4

Template:Short description {{#invoke:Hatnote|hatnote}} Template:Distinguish Template:Use dmy dates Template:Infobox number 4 (four) is a number, numeral and digit. It is the natural number following 3 and preceding 5. It is a square number, the smallest semiprime and composite number, and is considered unlucky in many East Asian cultures.

Evolution of the Hindu-Arabic digit

File:Evolution4glyph.png

File:Vier.jpg

Two modern handwritten fours

File:Algund Dominikanerinnenklosterkirche Portal bezeichnet 1481 (cropped).jpg

Sculpted date "1481" in the Convent church of Maria Steinach in Algund, South Tirol, Italy. The upward loop signifies the number 4.

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. The Shunga would add a horizontal line on top of the digit, and the Kshatrapa and Pallava evolved the digit to a point where the speed of writing was a secondary concern. The Arabs' 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.<ref>Georges Ifrah, The Universal History of Numbers: From Prehistory to the Invention of the Computer transl. David Bellos et al. London: The Harvill Press (1998): 394, Fig. 24.64</ref>

While the shape of the character for the digit 4 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in File:TextFigs148.svg.

On the seven-segment displays of pocket calculators and digital watches, as well as certain optical character recognition fonts, 4 is seen with an open top: File:Seven-segment 4.svg.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Television stations that operate on 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 letter ᔦ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4".<ref>Template:Cite news</ref>

Mathematics

There are four elementary arithmetic operations in mathematics: addition (+), subtraction (−), multiplication (×), and division (÷).<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four squares.<ref>Template:Citation</ref><ref>Template:Cite book</ref> Four is one of four all-Harshad numbers. 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>Template:Cite book</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>Template:Cite book</ref> Three colors are not, in general, sufficient to guarantee this.<ref>Template:Cite book</ref> The largest planar complete graph has four vertices.<ref>Template:Cite book</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>Template:Cite book</ref> The regular tetrahedron, also called a 3-simplex, is the simplest Platonic solid.<ref>Template:Cite book</ref> It has four regular triangles as faces that are themselves at dual positions with the vertices of another tetrahedron.<ref>Template:Cite book</ref>

The smallest non-cyclic group has four elements; it is the Klein four-group.<ref>Template:Cite book</ref> A_n alternating groups are not simple for values <math>n</math> ≤ <math>4</math>.

There are four Hopf fibrations of hyperspheres:

<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 fibrations 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 points and a circle).<ref>Template:Cite book</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>Template:Cite book</ref>

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	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num	Template:Num

Division	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16
4 ÷ x	4	2	1.Template:Overline	1	0.8	0.Template:Overline	0.Template:Overline	0.5	0.Template:Overline	0.4	0.Template:Overline	0.Template:Overline	0.Template:Overline	0.Template:Overline	0.2Template:Overline	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	Template:Num	Template:Num	256	1024	4096	16384	65536	262144	1048576	4194304	16777216	67108864	268435456	1073741824	4294967296
x4	1	Template:Num	81	256	625	1296	2401	4096	6561	10000	14641	20736	28561	38416	50625	65536

In culture

• Four is the sacred number of the Zia, an indigenous tribe located in the U.S. state of New Mexico.<ref>Template:Cite book</ref>
• The Chinese, the Koreans, and the Japanese are superstitious about the number four because it is a homonym for "death" in their languages.<ref>Template:Cite book</ref>

In logic and philosophy

File:Mugs of tea viewed from above.jpg

Four mugs

• The symbolic meanings of the number four are linked to those of the cross and the square. "Almost from prehistoric times, the number four was employed to signify what was solid, what could be touched and felt. Its relationship to the cross (four points) made it an outstanding symbol of wholeness and universality, a symbol which drew all to itself". Where lines of latitude and longitude intersect, they divide the earth into four proportions. Throughout the world kings and chieftains have been called "lord of the four suns" or "lord of the four quarters of the earth",<ref>Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402.</ref> which is understood to refer to the extent of their powers both territorially and in terms of total control of their subjects' doings.
• The Square of Opposition, in both its Aristotelian version and its Boolean version, consists of four forms: A ("All S is R"), I ("Some S is R"), E ("No S is R"), and O ("Some S is not R").

In technology

• In internet slang, "4" can replace the word "for" (as "four" and "for" are pronounced similarly). For example, typing "4u" instead of "for you".
• In Leetspeak, "4" may be used to replace the letter "A".

Other groups of four

• Approximately four weeks (4 times 7 days) to a lunar month (synodic month = 29.54 days). Thus the number four is universally an integral part of primitive sacred calendars.

References

• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 55–58

External links

• Marijn.Org on Why is everything four?
• A few thoughts on the number four, by Penelope Merritt at samuel-beckett.net
• The Number 4
• The Positive Integer 4
• Prime curiosities: 4