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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−1 0 1 2 3 4 5 6 7 8 9 → List of numbers Integers ← 0 10 20 30 40 50 60 70 80 90 →
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

5 is a Fermat prime, a 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).^[1]

5 is the first safe prime,^[2] and the first good prime.^[3] 11 forms the first pair of sexy primes with 5.^[4] 5 is the second Fermat prime, of a total of five known Fermat primes.^[5] 5 is also the first of three known Wilson primes (5, 13, 563).^[6]

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.^[7] A pentagram, or five-pointed polygram, is a star polygon constructed by connecting some non-adjacent of a regular pentagon as self-intersecting edges.^[8] The internal geometry of the pentagon and pentagram (represented by its Schläfli symbol {5/2}) appears prominently in Penrose tilings. Pentagrams are facets inside Kepler–Poinsot star polyhedra and Schläfli–Hess star polychora.

There are five regular Platonic solids the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron.^[9]

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.^[10] Five is a lower depending for the chromatic number of the plane, but this may depend on the choice of set-theoretical axioms:^[11]

The plane contains a total of five Bravais lattices, or arrays of points defined by discrete translation operations. Uniform tilings of the plane, are generated from combinations of only five regular polygons.^[12]

Higher dimensional geometry

A hypertetrahedron, or 5-cell, is the 4 dimensional analogue of the tetrahedron. It has five vertices. Its orthographic projection is homomorphic to the group K_5.^{[13]: p.120}

There are five fundamental mirror symmetry point group families in 4-dimensions. There are also 5 compact hyperbolic Coxeter groups, or 4-prisms, of rank 5, each generating uniform honeycombs in hyperbolic 4-space as permutations of rings of the Coxeter diagrams.^[14]

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

Algebra

The smallest non-trivial magic square

5 is the value of the central cell of the first non-trivial normal magic square, called the Luoshu square. All integers ${\displaystyle n\geq 34}$ can be expressed as the sum of five non-zero squares.^[15][16] There are five countably infinite Ramsey classes of permutations.^[17]: p.4 5 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.^[18]

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

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^[19] (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).^[20]

Unsolved problem in mathematics:

Is 5 the only odd, untouchable number?

(more unsolved problems in mathematics)

Group theory

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 , or K_3,3, the utility graph.^[21]

There are five complex exceptional Lie algebras. 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.^[22]: p.54 A centralizer of an element of order 5 inside the largest sporadic group ${\displaystyle \mathrm {F_{1}} }$ arises from the product between Harada–Norton sporadic group ${\displaystyle \mathrm {HN} }$ and a group of order 5.^[23][24]

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
5^x	5	25	125	625	3125	15625	78125	390625	1953125	9765625	48828125	244140625	1220703125	6103515625	30517578125
x^5	1	32	243	1024		7776	16807	32768	59049	100000	161051	248832	371293	537824	759375

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.^[25] 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 often 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. This makes it often indistinguishable from the letter S. Higher segment displays may sometimes may make use of a diagonal for one of the two.

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.^[26] Almost all amphibians, reptiles, and mammals which have fingers or toes have five of them on each extremity.^[27] Five is the number of appendages on most starfish, which exhibit pentamerism.^[28]

Computing

5 is the ASCII code of the Enquiry character, which is abbreviated to ENQ.^[29]

Literature

Poetry

A pentameter is verse with five repeating feet per line; the iambic pentameter was the most prominent form used by William Shakespeare.^[30]

Music

Modern musical notation uses a musical staff made of five horizontal lines.^[31] A scale with five notes per octave is called a pentatonic scale.^[32] A perfect fifth is the most consonant harmony, and is the basis for most western tuning systems.^[33] 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").^[34] 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.^[35]

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).^[36]

Islam

The Five Pillars of Islam.^[37] The five-pointed simple star ☆ is one of the five used in Islamic Girih tiles.^[38]

Mysticism

Gnosticism

The number five was an important symbolic number in Manichaeism, with heavenly beings, concepts, and others often grouped in sets of five.^{[citation needed]}

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.^[39] 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.^[40] 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.^[41]

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.^[42] 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).^[43]
• The number of dots in a quincunx.^[44]

See also

• 5 (disambiguation)

Notes

1. Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
2. Sloane, N. J. A. (ed.). "Sequence A005385 (Safe primes p: (p-1)/2 is also prime)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-02-14.
3. Sloane, N. J. A. (ed.). "Sequence A028388 (Good primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
4. Sloane, N. J. A. (ed.). "Sequence A023201 (Primes p such that p + 6 is also prime. (Lesser of a pair of sexy primes.))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-01-14.
5. Sloane, N. J. A. (ed.). "Sequence A019434 (Fermat primes)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2022-07-21.
6. Sloane, N. J. A. (ed.). "Sequence A007540 (Wilson primes: primes p such that (p-1)! is congruent -1 (mod p^2).)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-06.
7. Dixon, A. C. (March 1908). "The Conic through Five Given Points". The Mathematical Gazette. 4 (70). The Mathematical Association: 228–230. doi:10.2307/3605147. JSTOR 3605147. S2CID 125356690.
8. Sloane, N. J. A. (ed.). "Sequence A307681 (Difference between the number of sides and the number of diagonals of a convex n-gon.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
9. Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 61
10. de Grey, Aubrey D.N.J. (2018). "The Chromatic Number of the Plane is At Least 5". Geombinatorics. 28: 5–18. arXiv:1804.02385. MR 3820926. S2CID 119273214.
11. Exoo, Geoffrey; Ismailescu, Dan (2020). "The Chromatic Number of the Plane is At Least 5: A New Proof". Discrete & Computational Geometry. 64. New York, NY: Springer: 216–226. arXiv:1805.00157. doi:10.1007/s00454-019-00058-1. MR 4110534. S2CID 119266055. Zbl 1445.05040.
12. Grünbaum, Branko; Shepard, Geoffrey (November 1977). "Tilings by Regular Polygons" (PDF). Mathematics Magazine. 50 (5). Taylor & Francis, Ltd.: 227–236. doi:10.2307/2689529. JSTOR 2689529. S2CID 123776612. Zbl 0385.51006.
13. H. S. M. Coxeter (1973). Regular Polytopes (3rd ed.). New York: Dover Publications, Inc. pp. 1–368. ISBN 978-0-486-61480-9.
14. McMullen, Peter; Schulte, Egon (2002). Abstract Regular Polytopes. Encyclopedia of Mathematics and its Applications. Vol. 92. Cambridge: Cambridge University Press. pp. 162–164. doi:10.1017/CBO9780511546686. ISBN 0-521-81496-0. MR 1965665. S2CID 115688843.
15. Niven, Ivan; Zuckerman, Herbert S.; Montgomery, Hugh L. (1980). An Introduction to the Theory of Numbers (5th ed.). New York, NY: John Wiley. pp. 144, 145. ISBN 978-0-19-853171-5.
16. Sloane, N. J. A. (ed.). "Sequence A047701 (All positive numbers that are not the sum of 5 nonzero squares.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-09-20.

    Only twelve integers up to 33 cannot be expressed as the sum of five non-zero squares: {1, 2, 3, 4, 6, 7, 9, 10, 12, 15, 18, 33} where 2, 3 and 7 are the only such primes without an expression.

17. Böttcher, Julia; Foniok, Jan (2013). "Ramsey Properties of Permutations". The Electronic Journal of Combinatorics. 20 (1): P2. arXiv:1103.5686v2. doi:10.37236/2978. S2CID 17184541. Zbl 1267.05284.
18. Pomerance, Carl; Yang, Hee-Sung (14 June 2012). "On Untouchable Numbers and Related Problems" (PDF). math.dartmouth.edu. Dartmouth College: 1. S2CID 30344483. 2010 Mathematics Subject Classification. 11A25, 11Y70, 11Y16.
19. Helfgott, Harald Andres (2014). "The ternary Goldbach problem" (PDF). In Jang, Sun Young (ed.). Seoul International Congress of Mathematicians Proceedings. Vol. 2. Seoul, KOR: Kyung Moon SA. pp. 391–418. ISBN 978-89-6105-805-6. OCLC 913564239.
20. Tao, Terence (March 2014). "Every odd number greater than 1 has a representation is the sum of at most five primes" (PDF). Mathematics of Computation. 83 (286): 997–1038. doi:10.1090/S0025-5718-2013-02733-0. MR 3143702. S2CID 2618958.
21. Burnstein, Michael (1978). "Kuratowski-Pontrjagin theorem on planar graphs". Journal of Combinatorial Theory. Series B. 24 (2): 228–232. doi:10.1016/0095-8956(78)90024-2.
22. Robert L. Griess, Jr. (1998). Twelve Sporadic Groups. Springer Monographs in Mathematics. Berlin: Springer-Verlag. pp. 1−169. doi:10.1007/978-3-662-03516-0. ISBN 978-3-540-62778-4. MR 1707296. S2CID 116914446. Zbl 0908.20007.
23. Lux, Klaus; Noeske, Felix; Ryba, Alexander J. E. (2008). "The 5-modular characters of the sporadic simple Harada–Norton group HN and its automorphism group HN.2". Journal of Algebra. 319 (1). Amsterdam: Elsevier: 320–335. doi:10.1016/j.jalgebra.2007.03.046. MR 2378074. S2CID 120706746. Zbl 1135.20007.
24. Wilson, Robert A. (2009). "The odd local subgroups of the Monster". Journal of Australian Mathematical Society (Series A). 44 (1). Cambridge: Cambridge University Press: 12–13. doi:10.1017/S1446788700031323. MR 0914399. S2CID 123184319. Zbl 0636.20014.
25. 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. 24.65
26. Marcus, Jacqueline B. (2013-04-15). Culinary Nutrition: The Science and Practice of Healthy Cooking. Academic Press. p. 55. ISBN 978-0-12-391883-3. There are five basic tastes: sweet, salty, sour, bitter and umami...
27. Kisia, S. M. (2010), Vertebrates: Structures and Functions, Biological Systems in Vertebrates, CRC Press, p. 106, ISBN 978-1-4398-4052-8, The typical limb of tetrapods is the pentadactyl limb (Gr. penta, five) that has five toes. Tetrapods evolved from an ancestor that had limbs with five toes. ... Even though the number of digits in different vertebrates may vary from five, vertebrates develop from an embryonic five-digit stage.
28. Cinalli, G.; Maixner, W. J.; Sainte-Rose, C. (2012-12-06). Pediatric Hydrocephalus. Springer Science & Business Media. p. 19. ISBN 978-88-470-2121-1. The five appendages of the starfish are thought to be homologous to five human buds
29. Pozrikidis, Constantine (2012-09-17). XML in Scientific Computing. CRC Press. p. 209. ISBN 978-1-4665-1228-3. 5 5 005 ENQ (enquiry)
30. Veith (Jr.), Gene Edward; Wilson, Douglas (2009). Omnibus IV: The Ancient World. Veritas Press. p. 52. ISBN 978-1-932168-86-0. The most common accentual-syllabic lines are five-foot iambic lines (iambic pentameter)
31. "STAVE | meaning in the Cambridge English Dictionary". dictionary.cambridge.org. Retrieved 2020-08-02. the five lines and four spaces between them on which musical notes are written
32. Ricker, Ramon (1999-11-27). Pentatonic Scales for Jazz Improvisation. Alfred Music. p. 2. ISBN 978-1-4574-9410-9. Pentatonic scales, as used in jazz, are five note scales
33. Danneley, John Feltham (1825). An Encyclopaedia, Or Dictionary of Music ...: With Upwards of Two Hundred Engraved Examples, the Whole Compiled from the Most Celebrated Foreign and English Authorities, Interspersed with Observations Critical and Explanatory. editor, and pub. are the perfect fourth, perfect fifth, and the octave
34. Pelaia, Ariela. "Judaism 101: What Are the Five Books of Moses?". Learn Religions. Retrieved 2020-08-03.
35. Zenner, Walter P. (1988-01-01). Persistence and Flexibility: Anthropological Perspectives on the American Jewish Experience. SUNY Press. p. 284. ISBN 978-0-88706-748-8.
36. "CATHOLIC ENCYCLOPEDIA: The Five Sacred Wounds". www.newadvent.org. Retrieved 2020-08-02.
37. "PBS – Islam: Empire of Faith – Faith – Five Pillars". www.pbs.org. Retrieved 2020-08-03.
38. Sarhangi, Reza (2012). "Interlocking Star Polygons in Persian Architecture: The Special Case of the Decagram in Mosaic Designs" (PDF). Nexus Network Journal. 14 (2): 350. doi:10.1007/s00004-012-0117-5. S2CID 124558613.
39. Yoon, Hong-key (2006). The Culture of Fengshui in Korea: An Exploration of East Asian Geomancy. Lexington Books. p. 59. ISBN 978-0-7391-1348-6. The first category is the Five Agents [Elements] namely, Water, Fire, Wood, Metal, and Earth.
40. Walsh, Len (2008-11-15). Read Japanese Today: The Easy Way to Learn 400 Practical Kanji. Tuttle Publishing. ISBN 978-1-4629-1592-7. The Japanese names of the days of the week are taken from the names of the seven basic nature symbols
41. Chen, Yuan (2014). "Legitimation Discourse and the Theory of the Five Elements in Imperial China". Journal of Song-Yuan Studies. 44 (1): 325–364. doi:10.1353/sys.2014.0000. ISSN 2154-6665. S2CID 147099574.
42. Kronland-Martinet, Richard; Ystad, Sølvi; Jensen, Kristoffer (2008-07-19). Computer Music Modeling and Retrieval. Sense of Sounds: 4th International Symposium, CMMR 2007, Copenhagen, Denmark, August 2007, Revised Papers. Springer. p. 502. ISBN 978-3-540-85035-9. Plato and Aristotle postulated a fifth state of matter, which they called "idea" or quintessence" (from "quint" which means "fifth")
43. "Olympic Rings – Symbol of the Olympic Movement". International Olympic Committee. 2020-06-23. Retrieved 2020-08-02.
44. Laplante, Philip A. (2018-10-03). Comprehensive Dictionary of Electrical Engineering. CRC Press. p. 562. ISBN 978-1-4200-3780-7. quincunx five points

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