Wikipedia:WikiProject Numbers/Guidelines

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General guidelines

When should an article exist?

See Wikipedia:Notability (numbers) § Notability of specific individual numbers for guidelines on when an article on a number should exist.

The sections #Assessing mathematical number facts and #Assessing content outside of pure mathematics on this page have guidelines for what content should go on such articles.

Outline

Articles can be divided into sections based on how much content there is for each area they reference. A section devoted to mathematics is common, however it is optional when there is only mathematical content present (aside from references, or external links). Mathematical content should come before content that does not deal in pure mathematics, including practical uses in applied fields such as science, and extending to cultural associations, and the history of symbols associated with numbers. Mathematical coverage should open with simple mathematical properties of the number, and list some simple associated properties (whether they are prime, or not, abundant, etc). Then, articles should move on to discuss deeper mathematics that is associated with the numbers, such as properties arising from being a member of specific classes of primes, or properties that stem from relationships with particular geometric figures; any relevant mathematical history of the number as it relates to these properties is also worth covering.

The advise for assessing mathematical content specifically distinguishes between content that is simple (or basic), formulaic, and off-topic. Referencing the On-Line Encyclopedia of Integer Sequences (OEIS) is largely admissible and encouraged, within limits; criteria generally follows merits of uniqueness of the numbers in question within the sequences and any other relevant associated sequences. Assessing deeper connections of integers to other mathematical objects of interest is also relevant and an important part of coverage of mathematical properties ascribed to the numbers. This guideline also emphasizes, per Wikipedia policy, that original research (inclusive of synthesis of published information) is not permitted.

Infobox

Example number Infobox

← N − 1               N               N + 1 →
← x0  x1  x2  x3  x4  N  x6  x7  x8  x9 →
List of numbers ‧ Integers
← y  2y  3y  4y  5y  6y  7y  8y  9y  10y →
Cardinal	en
Ordinal	Nth
Factorization	1 × N (prime) or pa × pb × ...
Divisors	1, pa, pb, pc, ..., pa × pb, ..., N
Roman numeral	"N"
Binary	"N2"
...	...
Hexadecimal	"N16"

The Infobox — added at the top of the page using the template {{Infobox number}} — should include the symbol of the number in all known numeral systems for which Unicode characters exist, as long as it fits within a reasonable amount of space (examples include Egyptian, Roman, Tamil, Cyrillic, and Burmese; refer to the Infobox at the article for 1 for an example with relevant Wiki markup). Divisors should be listed in the Infobox if they fit in a reasonable amount of space, as they should generally be omitted from the article prose.

These are exceptions to the usual practice of only summarizing body text in the Infobox (see MOS:INFOBOXEXCEPTIONS).

Symbolic and linguistic representations

• For numbers which are also individual characters, information about etymology, history, and appearance (etc.) should remain in the articles about the specific numeral systems (e.g. Roman numerals, Tamil numerals), which should be linked from their symbols in the Infobox. An exception is made for the English numerals, where their etymology and history of symbolic representation should be written in summary style with a {{main}} template at the start of the section, e.g.:
  Main article: History of the Hindu–Arabic numeral system
• Translations (of which there are a huge number) should be deferred to Wiktionary. Check that there is a link (produced with {{wiktionary}}) in the "See also" section or lower (for numbers where a Wiktionary entry exists; numeral preferred - e.g. wikt:1 - but word or words if needed - e.g. wikt:one million).
• Explanation about how to construct number names in words to the articles Cardinal numeral and Ordinal numeral should not be specified either; these are automatically linked from the Infobox. (These cover all languages, so language-specific details are further deferred to articles like English numerals.)

Assessing mathematical number facts

The purpose of number articles is to provide a broad overview of a number's history, mathematical, scientific, or cultural properties and uses, without going into excessive detail (see WP:Summary). While there is an endless array of true mathematical facts about a given number, not all of them are appropriate for inclusion. This section provides several criteria for assessing the suitability of a mathematical number fact. Even if a fact is on topic, it may still not be interesting enough to warrant inclusion (see the interesting number paradox).

Simple facts

Simple facts are helpful, easily understood details about a number. They do not have to be sourced and should be included in a number article, although they are usually not sufficient reason to justify the existence of an article on their own. This criterion should be applied sparingly. A good rule of thumb is: "Would someone with no background in mathematics be expected to know this?".

Examples of simple facts:

• On 1729: 1729 is the natural number following 1728 and preceding 1730.
• On 1729: 1729 is a composite number with factors 1, 7, 13, 19, 91, 133, 247, and 1729.

Examples that are not simple facts:

• On 9: "The aliquot sum of 9 is 4".
• On 11: The rows of Pascal's triangle can be seen as representations of the powers of 11.

Routine facts

Routine facts are true for nearly every other number upon a slight modification. Generally, these come from mathematical expressions that are formulaic in nature, including algebraic formulae, or through recursive methods. In order for a sequence of integers to be considered non-routine, significant coverage of the sequence in the literature should exist.

The first (non-trivial) element of a non-routine integer sequence should be included. For finite sequences, the last element should be included as well. Integer entries from a large sequence that are not the of the first few or the last entries should generally not be included unless the specific number is covered in depth by the mathematical literature (routine mentions are not sufficient). Ideally, the specific integer entry being referenced should have applications in an unsolved or important solved problem in mathematics.

Examples that are routine:

• On 5: "5 is the third odd number."
• On 5: "5 is the first pentagonal number after 1."
• On 9: "The aliquot sum of 9 is 4".
• On 19: "The Collatz sequence for nine requires nineteen steps to return to one, more than any other number below it. On the other hand, nineteen requires twenty steps, like eighteen. Less than ten thousand, only thirty-one other numbers require nineteen steps to return to one."
• On 1729: "It is the tenth centered cube number, the nineteenth dodecagonal number, the thirteenth 24-gonal and the seventh 84-gonal number."
• On 197: "197 is the sum of the first twelve prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37"
• On 69: "69 is the third composite number in the 13-aliquot tree, following 27 and 35."
• On 744: "The number partitions of the square of seven (49) into prime parts is 744."

Examples that are non-routine:

• On 3: "3 is the first odd prime".
• On 19: "19 forms a twin prime with 17, a cousin prime with 23, and a sexy prime with 13".
• On 42: The 1806th Bernoulli number is the only Bernoulli number whose denominator is the same as its index (1806) in the sequence of Bernoulli numbers.

Off-topic facts

Off-topic facts are details which do not have a direct or substantial connection to the number $N$ itself. Content on a page for number $N$ should focus primarily on that specific number and not about another number or topic, even if they are related to $N$. Including such facts dilutes the focus of the content and can be confusing for readers.

Examples of Off-topic facts:

• On 5: "The factorial of five, 5! = 120, which is multiply perfect".
• On 5: "A magic constant of 505 is generated by a 10 × 10 normal magic square". (The only connection here being that 505 contains two 5s.)
• On 744: "The twentieth prime number is 71, where 31 is the eleventh; in turn, 20 is the eleventh composite number that is also the sixth self-convolution of Fibonacci numbers before 38, which is the prime index of 163."

Examples of On-topic facts:

• On 5: "5 is the first of the three known Wilson primes 5, 13, 563".
• On 9: "The aliquot sum of 9 is 4".
• On 19: "19 forms a twin prime with 17, a cousin prime with 23, and a sexy prime with 13".
• On 11: The rows of Pascal's triangle can be seen as representation of the powers of 11.

Note that even if a fact is on-topic, it may still not be interesting enough, too routine, or lack sufficient verification from reliable sources for inclusion in the article.

Connection

The existence of a connection between a number and an important or prominent object is usually acceptable.

Examples that are a connection to an important object:

• On 5: "In graph theory, all graphs with four or fewer vertices are planar, however, there is a graph with five vertices that is not: K₅, the complete graph with five vertices, where every pair of distinct vertices in a pentagon is joined by unique edges belonging to a pentagram."
• On 9: "A polygon with nine sides is called a nonagon".

No Wikipedia page

If a property is not important enough to have a Wikipedia page, it probably isn't important that the number in question satisfies it. This does not necessarily mean that if a property does have a Wikipedia page, it should automatically be included.

OEIS

OEIS is considered a reliable source. However, a mathematical fact appearing in OEIS is not by itself a good reason to include it in a Wikipedia article. OEIS has hundreds of thousands of sequences, most of which do not belong on Wikipedia. If an OEIS sequence has almost no mentions outside of OEIS, the number N being part of that sequence shouldn't be on the page for N. If N is a member of an OEIS sequence and this fact is discussed in depth in sources other than OEIS, this rule does not disqualify N being a member of that sequence from inclusion on the page for N.

No original research and no synthesis

Violations of WP:OR and WP:SYNTH should be removed even if they pass all of the previous tests. Most facts need to be stated in a reliable source, external to Wikipedia, prior to inclusion.

Assessing content outside of pure mathematics

It is preferable to have a stub article than to pad an article with trivial or tenuously related information. Normal Wikipedia policies, and guidelines, should be considered when choosing which content to include. Specifically, this also applies to scientific and other associations in fields benefiting from applications of mathematical knowledge, and as such they must also be verifiable, and covered with due weight.

Aside from WP:DUE and WP:UNDUE content as described in sections above for mathematics-specific content, metrics for information that is due and not strictly dealing in pure mathematics, yet is either applied mathematics or regarding specific associations that are philosophical, cultural, or from other specific traditional uses in society, can be outlined.

Applied mathematics

A scientific notion referenced in a number article depends on the mathematics of its properties and characteristics, and how substantial they are, as aforementioned. Examples include:

• a statement such as, there is theoretical and empirical evidence to suggest that brain computation is organized via power-of-two-based permutation logic,
• three soap films meet along a Plateau border at 120 degree angles.

Philosophy and symbolism

Symbolism in lucky or unlucky numbers (as a form of numerology) can be included, such as a direct reference to mathematical properties of the number, even if mystical (i.e. The One in the philosophy of neoplatonism), or by popular association (including associations with numerals only); for example, the number 666 as the number of the beast. Per WP:FRINGE, all such examples need reliable mainstream sourcing.

What not to include

Do not include content that relates to the article's title only as an identifier, or label, or simple enumeration, or measurement, or as a trivial mention. This includes:

• telephone number calling codes,
• route or road numbers, bus routes,
• firearms (i.e. Glock 17),
• military and transportation classes of vehicles, aircraft, or otherwise,
• album titles or song titles that include a numeral
• book titles (except for when they deal directly with mathematics),
• sports jerseys (identified with numbers, or letters and numbers), NASCAR car numbers, etc.,
• world records of any sort, aside from records that could be of cultural interest in mathematics, such as enumerating digits of pi by memory, etc.,
• marketed items, such as bubble gum 5,

or any other concept or item that is not directly related to the number itself (in this case, only its numeral is being referenced). If there is plausible ambiguity between the number and another use of the numeral, they should be mentioned on a disambiguation page for the number. If necessary, a hatnote can be added at the top of the article to link to another article for disambiguation. For example:

This page is about the number 1. For other uses, see One (disambiguation).

This enables the article to remain focused on content specifically related to the number.

Examples of items to not include:

• Route 66 which does not have a significant relationship with the number 66 - rather, the "66" is an identification code.
• 86 (novel series) is not about the number 86.
• The 35 in 35 mm film is a measurement, not a property of, or reference to, the number 35.
• The 9 in ISO 9 is a simple enumeration of a series of items, with no significant relationship with the number 9.
• The centre-forward in association football wearing the number 9 shirt is a trivial mention of the number 9.

Editing practice

Naming compliance

For most cases, articles should be named N (number), with the literal spelling of the numbers redirected accordingly (e.g., Four hundred and ninety-six redirects to 496 (number)).

Numbers deserving their own article that are greater than 999, should have the article title written in digits without any separators between the digits of the integer part. Writing the number with separators may of course be acknowledged in the body of the article. Thus, the article on the taxicab number 1,729 should be 1729 (number), though the article can mention that the number may be written "1,729" or "1.729."

Besides −1, there are no articles on negative integers. Adding information about a negative number therefore can fall under articles representative of numbers' positive, absolute values. One half is the chosen article name for 1⁄2.

Referencing

Just as with any information in Wikipedia, articles on numbers need to cite sources. (See Wikipedia:Citing sources for general information on citing sources.) Some statements that are easily verified with a pocket calculator might not need citations (e.g., the fact that 2×3 = 6), but anything slightly more difficult to verify does. It is acceptable for a number article to have few citations if the articles to which it links have primary and secondary sourcing. For example, to avoid footnote overkill, an article about a number p that is a certain type of prime does not need citations to background reading about that type of prime.

It is not the place of this project to prescribe a citation format. Until Wikipedia decides on a uniform citation format, number articles may use whatever citation format would be acceptable in a mathematics journal. Consistency within an article is generally preferred.

Web versions of respected professional journals are reliable sources. On the other hand, preprints on the arXiv that have not been peer-reviewed are generally unsuitable for our purposes. The On-Line Encyclopedia of Integer Sequences should be used with care: it is curated by experts and generally accurate, but it is by design rather indiscriminate about what it includes. From the standpoint of serious mathematical literature, some OEIS entries are mere curiosities. Those that are more noteworthy will include references into the literature that include more substantial discussions; consider supplementing citations to the OEIS with direct pointers to the peer-reviewed papers that it cites. MathWorld has a history of being less than reliable, for example by including neologisms that have not become accepted mathematical terminology. It should be avoided.

Edit summaries

Every project member (indeed anyone who edits Wikipedia) is encouraged to write brief but complete edit summaries. Please refer to Wikipedia:Edit summary for a guide on writing good edit summaries.