# 4

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This article is about the number. For the years, see [BC 4](/source/BC_4) and [4 AD](/source/4_AD). For other uses, see [4 (disambiguation)](/source/4_(disambiguation)), [IV (disambiguation)](/source/IV_(disambiguation)) and [Number 4](/source/Number_Four_(disambiguation)).

Not to be confused with [Cuatrillo](/source/Cuatrillo).

"2^2" redirects here. For the sequential power of two, see [Power of two](/source/Power_of_two).

**4** (**four**) is a [number](/source/Number), [numeral](/source/Numeral_(linguistics)) and [digit](/source/Numerical_digit). It is the [natural number](/source/Natural_number) following [3](/source/3) and preceding [5](/source/5). It is a [square number](/source/Square_number), the smallest [semiprime](/source/Semiprime) and [composite number](/source/Composite_number), and is [considered unlucky](/source/Tetraphobia) in many [East Asian](/source/East_Asia) cultures.

## Evolution of the Hindu-Arabic digit

[Brahmic numerals](/source/Brahmic_numerals) represented 1, 2, and 3 with as many lines. 4 was simplified by joining its four lines into a cross that looks like the modern plus sign.[1] The [Shunga](/source/Shunga_Empire) would add a horizontal line on top of the digit, and the [Kshatrapa](/source/Northern_Satraps) and [Pallava](/source/Pallava_dynasty) evolved the digit to a point where the speed of writing was a secondary concern. The [Arabs](/source/Arab)' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the digit less cursive, ending up with a digit very close to the original Brahmin cross.[2]

While the shape of the character for the digit 4 has an [ascender](/source/Ascender_(typography)) in most modern [typefaces](/source/Typeface), in typefaces with [text figures](/source/Text_figures) the glyph usually has a [descender](/source/Descender), as, for example, in .

On the [seven-segment displays](/source/Seven-segment_display) of pocket calculators and digital watches, as well as certain [optical character recognition](/source/Optical_character_recognition) fonts, 4 is seen with an open top: .[3]

[Television stations](/source/Television_station) that operate on [channel](/source/Television_channel) 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the [Canadian Aboriginal syllabics](/source/Canadian_Aboriginal_syllabics) letter ᔦ. The [magnetic ink character recognition](/source/Magnetic_ink_character_recognition) "CMC-7" font also uses this variety of "4".[4]

## Mathematics

[Lagrange's four-square theorem](/source/Lagrange's_four-square_theorem) states that every positive integer can be written as the sum of at most four [squares](/source/Square_number).[5][6] Four is one of four [all-Harshad numbers](/source/Harshad_number). Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x=y^{2}-z^{2}.

A four-sided plane figure is a [quadrilateral](/source/Quadrilateral) or quadrangle, sometimes also called a *tetragon*. It can be further classified as a [rectangle](/source/Rectangle) or *oblong*, [kite](/source/Kite), [rhombus](/source/Rhombus), and [square](/source/Square).

Four is the highest degree general [polynomial equation](/source/Polynomial_equation) for which there is a [solution in radicals](/source/Solution_in_radicals).[7]

Four is the only square number N=n\times n where N - 1 is a prime number.

The [four-color theorem](/source/Four-color_theorem) states that a [planar graph](/source/Planar_graph) (or, equivalently, a flat [map](/source/Map) of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors.[8] Three colors are not, in general, sufficient to guarantee this.[9] The largest planar [complete graph](/source/Complete_graph) has four vertices.[10]

A solid figure with four faces as well as four vertices is a [tetrahedron](/source/Tetrahedron), which is the smallest possible number of faces and vertices a [polyhedron](/source/Polyhedron) can have.[11] The regular tetrahedron, also called a 3-[simplex](/source/Simplex), is the simplest [Platonic solid](/source/Platonic_solid).[12] It has four [regular triangles](/source/Regular_triangle) as faces that are themselves at [dual positions](/source/Self-dual_polytope) with the vertices of another tetrahedron.[13]

The smallest non-[cyclic group](/source/Cyclic_group) has four elements; it is the [Klein four-group](/source/Klein_four-group).[14] *An* [alternating groups](/source/Alternating_group) are not [simple](/source/Simple_group) for values n ≤ 4.

There are four [Hopf fibrations](/source/Hopf_fibration) of [hyperspheres](/source/Hypersphere):

\begin{align}
S^0 & \hookrightarrow S^1 \to S^1, \\
S^1 & \hookrightarrow S^3 \to S^2, \\
S^3 & \hookrightarrow S^7 \to S^4, \\
S^7 & \hookrightarrow S^{15}\to S^8. \\
\end{align}

They are defined as locally trivial [fibrations](/source/Fibration) that map f : S^{2n-1} \rightarrow S^{n} for values of n=2,4,8 (aside from the trivial fibration mapping between two [points](/source/Point_(geometry)) and a [circle](/source/Circle)).[15]

In [Knuth's up-arrow notation](/source/Knuth's_up-arrow_notation), 2+2=2\times2=2^{2}=2\uparrow\uparrow 2=2\uparrow\uparrow\uparrow2=\;...\; = 4, and so forth, for any number of up arrows.[16]

There are four dimensions in the theory of [Minkowski space](/source/Minkowski_space), three of space and the one being time.

## List of basic calculations

Multiplication 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 50 100 1000 4 × x 4

Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4 ÷ x 4 2 1.3 1 0.8 0.6 0.571428 0.5 0.4 0.4 0.36 0.3 0.307692 0.285714 0.26 0.25 x ÷ 4 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4

Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 4x 4 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864 268435456 1073741824 4294967296 x4 1 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561 38416 50625 65536

## In culture

- Four is the sacred number of the [Zia](/source/Zia_(New_Mexico)), an indigenous tribe located in the U.S. state of [New Mexico](/source/New_Mexico).[17]
- [Superstition about the number four](/source/Tetraphobia) is most common in [East Asian](/source/East_Asia) nations, where it is a [homonym](/source/Homonym) for "death" in many languages.[18]

## In technology

- In [internet slang](/source/Internet_slang), "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, messaging "4 u" instead of "for you" when talking to someone.
- In [Leetspeak](/source/Leet), "4" may be used to replace the letter "A".

## References

1. Heller, Steven (January 3, 2017). *Graphic Design Rants and Raves Bon Mots on Persuasion, Entertainment, Education, Culture, and Practice*. Allworth Press. ISBN 9781621535393.

1. Ifrah, Georges (1998). *The Universal History of Numbers: From Prehistory to the Invention of the Computer*. Translated by Bellos, David. London: The Harvill Press. p. 394, Fig. 24.64.

1. ["Seven Segment Displays (7-Segment) | Pinout, Types and Applications"](https://www.electronicshub.org/seven-segment-displays/). *Electronics Hub*. 2019-04-22. [Archived](https://web.archive.org/web/20200728014108/https://www.electronicshub.org/seven-segment-displays/) 28 July 2020 at the Wayback Machine. Retrieved 2020-07-28.

1. ["Battle of the MICR Fonts: Which Is Better, E13B or CMC7? - Digital Check"](https://www.digitalcheck.com/battle-micr-fonts-better-e13b-cmc7/). *Digital Check*. 2017-02-02. [Archived](https://web.archive.org/web/20200803161254/https://www.digitalcheck.com/battle-micr-fonts-better-e13b-cmc7/) 3 August 2020 at the Wayback Machine. Retrieved 2020-07-28.

1. Spencer, Joel (1996), "Four Squares with Few Squares", *Number Theory: New York Seminar 1991–1995*, Chudnovsky, David V. (ed.), New York, NY: Springer US, pp. 295–297, [doi:10.1007/978-1-4612-2418-1_22](https://doi.org/10.1007/978-1-4612-2418-1_22). ISBN 978-1-4612-2418-1

1. Peterson, Ivars (2002). [*Mathematical Treks: From Surreal Numbers to Magic Circles*](https://books.google.com/books?id=4gWSAraVhtAC&q=7+for+instance+cannot+be+written+as+the+sum+of+three+squares.&pg=PA95). MAA. p. 95. ISBN 978-0-88385-537-9. 7 is an example of an integer that can't be written as the sum of three squares.

1. Bajnok, Béla (2013-05-13). [*An Invitation to Abstract Mathematics*](https://books.google.com/books?id=cNFzKnvxXoAC&q=Abel%E2%80%93Ruffini+theorem&pg=PT78). Springer Science & Business Media. ISBN 978-1-4614-6636-9. There is no algebraic formula for the roots of the general polynomial of degrees 5 or higher.

1. Bunch, Bryan (2000). *The Kingdom of Infinite Number*. New York: W. H. Freeman & Company. p. 48.

1. Ben-Menahem, Ari (2009-03-06). [*Historical Encyclopedia of Natural and Mathematical Sciences*](https://books.google.com/books?id=9tUrarQYhKMC&q=three+colors+map+not+enough&pg=PA2147). Springer Science & Business Media. p. 2147. ISBN 978-3-540-68831-0. (i.e. That there are maps for which three colors are not sufficient)

1. Molitierno, Jason J. (2016-04-19). [*Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs*](https://books.google.com/books?id=2kvNBQAAQBAJ&q=largest+planar+complete+graph+has+four+vertices&pg=PA197). CRC Press. p. 197. ISBN 978-1-4398-6339-8. ... The complete graph on the largest number of vertices that is planar is K4 and that a(K4) equals 4.

1. Grossnickle, Foster Earl & Reckzeh, John (1968). [*Discovering Meanings in Elementary School Mathematics*](https://books.google.com/books?id=Q2474oSAsc4C&q=4+is+the+smallest+possible+number+of+faces+(as+well+as+vertices)+of+a+polyhedron.). Holt, Rinehart and Winston. p. 337. ISBN 9780030676451. ...the smallest possible number of faces that a polyhedron may have is four

1. Grossnickle, Foster Earl & Reckzeh, John (1968). [*Discovering Meanings in Elementary School Mathematics*](https://books.google.com/books?id=Q2474oSAsc4C&q=4+is+the+smallest+possible+number+of+faces+(as+well+as+vertices)+of+a+polyhedron.). Holt, Rinehart and Winston. p. 337. ISBN 9780030676451. ...face of the platonic solid. The simplest of these shapes is the tetrahedron...

1. Hilbert, David & Cohn-Vossen, Stephan (1999). [*Geometry and the Imagination*](https://books.google.com/books?id=7WY5AAAAQBAJ&q=self-dual+regular+polyhedron&pg=PA143). American Mathematical Soc. p. 143. ISBN 978-0-8218-1998-2. ...the tetrahedron plays an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs...

1. Horne, Jeremy (2017-05-19). [*Philosophical Perceptions on Logic and Order*](https://books.google.com/books?id=ZfYoDwAAQBAJ&pg=PA299). IGI Global. p. 299. ISBN 978-1-5225-2444-1. [Archived](https://web.archive.org/web/20221031005437/https://books.google.com/books?id=ZfYoDwAAQBAJ&pg=PA299) 31 October 2022 at the Wayback Machine. Retrieved 31 October 2022. The Klein four-group is the smallest noncyclic group,...

1. Shokurov, A.V. (2002). ["Hopf fibration"](https://encyclopediaofmath.org/wiki/Hopf_fibration). *Encyclopedia of Mathematics*. Michiel Hazewinkel (ed.). Helsinki: [European Mathematical Society](/source/European_Mathematical_Society). ISBN 1402006098. [OCLC 1013220521](https://www.worldcat.org/oclc/1013220521). [Archived](https://web.archive.org/web/20230501005558/https://encyclopediaofmath.org/wiki/Hopf_fibration) 1 May 2023 at the Wayback Machine. Retrieved 2023-04-30.

1. Hodges, Andrew (2008-05-17). [*One to Nine: The Inner Life of Numbers*](https://books.google.com/books?id=HOcpgfiDu40C&q=2+%E2%86%91%E2%86%91+2&pg=PA249). W. W. Norton & Company. p. 249. ISBN 978-0-393-06863-4. 2 ↑↑ ... ↑↑ 2 is always 4

1. [*Bulletin - State Department of Education*](https://books.google.com/books?id=GZpLAQAAMAAJ&q=Four+is+the+sacred+number+of+the+Zia). Department of Education. 1955. p. 151. Four was a sacred number of Zia

1. Lachenmeyer, Nathaniel (2005). [*13: The Story of the World's Most Notorious Superstition*](https://books.google.com/books?id=sDXJ1s0YNAgC&q=+homonym+for+%22death%22). Penguin Group (USA) Incorporated. p. 187. ISBN 978-0-452-28496-8. In Chinese, Japanese, and Korean, the word for four is, unfortunately, an exact homonym for death

- Wells, D. *[The Penguin Dictionary of Curious and Interesting Numbers](/source/The_Penguin_Dictionary_of_Curious_and_Interesting_Numbers)* London: Penguin Group. (1987): 55–58

## External links

- [Marijn.Org on Why is everything four?](http://www.marijn.org/everything-is-4)
- [A few thoughts on the number four](https://web.archive.org/web/20180303094710/http://www.samuel-beckett.net/Penelope/four_symbolism.html), by Penelope Merritt at samuel-beckett.net
- [The Number 4](http://www.numdic.com/4)
- [The Positive Integer 4](http://www.positiveintegers.org/4)
- [Prime curiosities: 4](http://primes.utm.edu/curios/page.php/4.html)

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