Editing 4 (section)

==Mathematics ==
There are four elementary arithmetic [[Operation (mathematics)|operations]] in mathematics: [[addition]] ('''+'''), [[subtraction]] ('''−'''), [[multiplication]] ('''×'''), and [[Division (mathematics)|division]] ('''÷''').<ref>{{Cite web |last=Tiwari |first=Arvind Kumar |date=2023 |title=What are the four basic mathematical operations, and what do they mean? |url=https://www.quora.com/What-are-the-four-basic-mathematical-operations-and-what-do-they-mean |access-date=30 September 2024 |website=Quora}}</ref>

[[Lagrange's four-square theorem]] states that every positive integer can be written as the sum of at most four [[square number|square]]s.<ref>{{Citation|last=Spencer|first=Joel|title=Four Squares with Few Squares|year=1996|work=Number Theory: New York Seminar 1991–1995|pages=295–297|editor-last=Chudnovsky|editor-first=David V.|place=New York, NY|publisher=Springer US| language=en|doi=10.1007/978-1-4612-2418-1_22|isbn=978-1-4612-2418-1|editor2-last=Chudnovsky|editor2-first=Gregory V.|editor3-last=Nathanson|editor3-first=Melvyn B.}}</ref><ref>{{Cite book|last=Peterson|first=Ivars|url=https://books.google.com/books?id=4gWSAraVhtAC&q=7+for+instance+cannot+be+written+as+the+sum+of+three+squares.&pg=PA95|title=Mathematical Treks: From Surreal Numbers to Magic Circles|date=2002|publisher=MAA|isbn=978-0-88385-537-9|pages=95|language=en|quote=7 is an example of an integer that can't be written as the sum of three squares.}}</ref> Four is one of four [[Harshad number|all-Harshad number]]s. Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. <math>4x=y^{2}-z^{2}</math>.

A four-sided plane figure is a [[quadrilateral]] or quadrangle, sometimes also called a ''tetragon''. It can be further classified as a [[rectangle]] or ''oblong'', [[kite]], [[rhombus]], and [[square]].

Four is the highest degree general [[polynomial equation]] for which there is a [[solution in radicals]].<ref>{{Cite book| last=Bajnok|first=Béla|url=https://books.google.com/books?id=cNFzKnvxXoAC&q=Abel%E2%80%93Ruffini+theorem&pg=PT78|title=An Invitation to Abstract Mathematics|date=2013-05-13|publisher=Springer Science & Business Media| isbn=978-1-4614-6636-9| language=en|quote=There is no algebraic formula for the roots of the general polynomial of degrees 5 or higher.}}</ref>

Four is the only square number <math>I=i\times i</math> where <math>I - 1</math> is a prime number.

The [[four-color theorem]] states that a [[planar graph]] (or, equivalently, a flat [[map]] of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors.<ref>{{cite book |first=Bryan |last=Bunch |title=The Kingdom of Infinite Number |location=New York |publisher=W. H. Freeman & Company |year=2000 |page=48}}</ref> Three colors are not, in general, sufficient to guarantee this.<ref>{{Cite book|last=Ben-Menahem|first=Ari|url=https://books.google.com/books?id=9tUrarQYhKMC&q=three+colors+map+not+enough&pg=PA2147|title=Historical Encyclopedia of Natural and Mathematical Sciences|date=2009-03-06|publisher=Springer Science & Business Media| isbn=978-3-540-68831-0|pages=2147|language=en|quote=(i.e. That there are maps for which three colors are not sufficient)}}</ref> The largest planar [[complete graph]] has four vertices.<ref>{{Cite book|last=Molitierno|first=Jason J.| url=https://books.google.com/books?id=2kvNBQAAQBAJ&q=largest+planar+complete+graph+has+four+vertices&pg=PA197| title=Applications of Combinatorial Matrix Theory to Laplacian Matrices of Graphs|date=2016-04-19|publisher=CRC Press| isbn=978-1-4398-6339-8|pages=197|language=en|quote=... The complete graph on the largest number of vertices that is planar is K4 and that a(K4) equals 4.}}</ref>

A solid figure with four faces as well as four vertices is a [[tetrahedron]], which is the smallest possible number of faces and vertices a [[polyhedron]] can have.<ref>{{Cite book|last1=Grossnickle|first1=Foster Earl|url=https://books.google.com/books?id=Q2474oSAsc4C&q=4+is+the+smallest+possible+number+of+faces+(as+well+as+vertices)+of+a+polyhedron.|title=Discovering Meanings in Elementary School Mathematics|last2=Reckzeh|first2=John|date=1968|publisher=Holt, Rinehart and Winston|pages=337|isbn=9780030676451|language=en|quote=...the smallest possible number of faces that a polyhedron may have is four}}</ref> The regular tetrahedron, also called a 3-[[simplex]], is the simplest [[Platonic solid]].<ref>{{Cite book|last1=Grossnickle|first1=Foster Earl|url=https://books.google.com/books?id=Q2474oSAsc4C&q=4+is+the+smallest+possible+number+of+faces+(as+well+as+vertices)+of+a+polyhedron.|title=Discovering Meanings in Elementary School Mathematics|last2=Reckzeh|first2=John|date=1968|publisher=Holt, Rinehart and Winston|pages=337|isbn=9780030676451|language=en|quote=...face of the platonic solid. The simplest of these shapes is the tetrahedron...}}</ref> It has four [[regular triangle]]s as faces that are themselves at [[self-dual polytope|dual positions]] with the vertices of another tetrahedron.<ref>{{Cite book|last1=Hilbert|first1=David|url=https://books.google.com/books?id=7WY5AAAAQBAJ&q=self-dual+regular+polyhedron&pg=PA143|title=Geometry and the Imagination|last2=Cohn-Vossen |first2=Stephan |date=1999|publisher=American Mathematical Soc.|isbn=978-0-8218-1998-2|pages=143|language=en|quote=...the tetrahedron plays an anomalous role in that it is self-dual, whereas the four remaining polyhedra are mutually dual in pairs...}}</ref>

The smallest non-[[cyclic group]] has four elements; it is the [[Klein four-group]].<ref>{{Cite book|first=Jeremy|last=Horne|url=https://books.google.com/books?id=ZfYoDwAAQBAJ&pg=PA299|title=Philosophical Perceptions on Logic and Order|date=2017-05-19|publisher=IGI Global|isbn=978-1-5225-2444-1|pages=299|language=en|quote=The Klein four-group is the smallest noncyclic group,...|access-date=31 October 2022|archive-date=31 October 2022|archive-url=https://web.archive.org/web/20221031005437/https://books.google.com/books?id=ZfYoDwAAQBAJ&pg=PA299|url-status=live}}</ref> ''A{{sub|n}}'' [[alternating group]]s are not [[simple group|simple]] for values <math>n</math> ≤ <math>4</math>.

There are four [[Hopf fibration]]s of [[hypersphere]]s:

<math display=block>
\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}</math>

They are defined as locally trivial [[fibration]]s that map <math>f : S^{2n-1} \rightarrow S^{n}</math> for values of <math>n=2,4,8</math> (aside from the trivial fibration mapping between two [[Point (geometry)|points]] and a [[circle]]).<ref>{{Cite book |last=Shokurov |first= A.V. |editor=Michiel Hazewinkel |editor-link=Michiel Hazewinkel |chapter=Hopf fibration |title=Encyclopedia of Mathematics |publisher=[[European Mathematical Society]] |location=Helsinki |chapter-url=https://encyclopediaofmath.org/wiki/Hopf_fibration |year=2002 |isbn=1402006098 |oclc=1013220521 |access-date=2023-04-30 |archive-date=1 May 2023 |archive-url=https://web.archive.org/web/20230501005558/https://encyclopediaofmath.org/wiki/Hopf_fibration |url-status=live }}</ref>

In [[Knuth's up-arrow notation]], <math>2+2=2\times2=2^{2}=2\uparrow\uparrow 2=2\uparrow\uparrow\uparrow2=\;...\; = 4</math>, and so forth, for any number of up arrows.<ref>{{Cite book |last=Hodges |first=Andrew |url=https://books.google.com/books?id=HOcpgfiDu40C&q=2+%E2%86%91%E2%86%91+2&pg=PA249 |title=One to Nine: The Inner Life of Numbers |date=2008-05-17 |publisher=W. W. Norton & Company |isbn=978-0-393-06863-4 |pages=249 |language=en |quote=2 ↑↑ ...  ↑↑ 2 is always 4}}</ref>