5

This article is about the number. For the years, see 5 BC and AD 5. For other uses, see 5 (disambiguation), Number Five (disambiguation), and The Five (disambiguation).

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Cardinal	five
Ordinal	5th (fifth)
Numeral system	quinary
Factorization	prime
Prime	3rd
Divisors	1, 5
Greek numeral	Ε´
Roman numeral	V, v
Greek prefix	penta-/pent-
Latin prefix	quinque-/quinqu-/quint-
Binary	1012
Ternary	123
Senary	56
Octal	58
Duodecimal	512
Hexadecimal	516
Greek	ε (or Ε)
Arabic, Kurdish	٥
Persian, Sindhi, Urdu	۵
Ge'ez	፭
Bengali	৫
Kannada	೫
Punjabi	੫
Chinese numeral	五
Armenian	Ե
Devanāgarī	५
Hebrew	ה
Khmer	៥
Telugu	౫
Malayalam	൫
Tamil	௫
Thai	๕
Babylonian numeral	𒐙
Egyptian hieroglyph, Chinese counting rod	|||||
Maya numerals	𝋥
Morse code	.....

5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number.

Humans, and many other animals, have 5 digits on their limbs.

Mathematics

The first Pythagorean triple

Five is the second Fermat prime, the third Mersenne prime exponent, as well as a Fibonacci number. 5 is the first congruent number, as well as the length of the hypotenuse of the smallest integer-sided right triangle, making part of the smallest Pythagorean triple (3, 4, 5).

Geometry

A shape with five sides is called a pentagon. The pentagon is the first regular polygon that does not tile the plane with copies of itself. It is the largest face any of the five regular three-dimensional regular Platonic solid can have.

A conic is determined using five points in the same way that two points are needed to determine a line.A pentagram, or five-pointed polygram, is a star polygon constructed by connecting some non-adjacent of a regular pentagon as self-intersecting edges.

5 is the first safe prime where $(p-1)/2$ for a prime $p$ is also prime (2), and the first good prime, since it is the first prime number whose square (25) is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes (i.e., 3 × 7 = 21 and 11 × 2 = 22 are less than 25). 11, the fifth prime number, is the next good prime, that also forms the first pair of sexy primes with 5. 5 is the second Fermat prime of the form $2^{2^{n}}+1$, of a total of five known Fermat primes.

The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol {5/2}) appears prominently in Penrose tilings, and they are facets inside Kepler–Poinsot star polyhedra and Schläfli–Hess star polychora. A similar figure to the pentagram is a five-pointed simple isotoxal star ☆ without self-intersecting edges, often found inside Islamic Girih tiles (there are five different rudimentary types).

Algebra

5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. 5 is also the first of three known Wilson primes (5, 13, 563), where the square of a prime $p^{2}$ divides $(p-1)!+1.$ As a consequence of Fermat's little theorem and Euler's criterion, all squares are congruent to 0, 1, 4 (or −1) modulo 5. All integers $n\geq 34$ can be expressed as the sum of five non-zero squares. There are five countably infinite Ramsey classes of permutations.

Five is conjectured to be the only odd, untouchable number; if this is the case, then five will be the only odd prime number that is not the base of an aliquot tree.

Every odd number greater than five is conjectured to be expressible as the sum of three prime numbers; Helfgott has provided a proof of this (also known as the odd Goldbach conjecture) that is already widely acknowledged by mathematicians as it still undergoes peer-review. On the other hand, every odd number greater than one is the sum of at most five prime numbers (as a lower limit).

Unsolved problem in mathematics:

Is 5 the only odd, untouchable number?

(more unsolved problems in mathematics)

The smallest non-trivial magic square

Graph theory, and planar geometry

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,. By Kuratowski's theorem, a finite graph is planar if and only if it does not contain a subgraph that is a subdivision of K₅, or K_3,3, the utility graph.

The chromatic number of the plane is the minimum number of colors required to color the plane such that no pair of points at a distance of 1 has the same color. Five is a lower depending for the chromatic number of the plane, but this may depend on the choice of set-theoretical axioms:

The plane contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations: hexagonal, oblique, rectangular, centered rectangular, and square lattices. Uniform tilings of the plane, furthermore, are generated from combinations of only five regular polygons: the triangle, square, hexagon, octagon, and the dodecagon.

Polyhedral geometry

Illustration by Leonardo da Vinci of a regular dodecahedron, from Luca Pacioli's Divina proportione

There are five Platonic solids in three-dimensional space that are regular: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The dodecahedron in particular contains pentagonal faces, while the icosahedron, its dual polyhedron, has a vertex figure that is a regular pentagon. These five regular solids are responsible for generating thirteen figures that classify as semi-regular, which are called the Archimedean solids. There are also five:

• Regular polyhedron compounds: the compound of five tetrahedra, compound of ten tetrahedra, compound of five cubes, compound of five octahedra, and stella octangula. Icosahedral symmetry $\mathrm {I} _{h}$ is isomorphic to the alternating group on five letters $\mathrm {A} _{5}$ of order 120, realized by actions on these uniform polyhedron compounds (aside from the regular compound of two tetrahedra). All fifteen mirror planes of $\mathrm {I} _{h}$ pass through the edges of a regular spherical compound of five octahedra, whose sets of three orthogonal great circles use five colors.
• Space-filling convex polyhedra with regular faces: the triangular prism, hexagonal prism, cube, truncated octahedron, and gyrobifastigium. The cube is the only Platonic solid that can tessellate space on its own, and the truncated octahedron and gyrobifastigium are the only Archimedean and Johnson solids, respectively, that can tessellate space with their own copies.
• Cell-transitive parallelohedra: any parallelepiped, as well as the rhombic dodecahedron, the elongated dodecahedron, the hexagonal prism and the truncated octahedron. The cube is a special case of a parallelepiped, and the rhombic dodecahedron (with five stellations per Miller's rules) is the only Catalan solid to tessellate space on its own.
• Regular abstract polyhedra, which include the excavated dodecahedron and the dodecadodecahedron. They have combinatorial symmetries transitive on flags of their elements, with topologies equivalent to that of toroids and the ability to tile the hyperbolic plane.

Moreover, there are also precisely five uniform prisms and antiprisms that contain pentagons or pentagrams as faces — the pentagonal prism and antiprism, and the pentagrammic prism, antiprism, and crossed-antiprism.

Four-dimensional space

The four-dimensional 5-cell is the simplest regular polychoron.

The pentatope, or 5-cell, is the self-dual fourth-dimensional analogue of the tetrahedron, with Coxeter group symmetry $\mathrm {A} _{4}$ of order 120 = 5! and $\mathrm {S} _{5}$ group structure. Made of five tetrahedra, its Petrie polygon is a regular pentagon and its orthographic projection is equivalent to the complete graph K₅. It is one of six regular 4-polytopes, made of thirty-one elements: five vertices, ten edges, ten faces, five tetrahedral cells and one 4-face.

• A regular 120-cell, the dual polychoron to the regular 600-cell, can fit one hundred and twenty 5-cells. Also, five 24-cells fit inside a small stellated 120-cell, the first stellation of the 120-cell.
  A subset of the vertices of the small stellated 120-cell are matched by the great duoantiprism star, which is the only uniform nonconvex duoantiprismatic solution in the fourth dimension, constructed from the polytope cartesian product $\{5\}\otimes \{5/3\}$ and made of fifty tetrahedra, ten pentagrammic crossed antiprisms, ten pentagonal antiprisms, and fifty vertices.
• The abstract four-dimensional 57-cell is made of fifty-seven hemi-icosahedral cells, in-which five surround each edge. The 11-cell, another abstract 4-polytope with eleven vertices and fifty-five edges, is made of eleven hemi-dodecahedral cells each with fifteen edges. The skeleton of the hemi-dodecahedron is the Petersen graph.

Overall, the fourth dimension contains five fundamental Weyl groups that form a finite number of uniform polychora based on only twenty-five uniform polyhedra: $\mathrm {A} _{4}$, $\mathrm {B} _{4}$, $\mathrm {D} _{4}$, $\mathrm {F} _{4}$, and $\mathrm {H} _{4}$, accompanied by a fifth or sixth general group of unique 4-prisms of Platonic and Archimedean solids. There are also a total of five Coxeter groups that generate non-prismatic Euclidean honeycombs in 4-space, alongside five compact hyperbolic Coxeter groups that generate five regular compact hyperbolic honeycombs with finite facets, as with the order-5 5-cell honeycomb and the order-5 120-cell honeycomb, both of which have five cells around each face. Compact hyperbolic honeycombs only exist through the fourth dimension, or rank 5, with paracompact hyperbolic solutions existing through rank 10. Likewise, analogues of four-dimensional $\mathrm {H} _{4}$ hexadecachoric or $\mathrm {F} _{4}$ icositetrachoric symmetry do not exist in dimensions $n$ ⩾ $5$; however, there are prismatic groups in the fifth dimension which contains prisms of regular and uniform 4-polytopes that have $\mathrm {H} _{4}$ and $\mathrm {F} _{4}$ symmetry. There are also five regular projective 4-polytopes in the fourth dimension, all of which are hemi-polytopes of the regular 4-polytopes, with the exception of the 5-cell. Only two regular projective polytopes exist in each higher dimensional space.

Generally, star polytopes that are regular only exist in dimensions $2$ ⩽ $n$ < $5$, and can be constructed using five Miller rules for stellating polyhedra or higher-dimensional polytopes.

Five-dimensional space

The 5-simplex or hexateron is the five-dimensional analogue of the 5-cell, or 4-simplex. It has Coxeter group $\mathrm {A} _{5}$ as its symmetry group, of order 720 = 6!, whose group structure is represented by the symmetric group $\mathrm {S} _{6}$, the only finite symmetric group which has an outer automorphism. The 5-cube, made of ten tesseracts and the 5-cell as its vertex figure, is also regular and one of thirty-one uniform 5-polytopes under the Coxeter $\mathrm {B} _{5}$ hypercubic group. The demipenteract, with one hundred and twenty cells, is the only fifth-dimensional semi-regular polytope, and has the rectified 5-cell as its vertex figure, which is one of only three semi-regular 4-polytopes alongside the rectified 600-cell and the snub 24-cell. In the fifth dimension, there are five regular paracompact honeycombs, all with infinite facets and vertex figures; no other regular paracompact honeycombs exist in higher dimensions. There are also exclusively twelve complex aperiotopes in $\mathbb {C} ^{n}$ complex spaces of dimensions $n$ ⩾ $5$; alongside complex polytopes in $\mathbb {C} ^{5}$ and higher under simplex, hypercubic and orthoplex groups (with van Oss polytopes).

Veronese surface

A Veronese surface in the projective plane $\mathbb {P} ^{5}$ generalizes a linear condition $\nu :\mathbb {P} ^{2}\to \mathbb {P} ^{5}$ for a point to be contained inside a conic, where five points determine a conic.

In finite simple groups

Lie groups

There are five complex exceptional Lie algebras: ${\mathfrak {g}}_{2}$, ${\mathfrak {f}}_{4}$, ${\mathfrak {e}}_{6}$, ${\mathfrak {e}}_{7}$, and ${\mathfrak {e}}_{8}$. The smallest of these, ${\mathfrak {g}}_{2}$ of real dimension 28, can be represented in five-dimensional complex space and projected as a ball rolling on top of another ball, whose motion is described in two-dimensional space. ${\mathfrak {e}}_{8}$ is the largest, and holds the other four Lie algebras as subgroups, with a representation over $\mathbb {R}$ in dimension 496. It contains an associated lattice that is constructed with one hundred and twenty quaternionic unit icosians that make up the vertices of the 600-cell, whose Euclidean norms define a quadratic form on a lattice structure isomorphic to the optimal configuration of spheres in eight dimensions. This sphere packing $\mathrm {E} _{8}$ lattice structure in 8-space is held by the vertex arrangement of the 5₂₁ honeycomb, one of five Euclidean honeycombs that admit Gosset's original definition of a semi-regular honeycomb, which includes the three-dimensional alternated cubic honeycomb. The smallest simple isomorphism found inside finite simple Lie groups is $\mathrm {A_{5}} \cong A_{1}(4)\cong A_{1}(5)$, where here $\mathrm {A_{n}}$ represents alternating groups and $A_{n}(q)$ classical Chevalley groups. In particular, the smallest non-solvable group is the alternating group on five letters, which is also the smallest simple non-abelian group.

Sporadic groups

This diagram shows the subquotient relations of the twenty-six sporadic groups; the five Mathieu groups form the simplest class (colored red ).

Mathieu groups

The five Mathieu groups constitute the first generation in the happy family of sporadic groups. These are also the first five sporadic groups to have been described, defined as $\mathrm {M} _{n}$ multiply transitive permutation groups on $n$ objects, with $n$ ∈ {11, 12, 22, 23, 24}. In particular, $\mathrm {M} _{11}$, the smallest of all sporadic groups, has a rank 3 action on fifty-five points from an induced action on unordered pairs, as well as two five-dimensional faithful complex irreducible representations over the field with three elements, which is the lowest irreducible dimensional representation of all sporadic group over their respective fields with $n$ elements. Of precisely five different conjugacy classes of maximal subgroups of $\mathrm {M} _{11}$, one is the almost simple symmetric group $\mathrm {S} _{5}$ (of order 5!), and another is $\mathrm {M} _{10}$, also almost simple, that functions as a point stabilizer which contains five as its largest prime factor in its group order: 2⁴·3²·5 = 2·3·4·5·6 = 8·9·10 = 720. On the other hand, whereas $\mathrm {M} _{11}$ is sharply 4-transitive, $\mathrm {M} _{12}$ is sharply 5-transitive and $\mathrm {M} _{24}$ is 5-transitive, and as such they are the only two 5-transitive groups that are not symmetric groups or alternating groups. $\mathrm {M} _{22}$ has the first five prime numbers as its distinct prime factors in its order of 2⁷·3²·5·7·11; all Mathieu groups are subgroups of $\mathrm {M} _{24}$, which under the Witt design $\mathrm {W} _{24}$ of Steiner system $\operatorname {S(5,8,24)}$ emerges a construction of the extended binary Golay code $\mathrm {B} _{24}$ that has $\mathrm {M} _{24}$ as its automorphism group. $\mathrm {W} _{24}$ generates octads from code words of Hamming weight 8 from the extended binary Golay code, one of five different Hamming weights the extended binary Golay code uses: 0, 8, 12, 16, and 24. The Witt design and the extended binary Golay code in turn can be used to generate a faithful construction of the 24-dimensional Leech lattice Λ₂₄, which is primarily constructed using the Weyl vector $(0,1,2,3,\dots ,24;70)$ that admits the only non-unitary solution to the cannonball problem, where the sum of the squares of the first twenty-four integers is equivalent to the square of another integer, the fifth pentatope number (70). The subquotients of the automorphism of the Leech lattice, Conway group $\mathrm {Co} _{0}$, is in turn the subject of the second generation of seven sporadic groups.

Harada-Norton group

A centralizer of an element of order 5 inside the largest sporadic group $\mathrm {F_{1}}$ arises from the product between Harada–Norton sporadic group $\mathrm {HN}$ and a group of order 5. On its own, $\mathrm {HN}$ can be represented using standard generators $(a,b,ab)$ that further dictate a condition where $o([a,b])=5$. This condition is also held by other generators that belong to the Tits group $\mathrm {T}$, the only finite simple group that is a non-strict group of Lie type that can also classify as sporadic. Furthermore, over the field with five elements, $\mathrm {HN}$ holds a 133-dimensional representation where 5 acts on a commutative yet non-associative product as a 5-modular analogue of the Griess algebra $V_{2}$^♮, which holds $\mathrm {F_{1}}$ as its automorphism group.

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
5 × x	5	10	15	20	25	30	35	40	45	50		55	60	65	70	75	80	85	90	95	100

Division	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
5 ÷ x	5	2.5	1.6	1.25	1	0.83	0.714285	0.625	0.5	0.5		0.45	0.416	0.384615	0.3571428	0.3
x ÷ 5	0.2	0.4	0.6	0.8		1.2	1.4	1.6	1.8	2		2.2	2.4	2.6	2.8	3

Exponentiation	1	2	3	4	5	6	7	8	9	10		11	12	13	14	15
5x	5	25	125	625	3125	15625	78125	390625	1953125	9765625		48828125	244140625	1220703125	6103515625	30517578125
x5	1	32	243	1024		7776	16807	32768	59049	100000		161051	248832	371293	537824	759375

Decimal properties

All multiples of 5 will end in either 5 or 0, and vulgar fractions with 5 or 2 in the denominator do not yield infinite decimal expansions because they are prime factors of 10, the base.

In the powers of 5, every power ends with the number five, and from 5³ onward, if the exponent is odd, then the hundreds digit is 1, and if it is even, the hundreds digit is 6.

A number $n$ raised to the fifth power always ends in the same digit as $n$.

Evolution of the Arabic digit

The evolution of the modern Western digit for the numeral for five is traced back to the Indian system of numerals, where on some earlier versions, the numeral bore resemblance to variations of the number four, rather than "5" (as it is represented today). The Kushana and Gupta empires in what is now India had among themselves several forms that bear no resemblance to the modern digit. Later on, Arabic traditions transformed the digit in several ways, producing forms that were still similar to the numeral for four, with similarities to the numeral for three; yet, still unlike the modern five. It was from those digits that Europeans finally came up with the modern 5 (represented in writings by Dürer, for example).

While the shape of the character for the digit 5 has an ascender in most modern typefaces, in typefaces with text figures the glyph usually has a descender, as, for example, in .

On the seven-segment display of a calculator and digital clock, it is represented by five segments at four successive turns from top to bottom, rotating counterclockwise first, then clockwise, and vice-versa. It is one of three numbers, along with 4 and 6, where the number of segments matches the number.

Other fields

Astronomy

There are five Lagrangian points in a two-body system.

Biology

There are usually considered to be five senses (in general terms); the five basic tastes are sweet, salty, sour, bitter, and umami. Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity. Five is the number of appendages on most starfish, which exhibit pentamerism.

Computing

5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.

Literature

Poetry

A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.

Music

Modern musical notation uses a musical staff made of five horizontal lines. A scale with five notes per octave is called a pentatonic scale. A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems. In harmonics, the fifth partial (or 4th overtone) of a fundamental has a frequency ratio of 5:1 to the frequency of that fundamental. This ratio corresponds to the interval of 2 octaves plus a pure major third. Thus, the interval of 5:4 is the interval of the pure third. A major triad chord when played in just intonation (most often the case in a cappella vocal ensemble singing), will contain such a pure major third.

Five is the lowest possible number that can be the top number of a time signature with an asymmetric meter.

Religion

Judaism

The Book of Numbers is one of five books in the Torah; the others being the books of Genesis, Exodus, Leviticus, and Deuteronomy. They are collectively called the Five Books of Moses, the Pentateuch (Greek for "five containers", referring to the scroll cases in which the books were kept), or Humash (חומש, Hebrew for "fifth"). The Khamsa, an ancient symbol shaped like a hand with four fingers and one thumb, is used as a protective amulet by Jews; that same symbol is also very popular in Arabic culture, known to protect from envy and the evil eye.

Christianity

There are traditionally five wounds of Jesus Christ in Christianity: the nail wounds in Christ's two hands, the nail wounds in Christ's two feet, and the Spear Wound of Christ (respectively at the four extremities of the body, and the head).

Islam

The Five Pillars of Islam.

Mysticism

Gnosticism

The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.

Alchemy

According to ancient Greek philosophers such as Aristotle, the universe is made up of five classical elements: water, earth, air, fire, and ether. This concept was later adopted by medieval alchemists and more recently by practitioners of Neo-Pagan religions such as Wicca. There are five elements in the universe according to Hindu cosmology: dharti, agni, jal, vayu evam akash (earth, fire, water, air and space, respectively). In East Asian tradition, there are five elements: water, fire, earth, wood, and metal. The Japanese names for the days of the week, Tuesday through Saturday, come from these elements via the identification of the elements with the five planets visible with the naked eye. Also, the traditional Japanese calendar has a five-day weekly cycle that can be still observed in printed mixed calendars combining Western, Chinese-Buddhist, and Japanese names for each weekday. There are also five elements in the traditional Chinese Wuxing.

Quintessence, meaning "fifth element", refers to the elusive fifth element that completes the basic four elements (water, fire, air, and earth), as a union of these. The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.

Miscellaneous fields

The fives of all four suits in playing cards

• "Give me five" is a common phrase used preceding a high five.
• The Olympic Games have five interlocked rings as their symbol, representing the number of inhabited continents represented by the Olympians (Europe, Asia, Africa, Australia and Oceania, and the Americas).
• The number of dots in a quincunx.

See also

5 (disambiguation)

References

Further reading

• Wells, D. (1987). The Penguin Dictionary of Curious and Interesting Numbers. London, UK: Penguin Group. pp. 58–67.

External links

• Prime curiosities: 5
• Media related to 5 (number) at Wikimedia Commons