mathcal A


\(\Add\)  $\quad:\quad$\Add

$\qquad$Addition as a Primitive Recursive Function  
\(\adj {\mathbf A}\)  $\quad:\quad$\adj {\mathbf A}

$\qquad$Adjugate Matrix  
\(\map \Ai {x}\)  $\quad:\quad$\map \Ai {x}

$\qquad$Airy Function of the First Kind  
\(\am z\)  $\quad:\quad$\am z

$\qquad$Amplitude  
\(\arccot\)  $\quad:\quad$\arccot

$\qquad$Arccotangent  
\(\arccsc\)  $\quad:\quad$\arccsc

$\qquad$Arccosecant  
\(\arcosh\)  $\quad:\quad$\arcosh

$\qquad$Area Hyperbolic Cosine  
\(\Arcosh\)  $\quad:\quad$\Arcosh

$\qquad$Complex Area Hyperbolic Cosine  
\(\arcoth\)  $\quad:\quad$\arcoth

$\qquad$Area Hyperbolic Cotangent  
\(\Arcoth\)  $\quad:\quad$\Arcoth

$\qquad$Complex Area Hyperbolic Cotangent  
\(\arcsch\)  $\quad:\quad$\arcsch

$\qquad$Area Hyperbolic Cosecant  
\(\Arcsch\)  $\quad:\quad$\Arcsch

$\qquad$Complex Area Hyperbolic Cosecant  
\(\arcsec\)  $\quad:\quad$\arcsec

$\qquad$Arcsecant  
\(\arsech\)  $\quad:\quad$\arsech

$\qquad$Area Hyperbolic Secant  
\(\Arsech\)  $\quad:\quad$\Arsech

$\qquad$Complex Area Hyperbolic Secant  
\(\arsinh\)  $\quad:\quad$\arsinh

$\qquad$Area Hyperbolic Sine  
\(\Arsinh\)  $\quad:\quad$\Arsinh

$\qquad$Complex Area Hyperbolic Sine  
\(\artanh\)  $\quad:\quad$\artanh

$\qquad$Area Hyperbolic Tangent  
\(\Artanh\)  $\quad:\quad$\Artanh

$\qquad$Complex Area Hyperbolic Tangent  
\(\Area\)  $\quad:\quad$\Area

$\qquad$Area of Plane Figure  
\(\Arg z\)  $\quad:\quad$\Arg z

$\qquad$Principal Argument of Complex Number  
\(\Aut {S}\)  $\quad:\quad$\Aut {S}

$\qquad$Automorphism Group  
\(\BB\)  $\quad:\quad$\BB

$\qquad$that is: \mathcal B


\(\Bei\)  $\quad:\quad$\Bei

$\qquad$Bei Function  
\(\Ber\)  $\quad:\quad$\Ber

$\qquad$Ber Function  
\(\Bernoulli {p}\)  $\quad:\quad$\Bernoulli {p}

$\qquad$Bernoulli Distribution  
\(\BetaDist {\alpha} {\beta}\)  $\quad:\quad$\BetaDist {\alpha} {\beta}

$\qquad$Beta Distribution  
\(\bigintlimits {\map f s} {s \mathop = 0} {s \mathop = a}\)  $\quad:\quad$\bigintlimits {\map f s} {s \mathop = 0} {s \mathop = a}

$\qquad$Limits of Integration  
\(\bigsize {x}\)  $\quad:\quad$\bigsize {x}

$\qquad$Absolute Value  
\(\bigvalueat {\delta x} {x \mathop = x_j} \)  $\quad:\quad$\bigvalueat {\delta x} {x \mathop = x_j}


\(\Binomial {n} {p}\)  $\quad:\quad$\Binomial {n} {p}

$\qquad$Binomial Distribution  
\(\braket {a} {b}\)  $\quad:\quad$\braket {a} {b}

$\qquad$Dirac Notation  
\(\bsalpha\)  $\quad:\quad$\bsalpha


\(\bsbeta\)  $\quad:\quad$\bsbeta


\(\bschi\)  $\quad:\quad$\bschi


\(\bsDelta\)  $\quad:\quad$\bsDelta

$\qquad$a vector '$\Delta$'  
\(\bsdelta\)  $\quad:\quad$\bsdelta


\(\bsepsilon\)  $\quad:\quad$\bsepsilon


\(\bseta\)  $\quad:\quad$\bseta


\(\bsgamma\)  $\quad:\quad$\bsgamma


\(\bsiota\)  $\quad:\quad$\bsiota


\(\bskappa\)  $\quad:\quad$\bskappa


\(\bslambda\)  $\quad:\quad$\bslambda


\(\bsmu\)  $\quad:\quad$\bsmu


\(\bsnu\)  $\quad:\quad$\bsnu


\(\bsomega\)  $\quad:\quad$\bsomega


\(\bsomicron\)  $\quad:\quad$\bsomicron


\(\bsone\)  $\quad:\quad$\bsone

$\qquad$vector of ones  
\(\bsphi\)  $\quad:\quad$\bsphi


\(\bspi\)  $\quad:\quad$\bspi


\(\bspsi\)  $\quad:\quad$\bspsi


\(\bsrho\)  $\quad:\quad$\bsrho


\(\bssigma\)  $\quad:\quad$\bssigma


\(\bst\)  $\quad:\quad$\bst

$\qquad$a vector 't'  
\(\bstau\)  $\quad:\quad$\bstau


\(\bstheta\)  $\quad:\quad$\bstheta


\(\bsupsilon\)  $\quad:\quad$\bsupsilon


\(\bsv\)  $\quad:\quad$\bsv

$\qquad$a vector 'v'  
\(\bsw\)  $\quad:\quad$\bsw

$\qquad$a vector 'w'  
\(\bsx\)  $\quad:\quad$\bsx

$\qquad$a vector 'x'  
\(\bsxi\)  $\quad:\quad$\bsxi


\(\bsy\)  $\quad:\quad$\bsy

$\qquad$a vector 'y'  
\(\bsz\)  $\quad:\quad$\bsz

$\qquad$a vector 'z'  
\(\bszero\)  $\quad:\quad$\bszero

$\qquad$vector of zeros  
\(\bszeta\)  $\quad:\quad$\bszeta


\(\map \Card {S}\)  $\quad:\quad$\map \Card {S}

$\qquad$Cardinality  
\(\card {S}\)  $\quad:\quad$\card {S}

$\qquad$Cardinality  
\(\Cauchy {x_0} {\gamma}\)  $\quad:\quad$\Cauchy {x_0} {\gamma}

$\qquad$Cauchy Distribution  
\(\CC\)  $\quad:\quad$\CC

$\qquad$that is: \mathcal C


\(\Cdm {f}\)  $\quad:\quad$\Cdm {f}

$\qquad$Codomain of Mapping  
\(\ceiling {11.98}\)  $\quad:\quad$\ceiling {11.98}

$\qquad$Ceiling Function  
\(30 \cels\)  $\quad:\quad$30 \cels

$\qquad$Degrees Celsius  
\(15 \cents\)  $\quad:\quad$15 \cents

$\qquad$Cent  
\(\Char {R}\)  $\quad:\quad$\Char {R}

$\qquad$Characteristic of Ring, etc.  
\(\Ci\)  $\quad:\quad$\Ci

$\qquad$Cosine Integral Function  
\(\cis \theta\)  $\quad:\quad$\cis \theta

$\qquad$$\cos \theta + i \sin \theta$  
\(\cl {S}\)  $\quad:\quad$\cl {S}

$\qquad$Closure (Topology)  
\(\closedint {a} {b}\)  $\quad:\quad$\closedint {a} {b}

$\qquad$Closed Interval  
\(\cmod {z^2}\)  $\quad:\quad$\cmod {z^2}

$\qquad$Complex Modulus  
\(\cn u\)  $\quad:\quad$\cn u

$\qquad$Elliptic Function  
\(\condprob {A} {B}\)  $\quad:\quad$\condprob {A} {B}

$\qquad$Conditional Probability  
\(\conjclass {x}\)  $\quad:\quad$\conjclass {x}

$\qquad$Conjugacy Class  
\(\cont {f}\)  $\quad:\quad$\cont {f}

$\qquad$Content of Polynomial  
\(\ContinuousUniform {a} {b}\)  $\quad:\quad$\ContinuousUniform {a} {b}

$\qquad$Continuous Uniform Distribution  
\(\cosec\)  $\quad:\quad$\cosec

$\qquad$Cosecant (alternative form)  
\(\Cosh\)  $\quad:\quad$\Cosh

$\qquad$Hyperbolic Cosine  
\(\Coth\)  $\quad:\quad$\Coth

$\qquad$Hyperbolic Cotangent  
\(\cov {X, Y}\)  $\quad:\quad$\cov {X, Y}

$\qquad$Covariance  
\(\csch\)  $\quad:\quad$\csch

$\qquad$Hyperbolic Cosecant  
\(\Csch\)  $\quad:\quad$\Csch

$\qquad$Hyperbolic Cosecant  
\(\curl\)  $\quad:\quad$\curl

$\qquad$Curl Operator  
\(\DD\)  $\quad:\quad$\DD

$\qquad$that is: \mathcal D


\(\dfrac {\d x} {\d y}\)  $\quad:\quad$\dfrac {\d x} {\d y}

$\qquad$Roman $\d$ for Derivatives  
\(30 \degrees\)  $\quad:\quad$30 \degrees

$\qquad$Degrees of Angle  
\(\diam\)  $\quad:\quad$\diam

$\qquad$Diameter  
\(\Dic n\)  $\quad:\quad$\Dic n

$\qquad$Dicyclic Group  
\(\DiscreteUniform {n}\)  $\quad:\quad$\DiscreteUniform {n}

$\qquad$Discrete Uniform Distribution  
\(a \divides b\)  $\quad:\quad$a \divides b

$\qquad$Divisibility  
\(\dn u\)  $\quad:\quad$\dn u

$\qquad$Elliptic Function  
\(\Dom {f}\)  $\quad:\quad$\Dom {f}

$\qquad$Domain of Mapping  
\(\dr {a}\)  $\quad:\quad$\dr {a}

$\qquad$Digital Root  
\(\E\)  $\quad:\quad$\E

$\qquad$Elementary Charge  
\(\EE\)  $\quad:\quad$\EE

$\qquad$that is: \mathcal E


\(\Ei\)  $\quad:\quad$\Ei

$\qquad$Exponential Integral Function  
\(\empty\)  $\quad:\quad$\empty

$\qquad$Empty Set  
\(\eqclass {x} {\RR}\)  $\quad:\quad$\eqclass {x} {\RR}

$\qquad$Equivalence Class  
\(\erf\)  $\quad:\quad$\erf

$\qquad$Error Function  
\(\erfc\)  $\quad:\quad$\erfc

$\qquad$Complementary Error Function  
\(\expect {X}\)  $\quad:\quad$\expect {X}

$\qquad$Expectation  
\(\Exponential {\beta}\)  $\quad:\quad$\Exponential {\beta}

$\qquad$Exponential Distribution  
\(\Ext {\gamma}\)  $\quad:\quad$\Ext {\gamma}

$\qquad$Exterior  
\(\F\)  $\quad:\quad$\F

$\qquad$False  
\(30 \fahr\)  $\quad:\quad$30 \fahr

$\qquad$Degrees Fahrenheit  
\(\family {S_i}\)  $\quad:\quad$\family {S_i}

$\qquad$Indexed Family  
\(\FF\)  $\quad:\quad$\FF

$\qquad$that is: \mathcal F


\(\Field {\RR}\)  $\quad:\quad$\Field {\RR}

$\quad$AMSsymbols$\quad$Custom $\mathsf{Pr} \infty \mathsf{fWiki}$  
\(\Fix {\pi}\)  $\quad:\quad$\Fix {\pi}

$\qquad$Set of Fixed Elements  
\(\floor {11.98}\)  $\quad:\quad$\floor {11.98}

$\qquad$Floor Function  
\(\fractpart {x}\)  $\quad:\quad$\fractpart {x}

$\qquad$Fractional Part  
\(\Frob {R}\)  $\quad:\quad$\Frob {R}

$\qquad$Frobenius Endomorphism  
\(\Gal {S}\)  $\quad:\quad$\Gal {S}

$\qquad$Galois Group  
\(\Gaussian {\mu} {\sigma^2}\)  $\quad:\quad$\Gaussian {\mu} {\sigma^2}

$\qquad$Gaussian Distribution  
\(\gen {S}\)  $\quad:\quad$\gen {S}

$\qquad$Generator  
\(\Geometric {p}\)  $\quad:\quad$\Geometric {p}

$\qquad$Geometric Distribution  
\(\GF\)  $\quad:\quad$\GF

$\qquad$Galois Field  
\(\GG\)  $\quad:\quad$\GG

$\qquad$that is: \mathcal G


\(\GL {n, \R}\)  $\quad:\quad$\GL {n, \R}

$\qquad$General Linear Group  
\(\grad {p}\)  $\quad:\quad$\grad {p}

$\qquad$Gradient  
\(\harm {r} {z}\)  $\quad:\quad$\harm {r} {z}

$\qquad$General Harmonic Numbers  
\(\hav \theta\)  $\quad:\quad$\hav \theta

$\qquad$Haversine  
\(\hcf\)  $\quad:\quad$\hcf

$\qquad$Highest Common Factor  
\(\H\)  $\quad:\quad$\H

$\qquad$Set of Quaternions  
\(\HH\)  $\quad:\quad$\HH

$\qquad$Hilbert Space  
\(\hointl {a} {b}\)  $\quad:\quad$\hointl {a} {b}

$\qquad$Left HalfOpen Interval  
\(\hointr {a} {b}\)  $\quad:\quad$\hointr {a} {b}

$\qquad$Right HalfOpen Interval  
\(\horectl a b\)  $\quad:\quad$\horectl a b

$\qquad$HalfOpen Rectangle (on the left)  
\(\horectr c d\)  $\quad:\quad$\horectr c d

$\qquad$HalfOpen Rectangle (on the right)  
\(\ideal {a}\)  $\quad:\quad$\ideal {a}

$\qquad$Ideal of Ring  
\(\II\)  $\quad:\quad$\II

$\qquad$that is: \mathcal I


\(\map \Im z\)  $\quad:\quad$\map \Im z

$\qquad$Imaginary Part  
\(\Img {f}\)  $\quad:\quad$\Img {f}

$\qquad$Image of Mapping  
\(\index {G} {H}\)  $\quad:\quad$\index {G} {H}

$\qquad$Index of Subgroup  
\(\inj\)  $\quad:\quad$\inj

$\qquad$Canonical Injection  
\(\Inn {S}\)  $\quad:\quad$\Inn {S}

$\qquad$Group of Inner Automorphisms  
\(\innerprod {x} {y}\)  $\quad:\quad$\innerprod {x} {y}

$\qquad$Inner Product  
\(\Int {\gamma}\)  $\quad:\quad$\Int {\gamma}

$\qquad$Interior  
\(\intlimits {\dfrac {\map f s} s} {s \mathop = 1} {s \mathop = a}\)  $\quad:\quad$\intlimits {\dfrac {\map f s} s} {s \mathop = 1} {s \mathop = a}

$\qquad$Limits of Integration  
\(\invlaptrans {F}\)  $\quad:\quad$\invlaptrans {F}

$\qquad$Inverse Laplace Transform  
\(\JJ\)  $\quad:\quad$\JJ

$\qquad$that is: \mathcal J


\(\Kei\)  $\quad:\quad$\Kei

$\qquad$Kei Function  
\(\Ker\)  $\quad:\quad$\Ker

$\qquad$Ker Function  
\(\KK\)  $\quad:\quad$\KK

$\qquad$that is: \mathcal K


\(\laptrans {f}\)  $\quad:\quad$\laptrans {f}

$\qquad$Laplace Transform  
\(\lcm \set {x, y, z}\)  $\quad:\quad$\lcm \set {x, y, z}

$\qquad$Lowest Common Multiple  
\(\leadstoandfrom\)  $\quad:\quad$\leadstoandfrom


\(\leftset {a, b, c}\)  $\quad:\quad$\leftset {a, b, c}

$\qquad$Conventional set notation (left only)  
\(\leftparen {a + b + c}\)  $\quad:\quad$\leftparen {a + b + c}

$\qquad$Parenthesis (left only)  
\(\len {AB}\)  $\quad:\quad$\len {AB}

$\qquad$Length Function: various  
\(\Li\)  $\quad:\quad$\Li

$\qquad$Eulerian Logarithmic Integral  
\(\li\)  $\quad:\quad$\li

$\qquad$Logarithmic Integral  
\(\LL\)  $\quad:\quad$\LL

$\qquad$that is: \mathcal L


\(\Ln\)  $\quad:\quad$\Ln

$\qquad$Principal Branch of Complex Natural Logarithm  
\(\Log\)  $\quad:\quad$\Log

$\qquad$Principal Branch of Complex Natural Logarithm  
\(\map {f} {x}\)  $\quad:\quad$\map {f} {x}

$\qquad$Mapping or Function  
\(\meta {metasymbol}\)  $\quad:\quad$\meta {metasymbol}

$\qquad$Metasymbol  
\(27 \minutes\)  $\quad:\quad$27 \minutes

$\qquad$Minutes of Angle or Minutes of Time  
\(\MM\)  $\quad:\quad$\MM

$\qquad$that is: \mathcal M


\(\Mult\)  $\quad:\quad$\Mult

$\qquad$Multiplication as a Primitive Recursive Function  
\(\multiset {a, b, c}\)  $\quad:\quad$\multiset {a, b, c}

$\qquad$Multiset  
\(\NegativeBinomial {n} {p}\)  $\quad:\quad$\NegativeBinomial {n} {p}

$\qquad$Negative Binomial Distribution  
\(\Nil {R}\)  $\quad:\quad$\Nil {R}

$\qquad$Nilradical of Ring  
\(\nint {11.98}\)  $\quad:\quad$\nint {11.98}

$\qquad$Nearest Integer Function  
\(\NN\)  $\quad:\quad$\NN

$\qquad$that is: \mathcal N


\(\norm {z^2}\)  $\quad:\quad$\norm {z^2}

$\qquad$Norm  
\(\O\)  $\quad:\quad$\O

$\qquad$Empty Set  
\(\OO\)  $\quad:\quad$\OO

$\qquad$that is: \mathcal O


\(\oo\)  $\quad:\quad$\oo

$\qquad$that is: \mathcal o


\(\oldpence\)  $\quad:\quad$\oldpence

$\qquad$old pence  
\(\On\)  $\quad:\quad$\On

$\qquad$Class of All Ordinals  
\(\openint {a} {b}\)  $\quad:\quad$\openint {a} {b}

$\qquad$Open Interval  
\(\Orb S\)  $\quad:\quad$\Orb S

$\qquad$Orbit  
\(\Ord {S}\)  $\quad:\quad$\Ord {S}

$\qquad$$S$ is an Ordinal  
\(\order {G}\)  $\quad:\quad$\order {G}

$\qquad$Order of Structure, and so on  
\(\ot\)  $\quad:\quad$\ot

$\qquad$Order Type  
\(\Out {G}\)  $\quad:\quad$\Out {G}

$\qquad$Group of Outer Automorphisms  
\(\paren {a + b + c}\)  $\quad:\quad$\paren {a + b + c}

$\qquad$Parenthesis  
\(\ph z\)  $\quad:\quad$\ph z

$\qquad$Phase  
\(\Poisson {\lambda}\)  $\quad:\quad$\Poisson {\lambda}

$\qquad$Poisson Distribution  
\(\polar {r, \theta}\)  $\quad:\quad$\polar {r, \theta}

$\qquad$Polar Form of Complex Number  
\(\pounds\)  $\quad:\quad$\pounds

$\qquad$Pound Sterling  
\(\powerset {S}\)  $\quad:\quad$\powerset {S}

$\qquad$Power Set  
\(\PP\)  $\quad:\quad$\PP

$\qquad$that is: \mathcal P


\(\map {\pr_j} {F}\)  $\quad:\quad$\map {\pr_j} {F}

$\qquad$Projection  
\(\Preimg {f}\)  $\quad:\quad$\Preimg {f}

$\qquad$Preimage of Mapping  
\(\map {\proj_\mathbf v} {\mathbf u}\)  $\quad:\quad$\map {\proj_\mathbf v} {\mathbf u}

$\qquad$Vector Projection  
\(\PV\)  $\quad:\quad$\PV

$\qquad$Cauchy Principal Value  
\(\QQ\)  $\quad:\quad$\QQ

$\qquad$that is: \mathcal Q


\(\radians\)  $\quad:\quad$\radians

$\qquad$Radian  
\(\Rad\)  $\quad:\quad$\Rad

$\qquad$Radical of Ideal of Ring  
\(\ds \int \map f x \rd x\)  $\quad:\quad$\ds \int \map f x \rd x

$\qquad$Roman $\d$ for use in Integrals  
\(\rD\)  $\quad:\quad$\rD

$\qquad$Differential Operator  
\(y \rdelta x\)  $\quad:\quad$y \rdelta x

$\qquad$$\delta$ operator for use in sums  
\(30 \rankine\)  $\quad:\quad$30 \rankine

$\qquad$Degrees Rankine  
\(\map \Re z\)  $\quad:\quad$\map \Re z

$\qquad$Real Part  
\(\relcomp {S} {A}\)  $\quad:\quad$\relcomp {S} {A}

$\qquad$Relative Complement  
\(\rem\)  $\quad:\quad$\rem

$\qquad$Remainder  
\(\Res {f} {z_0}\)  $\quad:\quad$\Res {f} {z_0}

$\qquad$Residue  
\(\rightparen {a + b + c}\)  $\quad:\quad$\rightparen {a + b + c}

$\qquad$Parenthesis (right only)  
\(\rightset {a, b, c}\)  $\quad:\quad$\rightset {a, b, c}

$\qquad$Conventional set notation (right only)  
\(\Rng {f}\)  $\quad:\quad$\Rng {f}

$\qquad$Range of Mapping  
\(\RR\)  $\quad:\quad$\RR

$\qquad$that is: \mathcal R


\(\sech\)  $\quad:\quad$\sech

$\qquad$Hyperbolic Secant  
\(\Sech\)  $\quad:\quad$\Sech

$\qquad$Hyperbolic Secant  
\(53 \seconds\)  $\quad:\quad$53 \seconds

$\qquad$Seconds of Angle or Seconds of Time  
\(\sequence {a_n}\)  $\quad:\quad$\sequence {a_n}

$\qquad$Sequence  
\(\set {a, b, c}\)  $\quad:\quad$\set {a, b, c}

$\qquad$Conventional set notation  
\(\ShiftedGeometric {p}\)  $\quad:\quad$\ShiftedGeometric {p}

$\qquad$Shifted Geometric Distribution  
\(\shillings\)  $\quad:\quad$\shillings

$\qquad$shillings  
\(\Si\)  $\quad:\quad$\Si

$\qquad$Sine Integral Function  
\(\Sinh\)  $\quad:\quad$\Sinh

$\qquad$Hyperbolic Sine  
\(\size {x}\)  $\quad:\quad$\size {x}

$\qquad$Absolute Value, and so on  
\(\SL {n, \R}\)  $\quad:\quad$\SL {n, \R}

$\qquad$Special Linear Group  
\(\sn u\)  $\quad:\quad$\sn u

$\qquad$Elliptic Function  
\(\span\)  $\quad:\quad$\span

$\qquad$Linear Span  
\(\Spec {R}\)  $\quad:\quad$\Spec {R}

$\qquad$Spectrum of Ring  
\(\sqbrk {a} \)  $\quad:\quad$\sqbrk {a}


\(\SS\)  $\quad:\quad$\SS

$\qquad$that is: \mathcal S


\(\Stab x\)  $\quad:\quad$\Stab x

$\qquad$Stabilizer  
\(\stratgame {N} {A_i} {\succsim_i}\)  $\quad:\quad$\stratgame {N} {A_i} {\succsim_i}

$\qquad$Strategic Game  
\(\struct {G, \circ}\)  $\quad:\quad$\struct {G, \circ}

$\qquad$Algebraic Structure  
\(\StudentT {k}\)  $\quad:\quad$\StudentT {k}

$\qquad$Student's tDistribution  
\(\SU {n}\)  $\quad:\quad$\SU {n}

$\qquad$Unimodular Unitary Group  
\(\Succ\)  $\quad:\quad$\Succ

$\qquad$Successor Function  
\(\supp\)  $\quad:\quad$\supp

$\qquad$Support  
\(\Syl {p} {N}\)  $\quad:\quad$\Syl {p} {N}

$\qquad$Sylow $p$Subgroup  
\(\symdif\)  $\quad:\quad$\symdif

$\qquad$Symmetric Difference  
\(\T\)  $\quad:\quad$\T

$\qquad$True  
\(\Tanh\)  $\quad:\quad$\Tanh

$\qquad$Hyperbolic Tangent  
\(\tr\)  $\quad:\quad$\tr

$\qquad$Trace  
\(\TT\)  $\quad:\quad$\TT

$\qquad$that is: \mathcal T


\(\tuple {a, b, c}\)  $\quad:\quad$\tuple {a, b, c}

$\qquad$Ordered Tuple  
\(\UU\)  $\quad:\quad$\UU

$\qquad$that is: \mathcal U


\(\valueat {\dfrac {\delta y} {\delta x} } {x \mathop = \xi} \)  $\quad:\quad$\valueat {\dfrac {\delta y} {\delta x} } {x \mathop = \xi}


\(\var {X}\)  $\quad:\quad$\var {X}

$\qquad$Variance  
\(\vers \theta\)  $\quad:\quad$\vers \theta

$\qquad$Versed Sine  
\(\VV\)  $\quad:\quad$\VV

$\qquad$that is: \mathcal V


\(\weakconv\)  $\quad:\quad$\weakconv

$\qquad$Weak Convergence  
\(\weakstarconv\)  $\quad:\quad$\weakstarconv

$\qquad$Weak$*$ Convergence  
\(\WW\)  $\quad:\quad$\WW

$\qquad$that is: \mathcal W


\(\XX\)  $\quad:\quad$\XX

$\qquad$that is: \mathcal X


\(\YY\)  $\quad:\quad$\YY

$\qquad$that is: \mathcal Y


\(\ZZ\)  $\quad:\quad$\ZZ

$\qquad$that is: \mathcal Z

Aligned Equations
To include aligned equations, a set of templates has been written: begineqn, eqn and endeqn.
For more explanation, see Template:eqn.
Known issues
See Problem with Eqn template.
Specific Topics
Commutative diagrams
External references and manuals
It may not be exactly the same version of $\LaTeX$, but I always find this page helpful as a first, quick overview:
This is also a good reference page, pertaining to MediaWiki $\LaTeX$:
but be aware that not all commands are supported.
This is a link of all the currently supported commands available:
House Style
This page lists the various house style conventions that have been adopted on $\mathsf{Pr} \infty \mathsf{fWiki}$.
Active contributors are expected to gradually master these, but when you first start to contribute, seasoned editors will come in and tidy the pages you write or create.
You are encouraged to study the changes made, and hence to pick up the style as you progress.
Linking
Due to the desired standard of rigor on $\mathsf{Pr} \infty \mathsf{fWiki}$, there are a lot of concepts on any given (proof) page that have their own, dedicated Proof or Definition page on $\mathsf{Pr} \infty \mathsf{fWiki}$.
To ensure ease of reference and maximal clarity and consistency, the following rules for internal reference are to be adhered to.
For information on creating links, see this section.
References to Theorems and Axioms
Whenever a theorem is invoked or referred to, be it in a proof or, for example, a clarifying comment, it should be referenced by its full title.
Also, for ease of editing, there is no need to change the case of theorem names; the $\mathsf{Pr} \infty \mathsf{fWiki}$ page title will suffice.
Thus, for example, a valid reference to the result Union Distributes over Intersection is simply:
 "By Union Distributes over Intersection, $A \cup \left({B \cap C}\right) = \left({A \cup B}\right) \cap \left({A \cup C}\right)$."
This is achieved by simply putting the title of the page you want to reference between double square brackets, [[
and ]]
.
The same convention applies to axioms, except that the namespace identifier Axiom: should be removed.
The correct way to reference the page Axiom:Axiom of Choice thus is:
which is produced by:
[[Axiom:Axiom of ChoiceAxiom of Choice]]
References to Definitions
Whenever a concept (or part of it) is invoked on a page, it should be presented as a link to the definition of that concept.
It is preferred that every use of the word for that concept is presented as such a link.
These references are made in a nonintrusive way. Thus, we write:
 Let $R$ be a ring.
and not:
 Let $R$ be a Ring (Abstract Algebra).
Permanent Redirects to Definitions on Subpages
Many definitions have subpages, for example:
In such cases there is (or ought to be) a permanent redirect to such a page, which is to be used instead in all cases.
In this case we have:
If you find you need to link to such a subpage, use the "What links here" (under "Tools" in the menu on the left hand side) to find out what permanent redirect may be available.
If there is none, feel free to implement one.
 Note
This of course does not apply when the subpage is to handle multiple definitions, for example:
 Definition:Prime Number/Definition 1
 Definition:Prime Number/Definition 2
 Definition:Prime Number/Definition 3
... and so on.
The rule here is (unless there are specific reasons to invoke exactly that instance of the definition, either to avoid circular arguments or in proofs of the equivalence of such definitions) not to invoke the individual subpage, but merely the top level page (in this case Definition:Prime Number). Otherwise maintenance becomes significantly more difficult.
Mathematical Symbols
Symbol Set
The only symbols that are accepted in $\mathsf{Pr} \infty \mathsf{fWiki}$ $\LaTeX$ source code are the standard alphanumeric and punctuation characters that can be found on a standard Englishlanguage keyboard. Letters with diacritical marks such as "á" should not be used.
If nonEnglish characters are needed within $\LaTeX$ source code (the obvious instance being Greek), then the full $\LaTeX$ definition is to be used.
For example: $\alpha$ is to be rendered by the code $\alpha$
.
The same applies to specialised mathematical symbols. While it is appreciated that some contributors may have favourite techniques to allow them to place various mathematical symbols directly into the wiki text, such techniques are not portable and cause rendering issues in some browsers.
The only exceptions to this rule are:
 when reporting the name of a mathematician, for example: Wacław Franciszek Sierpiński
 when explaining the linguistic derivation of a term in, for example, a Language Note section. It is standard, for example, to use Greek characters directly here, rather than their $\LaTeX$ codes.
Inline Equations
Inline equations (that is, those that appear as part of a text sentence) merely need the dollar delimiters. For example:
 The semilinear wave equation $\partial_t U = A U + \map B U$ is Hamiltonian.
is produced by the input:
 The semilinear wave equation
$\partial_t U = A U + \map B U$
is Hamiltonian.
Displayed Equations
Displayed equations should be indented using a single colon, for example, a displayed equation should look like:
 $\ds \map H U = \int_0^{2 \pi} \frac {\paren {\partial_x u}^2} 2 + \frac {v^2} 2  \map F u \rd x$
which you can enter as:
:$\ds \map H U = \int_0^{2 \pi} \frac {\paren {\partial_x u}^2} 2 + \frac {v^2} 2  \map F u \rd x$
Note the $\mathsf{Pr} \infty \mathsf{fWiki}$ custom command \ds
which is a convenience abbreviation for the full command \displaystyle
.
Using a format that places the equation on the center of the page:
 $$E = m c^2$$
is discouraged, because with our "short sentence" house style, this breaks up the reading flow.
Big Operators
The \ds
command should be used at the front of expressions using the 'big operators' such as \sum
and \prod
, whether the equation is displayed or inline.
This includes (but may not be exclusive to) the commands \frac
, \binom
, \lim
, \bigcup
, \bigcap
, \int
, \sum
and \prod
.
For example:
 $\sum_{i \mathop = 1}^n$
 $\prod_{i \mathop = 1}^n$
 $\frac {b \pm \sqrt {b^2  4ac} } {2 a}$
 $\lim_{n \to \infty} \frac 1 n$
all look better as:
 $\ds \sum_{i \mathop = 1}^n$
 $\ds \prod_{i \mathop = 1}^n$
 $\ds \frac {b \pm \sqrt {b^2  4ac} } {2 a}$
 $\ds \lim_{n \mathop \to \infty} \frac 1 n$
and are produced by, respectively:
\ds \sum_{i \mathop = 1}^n
\ds \prod_{i \mathop = 1}^n
\ds \frac {b \pm \sqrt {b^2  4ac} } {2 a}
\ds \lim_{n \mathop \to \infty} \frac 1 n
Furthermore, to improve aesthetic appeal certain characters, such as $=$ and $\in$, when used in subscripts of big operators, must be endowed with the \mathop
command to enforce appropriate spacing.
As a contrast, compare:
 $\ds \sum_{i = 1}^n \quad \sum_{i \mathop = 1}^n$
 $\ds \bigcap_{n \in \N} \quad \bigcap_{n \mathop \in \N}$
The \mathop
command is to be used in the following manner (the code produces $\ds \sum_{i \mathop = 1}^n$):
\ds \sum_{i \mathop = 1}^n
Abbreviated Symbols
Certain symbols have abbreviated forms for their big versions:
\dfrac
can be used instead of\ds \frac
\dbinom
can be used instead of\ds \binom
where d
is for display
.
Of course, if other big operators are used in the same equation, the \ds
command is needed anyway.
However, it does no harm to include \dfrac
and \dbinom
inside a line defined as \ds
, and may make refactoring easier.
So feel free to develop the habit of using \dfrac
and \dbinom
throughout.
The d of Calculus
When writing calculus operators, use a nonitalic form for the $\d$. To achieve this, write it as \d
or \rd
(the latter includes a halfspace before it, for use in integrals).
So you would have:
 $\dfrac {\d y} {\d x}$
which would be produced by:
\dfrac {\d y} {\d x}
rather than:
 $\dfrac {d y} {d x}$
which would be produced by:
\dfrac {d y} {d x}
Fonts
We have several fonts available, many of which have particular conventional uses in mathematics.
Examples are:
 Calligraphy:
\mathcal
, which produces $\mathcal{ABCDE} ..., \mathcal {1234567890}$ (uppercase only, but also digits)  Blackboard:
\mathbb
or (preferably)\Bbb
, which produces $\Bbb{ABCDE} ...$ (uppercase only, no digits)  Script:
\mathscr
, which produces $\mathscr{ABCDE} ...$ (uppercase only, no digits)  Sans serif:
\mathsf
, which produces $\mathsf{ABCDE} ... \mathsf{abcde} ..., \mathsf {1234567890}$  Fraktur:
\mathfrak
, which produces $\mathfrak{ABCDE} ... \mathfrak{abcde} ..., \mathfrak {1234567890}$  Fixed Width:
\mathtt
, which produces $\mathtt{ABCDE} ... \mathtt{abcde} ..., \mathtt {1234567890}$
The use of Fraktur and Script are discouraged, as they are not so easy on the eye and can be difficult to decipher on certain browsers.
Also note that:
\T
and\F
implement $\T$ and $\F$ respectively, designed to be used for True and False respectively $\N, \Z, \Q, \R, \C$ have their own $\LaTeX$ codes:
\N, \Z, \Q, \R, \C
 We also have
\GF
implemented for $\GF$ (note that\F
cannot be used here, as it is already used for $\F$)  all
\mathcal
uppercase letters have custom $\mathsf{Pr} \infty \mathsf{fWiki}$ shortcut implementations, as\AA
,\BB
, and so on
Use of Logical Symbols in Mathematical Exposition
This applies mainly to the use of the conjunction symbol $\land$, that is $\text {and}$, and the disjunction symbol $\lor$, that is $\text {or}$.
It is convenient sometimes to write a statement in the style:
 $\forall y \in R: \lambda_y = y * I_{_R} \land \rho_y = I_{_R} * y$
However, it may not be immediately obvious to the reader exactly what $\land$ means.
In the various fields, for example abstract algebra and set theory, $\land$ and $\lor$ have a number of different meanings, for example meet and join.
If the reader has been studying such material, it can be irritating to have to change mental gears and suddenly have to adjust to the fact that $\land$ means $\text {and}$.
Hence it is strongly recommended that the above statement be written:
 $\forall y \in R: \lambda_y = y * I_{_R} \text { and } \rho_y = I_{_R} * y$
reserving $\land$ and $\lor$ for their use in the field of logic.
Punctuation niceties
A sentence broken by a displayed equation should be ended with a colon:
 $\dfrac {\text{display}} {\text{equation}}$
for a better presentation.
On the other hand, the displayed equation itself should not be ended with a full stop or comma.
That is, one should write:
 $\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S$
and not:
 $\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S$.
In particular, including the full stop inside the $\LaTeX$ it terminates is definitely incorrect, for readily apparent reasons. So please do not do this:
 $\displaystyle \bigcap_{S \mathop \in \Bbb S} \Bbb U \setminus S = \Bbb U \setminus \bigcup_{S \mathop \in \Bbb S} S.$
This is a style tip borrowed from Ian Stewart, from his Galois Theory, 3rd ed. of $2004$.
Use of commas is discouraged. This sort of structure is considered incorrect:
 Let $x$ be as follows,
 $x \in S$
as commas are reserved in mathematics for separation of elements of a list.
Q.E.D.
To end a proof, the template {{qed}}
should be used, which looks like:
$\blacksquare$
or if you wish to break your page up into subproofs, end those subproofs with {{qedlemma}}
, which looks like:
$\Box$
In a dash for consistent notation, it is understood that these templates should immediately succeed the last line of the proof, that is:
Hence the result. {{qed}}
and not:
Hence the result. {{qed}}
Tempting though it is to write "Q.E.D." at the bottom, this is so uncool as to be positively naff.
$\LaTeX$ Code Style
There are a few general source code conventions which make your code easier to read and maintain:
 Each variable, and each command beginning with a backslash should be preceded by a space, except (for some unexplained result of evolution) when enclosing things in brackets. See some of the above instance for a typical example.
 When enclosing brackets around an object, always use the
\paren { ... }
command, for example\paren {a + b}
rather than(a + b)
.
 There should be no need to use the commands
\big, \Big, \Bigg
etc. for specifying the sizes of parentheses. Using the\paren {...}
technique (as above) will almost always automatically size the brackets aesthetically.
 Punctuation should appear (if it is really necessary) outside the $\LaTeX$ environment, for example:
 Hence $\map f {\sin x}$. (produced by:
$\map f {\sin x}$.
)
 Hence $\map f {\sin x}$. (produced by:
 as opposed to:
 Hence $\map f {\sin x}.$ (produced by:
$\map f {\sin x}.$
)
 Hence $\map f {\sin x}.$ (produced by:
 Singlecharacter parameters to standard $\LaTeX$ constructs need not be put in curly braces. That is,
\dfrac 1 2
is preferred to\dfrac {1} {2}
. They both produce: $\dfrac 1 2$
It makes the source code significantly easier to read.
Having said that, please do not ignore the rule about spacing. The same effect can also be achieved by \dfrac12
(see, it still looks like $\dfrac12$) but that is significantly harder to interpret visually.
Aligned Material
If an equation includes multiple equalities or inequalities, it is best to place each equality on a new line.
For example:
 $\dfrac \d {\d t} \map H U = \d \map H U \cdot \dot U = \map \Omega {\map {X_H} U, \dot U} = \map \Omega {\map {X_H} U, \map {X_H} U} = 0$
would look better as an aligned equation. This is done using the following commands:
{{begineqn}} {{eqn  l =  r = }} ... {{eqn  l =  r = }} {{endeqn}}
Here, the section following  l =
is a $\LaTeX$ environment, and should contain anything you want to appear to the left of the equals sign.
The section following  r =
is the same, but will appear to the right of the equals sign.
 ll =
and  rr =
are similar, but produce material in columns further to the left and further to the right respectively. In particular, the ll
column is often used for an "implies" or "leads to" sign where the l
and r
are used for either side of a string of equations.
All these $\LaTeX$ environments are already in \displaystyle
mode, so there is no need to include that command in your equation.
When entering such an {{eqn}}
environment, it should globally look like this:
{{eqn  l = 1 + 1  r = 2 }}
That is, it adheres to the following principles:
 Every empty column should in general be omitted, except perhaps for
 c =
sections, which can be left as placeholders for possible future addition of comments  Nonempty columns are entered on separate lines, with the

and=
all aligned.
These conventions serve to optimize readability.
More options and abilities of the {{eqn}}
can be found on its page, {{eqn}}
.
The section following  c =
is not a $\LaTeX$ environment, and can be used to add any comments about the equation at this point.
So the example we gave above would be typeset as:
{{begineqn}} {{eqn  l = \frac \d {\d t} \map H U  r = \d \map H U \cdot \dot U  c = [[Chain Rule for Derivatives]] }} {{eqn  r = \map \Omega {\map {X_H} U, \dot U}  c = Definition of $X_H$ }} {{eqn  r = \map \Omega {\map {X_H} U, \map {X_H} U}  c = [[Hamilton's Equations]] }} {{eqn  r = 0  c = from above: $\Omega$ is [[Definition:SkewSymmetryskewsymmetric]] }} {{endeqn}}
The finished result will look like:
\(\ds \frac \d {\d t} \map H U\)  \(=\)  \(\ds \d \map H U \cdot \dot U\)  Chain Rule for Derivatives  
\(\ds \)  \(=\)  \(\ds \map \Omega {\map {X_H} U, \dot U}\)  Definition of $X_H$  
\(\ds \)  \(=\)  \(\ds \map \Omega {\map {X_H} U, \map {X_H} U}\)  Hamilton's Equations  
\(\ds \)  \(=\)  \(\ds 0\)  from above: $\Omega$ is skewsymmetric 
The operator that is displayed in this template can be changed using  o =
to show inequalities, etc.
Note the following:
 Do not include two consecutive open or close curly braces:
{{
or}}
anywhere in your{{eqn}}
templates. It will break the interpreter.
Put spaces in: { {
or } }
and it will be okay.
 Do not include the vertical line

(a.k.a. "pipe") in $\LaTeX$ expressions as this also breaks the interpreter. Use\vert
(or\lvert
and\rvert
) instead.
 In particular,
\
(used to produce $\$) has the same problem. Use\Vert
etc. instead.
These caveats apply only within the {{eqn}}
environment. Elsewhere on the page such constructs should be fine. To accommodate for the inevitable copypaste efforts, and for consistency's sake, it is however desirable to always use \vert
and \Vert
, and to insert a space between adjacent curly braces within $\LaTeX$ strings.
Linguistic Style
Language
This is an English language website, and so all pages are to be presented in English.^{[1]} Where there is a difference between spellings between US and restofworld English, the US version is generally used, with a few exceptions (the spelling of metre is under discussion).
Linguistic Style
During the presentation of a mathematical argument, a formal style is preferred.
For example:
 Suppose that ...
is preferred to:
 Let's suppose that ...
and:
 Hence the result.
is preferred to:
 ... and we're done.
As an attempt is being made for $\mathsf{Pr} \infty \mathsf{fWiki}$ to appeal to as wide an audience as possible worldwide, using colloquial language (except for example when illustrating logical concepts by means of everyday examples) is discouraged.
"Let" and "Suppose"
It is preferred that "Let" is used to introduce the existence of an entity in an argument, as follows:
 Let $S$ be a set.
 Let $x, y \in S: x \ne y$.
 $\ldots$
However, when introducing an entity whose existence is in question (for example, when constructing a Proof by Contradiction), the word "Suppose" is recommended:
 Suppose $T \subseteq S$ such that $\card T > \card S$.
 $\ldots$
"Any"
The word "any" can be ambiguous.
It is recommended that it not be used.
Instead, consider whether "every" or "an arbitrary" can be used instead.
Abbreviations
The difference between "e.g." (exempli gratia  for example) and "i.e." (id est  that is) is sadly falling into obscurity. It is all too common for "i.e." to be used when "for example" is meant, and vice versa.
So as to remove all confusion, such abbreviations are discouraged.
Also, beware the ubiquitous confusion between its and it's. The full version it is should be used instead of it's in any case, so it's should have no reason to appear.
Sentence Length
During the course of an argument to present a mathematical proof, follow these rules:
 Each sentence should be short.
 Each sentence should convey one step, either:
 One simple statement, or:
 One compound statement of the form: $P$, therefore $Q$.
 Each sentence should be on a separate line.
Compare the presentations:
$(1):$
 $S$, because of $R$ (we know this from Tom's Theorem), because of $Q$ (from above) which applies when $P$ holds (see Fred's Theorem), but we know $P$ holds because it's what we defined in the first place.
$(2):$
 Let $P$ hold.
 From Fred's Theorem, it follows that $Q$.
 From above, $R$.
 From Tom's Theorem, $S$.
The following is an example of the style of mathematical exposition which we believe has no place in $\mathsf{Pr} \infty \mathsf{fWiki}$, and indeed, the entire universe:
 The ($\implies$) is shown just the same as above, while the other direction easily follows, since $\MM$ satisfying the condition that for every $\LL$formula $\map \phi {x, \bar v}$ and for every $\bar a$ in $\MM$, if there is an $n$ in $\NN$ such that $\NN \models \map \phi {n, \bar a}$, then there is an $m$ in $\MM$ such that $\NN \models \map \phi {m, \bar a}$, is closed under functions (by directly applying the condition to formulae of the form $\map \phi {x, \bar y} = \paren {x = \map f {\bar y} }$), and hence the universe of a substructure, which reduces it to the statement above.
Here's an even worse example, posted up by an editor whose approach to contribution is so contrary to house style that appears to be deliberate trolling:
 If 24*k with k coprime to 6 has exactly 120 divisors, than k has exactly 15 divisors, thus k is a square number, thus k cannot be == 5, 7, 11 mod 12 (since 5, 7, 11 are not quadratic residues mod 12), thus a number == 120, 168, 264 mod 288 cannot have exactly 120 divisors (since such numbers can be written as 24*k with k coprime to 6 and k == 5, 7, 11 mod 12), thus if there are 120 consecutive integers with exactly 120 divisors, than the start number must be == 0, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287 mod 288, and hence == 0, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31 mod 32, thus there are 4 consecutive multiples of 32 among these 120 integers, and one of these 4 numbers must be == 64 mod 128, thus the number of divisors of this number must be divisible by 7 and cannot be 120, which is a contradiction!
Please, don't do this. Just don't.
Parenthetical Notes
It is often tempting to slide some further information or explanation into the middle of a sentence by slipping it into parenthesis.
It is highly recommended that this practice be avoided as far as possible.
Not only does this clash with the $\mathsf{Pr} \infty \mathsf{fWiki}$ house style of short, simple sentences, but also can cause confusion when the parenthesized material contains similar mathematical notation:
 Let $\map {f_n} x \in \Sigma$ (where $\map {f_n} x$ is a family (indexed by $\map I \alpha$) of functions in $\struct {\Sigma, \mu}$) be a function such that ...
So please don't do this.
A similar stylistic presentation to be avoided is the following
 The supremum of $f$, $\map \sup f$, is defined as:
 $\map \sup f = \map \max {\Img f}$
It is clear from the line following that $\map \sup f$ is how the supremum of $f$ is denoted, so there is no need to include it as a parenthetical explanation in the first line.
This is preferred:
 The supremum of $f$ is defined as:
 $\map \sup f = \map \max {\Img f}$
If you are uneasy about the ability of the reader to make that connection, feel free to write something like:
 The supremum of $f$ is defined and denoted as:
 $\map \sup f = \map \max {\Img f}$
Filler Words
Whether or not filler words are needed (it follows that, we have, hence etc.) is a stylistic decision. Fewer words are preferred, but clarity and completeness override every other consideration.
The general approach is to try to use as terse a form as possible.
Compare:
 We have that the ordinal subset of an ordinal is an initial segment of it, so it follows that:
with:
The latter form is preferred.
Definitions
In definitions, in particular, it is often tempting to fill up the internet with linguistic constructs like:
 $X$ is called a some object
 $X$ is known as some epithet
 $X$ is said to be some property
 $X$ is described as being some type
The following are preferred:
 $X$ is some object
and so on.
Just use is.
Empty Statements and Waffle
It is tempting to fill a page up with statements that do not actually impart any information, but which make the author look and feel good.
Such are to be avoided.
Examples:
 The first part of the proof is easy.
 We mention for the interested reader ...
 This is trivial:
See also the templates {{handwaving}}
and {{explain}}
.
This article is complete as far as it goes, but it could do with expansion. In particular: There are plenty more  these will be added as they are encountered. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. 
Capital Letters begin Sentences
This is raised as a particular point, because it crops up over and over again.
The sentence form in question is:
 Let (suchandsuch) hold, where (soandso) means (thus and so).
When (suchandsuch) is a statement in mathematical symbols, placed on its own line (as per house style recommendations), the temptation is to present the above sentence as:
 Let:
 $\ds S = \sum_{i \mathop \in \N} \frac 1 {2^i}$
 Where $\ds \sum$ denotes summation.
Just because it starts a new line does not mean that "where" is to be written with a capital W. It is the continuation of the previous sentence, which just happens to have, as part of its main clause, a mathematical expression.
It should be:
 Let:
 $\ds S = \sum_{i \mathop \in \N} \frac 1 {2^i}$
 where $\ds \sum$ denotes summation.
Breaking this linguistic rule can lead to confusion, especially when the "where" clause starts to get complicated:
 Let:
 $\ds S = \sum_{j \mathop \in \N} \lim_{x \mathop \to \infty} \cos j x + i \sin j x$
 Where $\ds \sum$ denotes summation and $\lim$ is the limit as $x$ tends to infinity and:
 $\cos j x + i \sin j x = e^{ijx}$
In the above, the reader, thinking that "where" starts the next sentence, and therefore a new thought, is left wondering:
 "Where this applies, and that means that, and this ... then what?"
whereas in fact the only reason for the "where" clause is to amplify the sense of the expression above it.
Similarly:
\(\ds x \in A \cap \paren {B \cap C}\)  \(\leadstoandfrom\)  \(\ds x \in A \land \paren {x \in B \land x \in C}\)  By definition of Set Intersection  
\(\ds \)  \(\leadstoandfrom\)  \(\ds \paren {x \in A \land x \in B} \land x \in C\)  Rule of Association: Conjunction  
\(\ds \)  \(\leadstoandfrom\)  \(\ds x \in \paren {A \cap B} \cap C\)  By definition of Set Intersection 
In the above, the "by definition" phrases in the comment column should not start with a capital letter, as they continue the "sentence" started on the left.
Thus the above structure is better rendered as:
\(\ds x \in A \cap \paren {B \cap C}\)  \(\leadstoandfrom\)  \(\ds x \in A \land \paren {x \in B \land x \in C}\)  by definition of Set Intersection  
\(\ds \)  \(\leadstoandfrom\)  \(\ds \paren {x \in A \land x \in B} \land x \in C\)  Rule of Association: Conjunction  
\(\ds \)  \(\leadstoandfrom\)  \(\ds x \in \paren {A \cap B} \cap C\)  by definition of Set Intersection 
Better still, lose the redundant fillerword "by", and render the entire structure elegantly as:
\(\ds x \in A \cap \paren {B \cap C}\)  \(\leadstoandfrom\)  \(\ds x \in A \land \left({x \in B \land x \in C}\right)\)  Definition of Set Intersection  
\(\ds \)  \(\leadstoandfrom\)  \(\ds \paren {x \in A \land x \in B} \land x \in C\)  Rule of Association: Conjunction  
\(\ds \)  \(\leadstoandfrom\)  \(\ds x \in \paren {A \cap B} \cap C\)  Definition of Set Intersection 
Now, as there is no fillerword "by", the comment is no longer implicitly part of a sentence, and so the comment is a standalone label which now merits an uppercase presentation.
Here, note that a further evolutionary step has been made: to replace the code Definition of [[Definition:Set IntersectionSet Intersection]]
with the template construct {{DefofSet Intersection}}
for further streamlining of the source.
 Beware
 This
{{Defof}}
template was designed specifically for thec
parameter of the{{Eqn}}
template.
It is not for using in the body of an exposition, specifically because of the fact that it has been designed to start with a capital letter.
Sources
It is good to indicate where the information comes from. This is done in $\mathsf{Pr} \infty \mathsf{fWiki}$ in the last of the page in a section called Sources.
Adding sources
If there are multiple sources, they are to be listed first in chronological order, then alphabetically on the name of the (first) author.
As stated on Help:Page Editing, the sources should be using a bulleted list, ordered by date of publication of the edition cited, and after that alphabetically, sorted on the surname of the (first) author.
For example (an excerpt of the Sources section of Definition:Set Union):
== Sources == * {{BookReferenceNaive Set Theory1960Paul R. Halmosprev = Union of Singletonnext = Union with Empty Set}}: $\S 4$: Unions and Intersections * {{BookReferenceAbstract Algebra1964W.E. Deskinsprev = Equality of Setsnext = Definition:Set Intersection}}: $\S 1.1$: Definition $1.2$ * {{BookReferencePoint Set Topology1964Steven A. Gaalprev = Definition:Set Union/General Definitionnext = Union is Commutative}}: Introduction to Set Theory: $1$. Elementary Operations on Sets * {{BookReferenceSets and Groups1965J.A. Greenprev = Empty Set Subset of Allnext = Intersection Subset Union}}: $\S 1.4$ * {{BookReferenceModern Algebra1965Seth Warnerprev = Associative and Anticommutativenext = Definition:Set Intersection}}: $\S 3$
Types of sources
There are several templates that can be used:
Hardcopy Sources
 Template:BookReference
 This is used to reference a specific book which will have been documented in the Books page. The idea of this is that if you have sourced the information for a page directly from a book, then it should be possible to provide the details of that book.
Example:
 1969: C.R.J. Clapham: Introduction to Abstract Algebra: $\S 4.17$: Theorem $28$
which can be found on the page Characteristic times Ring Element is Ring Zero.
 Template:Citation
 This is used to reference a specific article in a journal. This is still under development, as the individual Journal entries still need to be worked on.
Examples of their use can be found on various Mathematicians pages, for example:
 1908: Mathematical Logic as Based on the Theory of Types (Amer. J. Math. Vol. 30: pp. 222 – 262)
which appears on the page for Bertrand Russell.
The style of this is still evolving.
Online Sources
There are templates for the following online sources. Each one has been crafted so as to produce a reference in the style requested by the online source in question.
 Template:MathWorld
 This provides a direct link to a page on the https://mathworld.wolfram.com/ website.
Example:
 Weisstein, Eric W. "Circular Sector." From MathWorldA Wolfram Web Resource. https://mathworld.wolfram.com/CircularSector.html
which can be found in the page Area of Sector.
 Template:Planetmath
 This provides a direct link to a page on the https://planetmath.org/ website.
Example:
 This article incorporates material from Urysohn's Lemma on PlanetMath, which is licensed under the Creative Commons Attribution/ShareAlike License.
which can be found in the page Urysohn's Lemma.
 Template:MacTutor Biography
 This provides a direct link to a page on the https://mathshistory.standrews.ac.uk/ website.
Example:
which can be found in the page for Hanna Neumann. Note that the link presentation is taken from the page the template is invoked from.
 Template:KhanAcademy
 This provides a direct link to the Khan Academy.
Example:
 For a video presentation of the contents of this page, visit the Khan Academy.
which can be found in the page Limit of Sine of X over X at Zero/Geometric Proof.
 Template:Metamath
 This provides a direct link to Metamath.
Example:
which can be found in the page First Principle of Transfinite Recursion.
 Template:Mizar
 This provides a direct link to Mizar.
Example:
 Mizar article TOPGEN_1:10
which can be found in the page Characterization of Boundary by Basis.
 Template:OEIS
 This provides a direct link to the OnLine Encyclopedia of Integer Sequences.
Example:
 This sequence is A002193 in the OnLine Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
which can be found in the page Square Root of 2 is Irrational.
 Template:SpringerOnline
 This provides a direct link to a page on the Springer Online Encyclopedia of Mathematics.
Example:
 Ring. O.A. Ivanova (originator),Encyclopedia of Mathematics. URL: https://www.encyclopediaofmath.org/index.php?title=Ring
which can be found in the page Definition:Ring (Abstract Algebra).
 Template:Stackexchange
 This provides a direct link to a page on Mathematics Stack Exchange.
Example:
 robjohn (https://math.stackexchange.com/users/13854/robjohn), How to prove that $\ds \lim \limits_{x \mathop \to 0} \frac {\sin x} x = 1$?, URL (version: 20130619): https://math.stackexchange.com/q/75151
which can be found in the page Derivative of Sine Function/Proof 5.
Acceptability of Online Sources
NOTE: The above are currently the ONLY web resources which are to be used as general citation sources.
Others may be added to the above as and when they come to our attention as being particularly useful.
So feel free to challenge this assertion if you find something which appears to be a particularly rich and productive resource.
Scholarly papers which are available online may usually also be cited.
What are not generally acceptable include:
 Lecture notes for university courses available online (because they do not stay online forever, and this causes dead links)
 Links to pages in homework help forums
 Discussion pages in any web forum
 Wikipedia  not because we don't like them, but because as they are selfproclaimed tertiary source, there is no need to do so  we would rather go to the actual source works. See also Wikipedia:Citing Wikipedia.
Splitting sources
In some cases it is necessary to split a referenced theorem, proof or definition into multiple pages, because for example:
 a theorem contains multiple statements
 a proof contains in fact multiple proofs
 a definition defines multiple concepts at once.
If so, the source has to be referenced at every page, and its process flow is updated according to the order in which the elements appear in the source.
References
 ↑ Suggestions have been made as to how we may go about the exercise of internationalization, but progress in that direction is slow due to its perceived low priority.