5

Last updated
← 4 5 6 →
−1 0 1 2 3 4 5 6 7 8 9 →
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 .....
ASCII value ENQ

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.

Contents

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

Mathematics

The first Pythagorean triple Pythagoras' Special Triples.svg
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 K5. [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. Schlegel wireframe 5-cell.png
The four-dimensional 5-cell is the simplest regular polychoron.

Algebra

The smallest non-trivial magic square Magic Square Lo Shu.svg
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 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 ). SporadicGroups.png
This diagram shows the subquotient relations of the twenty-six sporadic groups; the five Mathieu groups form the simplest class (colored red EllipseSubqR.svg ).

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 K3,3, 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 arises from the product between Harada–Norton sporadic group and a group of order 5. [23] [24]

List of basic calculations

Multiplication 1234567891011121314151617181920
5 × x5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100
Division 123456789101112131415
5 ÷ x52.51.61.25 1 0.830.7142850.6250.50.50.450.4160.3846150.35714280.3
x ÷ 50.20.40.60.81.21.41.61.8 2 2.22.42.62.8 3
Exponentiation 123456789101112131415
5x5 25 125 625 3125156257812539062519531259765625488281252441406251220703125610351562530517578125
x5 1 32 243 1024 7776168073276859049100000161051248832371293537824759375

Evolution of the Arabic digit

Evolution5glyph.png

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 Text figures 256.svg .

Seven-segment 5.svg

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

Religion

Islam

The Five Pillars of Islam. [26] The five-pointed simple star ☆ is one of the five used in Islamic Girih tiles. [27]

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 ]

The pentagram, or five-pointed star, bears mystic significance in various belief systems including Baháʼí, Christianity, Freemasonry, Satanism, Taoism, Thelema, and Wicca.

Miscellaneous

The fives of all four suits in playing cards 5 playing cards.jpg
The fives of all four suits in playing cards

See also

Related Research Articles

7 (seven) is the natural number following 6 and preceding 8. It is the only prime number preceding a cube.

6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.

15 (fifteen) is the natural number following 14 and preceding 16.

17 (seventeen) is the natural number following 16 and preceding 18. It is a prime number.

19 (nineteen) is the natural number following 18 and preceding 20. It is a prime number.

33 (thirty-three) is the natural number following 32 and preceding 34.

23 (twenty-three) is the natural number following 22 and preceding 24.

25 (twenty-five) is the natural number following 24 and preceding 26.

84 (eighty-four) is the natural number following 83 and preceding 85. It is seven dozens.

73 (seventy-three) is the natural number following 72 and preceding 74. In English, it is the smallest natural number with twelve letters in its spelled out name.

32 (thirty-two) is the natural number following 31 and preceding 33.

31 (thirty-one) is the natural number following 30 and preceding 32. It is a prime number.

54 (fifty-four) is the natural number and positive integer following 53 and preceding 55. As a multiple of 2 but not of 4, 54 is an oddly even number and a composite number.

61 (sixty-one) is the natural number following 60 and preceding 62.

63 (sixty-three) is the natural number following 62 and preceding 64.

104 is the natural number following 103 and preceding 105.

1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.

700 is the natural number following 699 and preceding 701.

1728 is the natural number following 1727 and preceding 1729. It is a dozen gross, or one great gross. It is also the number of cubic inches in a cubic foot.

14 (fourteen) is the natural number following 13 and preceding 15.

References

  1. ↑ Sloane, N. J. A. (ed.). "SequenceA003273(Congruent numbers)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01.
  2. ↑ Sloane, N. J. A. (ed.). "SequenceA005385(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.). "SequenceA028388(Good primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2016-06-01.
  4. ↑ Sloane, N. J. A. (ed.). "SequenceA023201(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.). "SequenceA019434(Fermat primes)". The On-Line Encyclopedia of Integer Sequences . OEIS Foundation. Retrieved 2022-07-21.
  6. ↑ Sloane, N. J. A. (ed.). "SequenceA007540(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.). "SequenceA307681(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.). "SequenceA047701(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. ↑ "PBS – Islam: Empire of Faith – Faith – Five Pillars". www.pbs.org. Retrieved 2020-08-03.
  27. ↑ 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.

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