6

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🎲 the perfect number

Perfect number Cuisenaire rods 6.png

This article is about the number. For the years, see 6 BC and AD 6. For other uses, see 6 (disambiguation) and Number Six (disambiguation).
← 5 6 7 →
−1 0 1 2 3 4 5 6 7 8 9 →
Cardinalsix
Ordinal6th
(sixth)
Numeral systemsenary
Factorization2 × 3
Divisors1, 2, 3, 6
Greek numeralϚ´
Roman numeralVI, vi, ↅ
Greek prefixhexa-/hex-
Latin prefixsexa-/sex-
Binary1102
Ternary203
Senary106
Octal68
Duodecimal612
Hexadecimal616
Greekστ (or ΣΤ or ς)
Arabic, Kurdish, Sindhi, Urdu٦
Persian۶
Amharic፮
Bengali৬
Chinese numeral六，陸
Devanāgarī६
Gujarati૬
Hebrewו
Khmer៦
Thai๖
Telugu౬
Tamil௬
Saraiki٦
Malayalam൬
ArmenianԶ
Babylonian numeral𒐚
Egyptian hieroglyph𓏿
Morse code_ ....

6 (six) is the natural number following 5 and preceding 7. It is a composite number and the smallest perfect number.[1]

In mathematics

A six-sided polygon is a hexagon,[1] one of the three regular polygons capable of tiling the plane. A hexagon also has 6 edges as well as 6 internal and external angles.

6 is the second smallest composite number.[1] It is also the first number that is the sum of its proper divisors, making it the smallest perfect number.[2] 6 is the first unitary perfect number, since it is the sum of its positive proper unitary divisors, without including itself. Only five such numbers are known to exist.[3] 6 is the largest of the four all-Harshad numbers.[4]

6 is a pronic number,[5] a congruent number,[6] a harmonic divisor number, and a semiprime.[7] 6 is also the first Granville number, or 

 
   
     
       
         
           S
         
       
     
   
   {\displaystyle {\mathcal {S}}}

-perfect number. A Golomb ruler of length 6 is a "perfect ruler".[8] 6 is a harmonic divisor number.[9]

The six exponentials theorem guarantees that under certain conditions one of a set of six exponentials is transcendental.[10] The smallest non-abelian group is the symmetric group 

 
   
     
       
         
           S
           
             3
           
         
       
     
   
   {\displaystyle \mathrm {S_{3}} }

which has 3! = 6 elements.[1] 6 the answer to the two-dimensional kissing number problem.[11]

A regular cube, with six faces

A cube has 6 faces. A tetrahedron has 6 edges. In four dimensions, there are a total of six convex regular polytopes.

In the classification of finite simple groups, twenty of twenty-six sporadic groups in the happy family are part of three families of groups which divide the order of the friendly giant, the largest sporadic group: five first generation Mathieu groups, seven second generation subquotients of the Leech lattice, and eight third generation subgroups of the friendly giant. The remaining six sporadic groups do not divide the order of the friendly giant, which are termed the pariahs (Ly, O'N, Ru, J4, J3, and J1).[12]

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 25 50 100 1000
6 × x 6 12 18 24 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 150 300 600 6000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
6 ÷ x 6 3 2 1.5 1.2 1 0.857142 0.75 0.6 0.6 0.54 0.5 0.461538 0.428571 0.4
x ÷ 6 0.16 0.3 0.5 0.6 0.83 1 1.16 1.3 1.5 1.6 1.83 2 2.16 2.3 2.5
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
6x 6 36 216 1296 7776 46656 279936 1679616 10077696 60466176 362797056 2176782336 13060694016
x6 1 64 729 4096 15625 46656 117649 262144 531441 1000000 1771561 2985984 4826809

Greek and Latin word parts

Hexa

Hexa is classical Greek for "six".[1] Thus:

  • "Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16[13]
  • A hexagon is a regular polygon with six sides[14]
  • A hexahedron is a polyhedron with six faces, with a cube being a special case[15]
  • Hexameter is a poetic form consisting of six feet per line
  • A "hex nut" is a nut with six sides, and a hex bolt has a six-sided head
  • The prefix "hexa-" also occurs in the systematic name of many chemical compounds, such as hexane which has 6 carbon atoms (C6H14).

The prefix sex-

Sex- is a Latin prefix meaning "six".[1] Thus:

  • Senary is the ordinal adjective meaning "sixth"[16]
  • People with sexdactyly have six fingers on each hand
  • The measuring instrument called a sextant got its name because its shape forms one-sixth of a whole circle
  • A group of six musicians is called a sextet
  • Six babies delivered in one birth are sextuplets
  • Sexy prime pairs – Prime pairs differing by six are sexy, because sex is the Latin word for six.[17][18]

The SI prefix for 10006 is exa- (E), and for its reciprocal atto- (a).

Evolution of the Hindu-Arabic digit

The first appearance of 6 is in the Edicts of Ashoka c. 250 BCE. These are Brahmi numerals, ancestors of Hindu-Arabic numerals.
The first known digit "6" in the number "256" in Ashoka's Minor Rock Edict No.1 in Sasaram, c. 250 BCE

The evolution of our modern digit 6 appears rather simple when compared with the other digits. The modern 6 can be traced back to the Brahmi numerals of India, which are first known from the Edicts of Ashoka c. 250 BCE.[19][20][21][22] It was written in one stroke like a cursive lowercase e rotated 90 degrees clockwise. Gradually, the upper part of the stroke (above the central squiggle) became more curved, while the lower part of the stroke (below the central squiggle) became straighter. The Arabs dropped the part of the stroke below the squiggle. From there, the European evolution to our modern 6 was very straightforward, aside from a flirtation with a glyph that looked more like an uppercase G.[23]

On the seven-segment displays of calculators and watches, 6 is usually written with six segments. Some historical calculator models use just five segments for the 6, by omitting the top horizontal bar. This glyph variant has not caught on; for calculators that can display results in hexadecimal, a 6 that looks like a "b" is not practical.

Just as in most modern typefaces, in typefaces with text figures the character for the digit 6 usually has an ascender, as, for example, in .[24]

This digit resembles an inverted 9. To disambiguate the two on objects and documents that can be inverted, the 6 has often been underlined, both in handwriting and on printed labels.

In music

A standard guitar has six strings.

In artists

In instruments

  • A standard guitar has six strings[31]
  • Most woodwind instruments have six basic holes or keys (e.g., bassoon, clarinet, pennywhistle, saxophone); these holes or keys are usually not given numbers or letters in the fingering charts

In music theory

  • There are six whole tones in an octave.[32]
  • There are six semitones in a tritone.[33]

In works

In religion

See also: 666

Judaism

Islam

Indeed, We created the heavens and the earth and everything in between in six Days,1 and We were not ˹even˺ touched with fatigue.2

— Surah Qaf:38

Note 1: The word day is not always used in the Quran to mean a 24-hour period. According to Surah Al-Hajj (The Pilgrimage):47, a heavenly Day is 1000 years of our time. The Day of Judgment will be 50,000 years of our time - Surah Al-Maarij (The Ascending Stairways):4. Hence, the six Days of creation refer to six eons of time, known only by Allah.

Note 2: Some Islamic scholars believe this verse comes in response to Exodus 31:17, which says, "The Lord made the heavens and the earth in six days, but on the seventh day He rested and was refreshed."

Others

In science

Astronomy

Biology

The cells of a beehive are six-sided.

Chemistry

A molecule of benzene has a ring of six carbon and six hydrogen atoms.
A molecule of benzene has a ring of six carbon and six hydrogen atoms.

Medicine

  • There are six tastes in traditional Indian medicine (Ayurveda): sweet, sour, salty, bitter, pungent, and astringent. These tastes are used to suggest a diet based on the symptoms of the body.[57]
  • Phase 6 is one of six pandemic influenza phases.[58]

Physics

In the Standard Model of particle physics, there are six types of quarks and six types of leptons.

In technology

6 as a resin identification code, used in recycling.
6 as a resin identification code, used in recycling.

In calendars

Anthropology

See also

References

  1. ^ a b c d e f Weisstein, Eric W. "6". mathworld.wolfram.com. Retrieved 2020-08-03.
  2. ^ Higgins, Peter (2008). Number Story: From Counting to Cryptography. New York: Copernicus. p. 11. ISBN 978-1-84800-000-1.
  3. ^ Sloane, N. J. A. (ed.). "Sequence A002827 (Unitary perfect numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  4. ^ Weisstein, Eric W. "Harshad Number". mathworld.wolfram.com. Retrieved 2020-08-03.
  5. ^ "Sloane's A002378: Pronic numbers". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2020-11-30.
  6. ^ Sloane, N. J. A. (ed.). "Sequence A003273 (Congruent numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  7. ^ Sloane, N. J. A. (ed.). "Sequence A001358 (Semiprimes (or biprimes): products of two primes.)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2023-08-03.
  8. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 72
  9. ^ Sloane, N. J. A. (ed.). "Sequence A001599 (Harmonic or Ore numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation. Retrieved 2016-06-01.
  10. ^ Weisstein, Eric W. "Six Exponentials Theorem". mathworld.wolfram.com. Retrieved 2020-08-03.
  11. ^ Weisstein, Eric W. "Kissing Number". mathworld.wolfram.com. Retrieved 2020-08-03.
  12. ^ Griess, Jr., Robert L. (1982). "The Friendly Giant" (PDF). Inventiones Mathematicae. 69: 91–96. Bibcode:1982InMat..69....1G. doi:10.1007/BF01389186. hdl:2027.42/46608. MR 0671653. S2CID 123597150. Zbl 0498.20013.
  13. ^ Weisstein, Eric W. "Hexadecimal". mathworld.wolfram.com. Retrieved 2020-08-03.
  14. ^ Weisstein, Eric W. "Hexagon". mathworld.wolfram.com. Retrieved 2020-08-03.
  15. ^ Weisstein, Eric W. "Hexahedron". mathworld.wolfram.com. Retrieved 2020-08-03.
  16. ^ Weisstein, Eric W. "Base". mathworld.wolfram.com. Retrieved 2020-08-03.
  17. ^ Chris K. Caldwell; G. L. Honaker Jr. (2009). Prime Curios!: The Dictionary of Prime Number Trivia. CreateSpace Independent Publishing Platform. p. 11. ISBN 978-1-4486-5170-2.
  18. ^ Weisstein, Eric W. "Sexy Primes". mathworld.wolfram.com. Retrieved 2020-08-03.
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  21. ^ Katz, Victor J.; Parshall, Karen Hunger (2014). Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century. Princeton University Press. p. 105. ISBN 978-1-4008-5052-5.
  22. ^ Pillis, John de (2002). 777 Mathematical Conversation Starters. MAA. p. 286. ISBN 978-0-88385-540-9.
  23. ^ 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): 395, Fig. 24.66
  24. ^ Negru, John (1988). Computer Typesetting. Van Nostrand Reinhold. p. 59. ISBN 978-0-442-26696-7. slight ascenders that rise above the cap height ( in 4 and 6 )
  25. ^ Auric, Georges; Durey, Louis; Honegger, Arthur; Milhaud, Darius; Poulenc, Francis; Tailleferre, Germaine (2014-08-20). Caramel Mou and Other Great Piano Works of "Les Six": Pieces by Auric, Durey, Honegger, Milhaud, Poulenc and Tailleferre (in French). Courier Corporation. ISBN 978-0-486-49340-4.
  26. ^ "Six Organs of Admittance". www.sixorgans.com. Retrieved 2020-08-03.
  27. ^ "Electric Six | Biography, Albums, Streaming Links". AllMusic. Retrieved 2020-08-03.
  28. ^ "Sixpence None The Richer". GRAMMY.com. 2020-05-19. Retrieved 2020-08-04.
  29. ^ "Slant 6 | Biography & History". AllMusic. Retrieved 2020-08-04.
  30. ^ "You Me at Six | Biography & History". AllMusic. Retrieved 2020-08-04.
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  32. ^ D'Amante, Elvo (1994-01-01). Music Fundamentals: Pitch Structures and Rhythmic Design. Scarecrow Press. p. 194. ISBN 978-1-4616-6985-2. The division of an octave into six equal parts is referred to as the whole-tone scale
  33. ^ Horsley, Charles Edward (1876). A Text Book of Harmony: For the Use of Schools and Students. Sampson Low, Marston, Searle, & Rivington. p. 4. Like the Tritone, it contains six semitones
  34. ^ Tribble, Mimi (2004). 300 Ways to Make the Best Christmas Ever!: Decorations, Carols, Crafts & Recipes for Every Kind of Christmas Tradition. Sterling Publishing Company, Inc. p. 145. ISBN 978-1-4027-1685-0. Six geese a-laying
  35. ^ Staines, Joe (2010-05-17). The Rough Guide to Classical Music. Penguin. p. 393. ISBN 978-1-4053-8321-9. ...the six arias with variations collected under the title Hexachordum Apollinis (1699)...
  36. ^ Hegarty, Paul; Halliwell, Martin (2011-06-23). Beyond and Before: Progressive Rock since the 1960s. Bloomsbury Publishing USA. p. 169. ISBN 978-1-4411-1480-8. Six Degrees of Inner Turbulence
  37. ^ Curran, Angela (2015-10-05). Routledge Philosophy Guidebook to Aristotle and the Poetics. Routledge. p. 133. ISBN 978-1-317-67706-2. THE SIX QUALITATIVE ELEMENTS OF TRAGEDY
  38. ^ Plaut, W. Gunther (1991). The Magen David: How the Six-pointed Star Became an Emblem for the Jewish People. B'nai B'rith Books. ISBN 978-0-910250-16-0. How the Six-pointed Star Became an Emblem for the Jewish People
  39. ^ Lauterbach, Jacob Zallel (1916). Midrash and Mishnah: A Study in the Early History of the Halakah. Bloch. p. 9. Six orders of Mishnah
  40. ^ Rosen, Ceil; Rosen, Moishe (2006-05-01). Christ in the Passover. Moody Publishers. p. 79. ISBN 978-1-57567-480-3. Six symbolic foods
  41. ^ Repcheck, Jack (2008-12-15). The Man Who Found Time: James Hutton And The Discovery Of Earth's Antiquity. Basic Books. ISBN 978-0-7867-4399-5. it actually took only six days to create the earth
  42. ^ "CHURCH FATHERS: City of God, Book XI (St. Augustine)". www.newadvent.org. Retrieved 2020-08-04. These works are recorded to have been completed in six days (the same day being six times repeated), because six is a perfect number
  43. ^ Grossman, Grace Cohen; Ahlborn, Richard E.; Institution, Smithsonian (1997). Judaica at the Smithsonian: Cultural Politics as Cultural Model. Smithsonian Institution Press. p. 228. Shavuot falls on the sixth day of the Hebrew month of Sivan
  44. ^ Robertson, William Archibald Scott (1880). The crypt of Canterbury cathedral; its architecture, its history, and its frescoes. Mitchell & Hughes. p. 91. ...but seraphs, with six wings
  45. ^ Shapera, Paul M. (2009-08-15). Iran's Religious Leaders. The Rosen Publishing Group, Inc. p. 10. ISBN 978-1-4358-5283-9. Islam has six articles of faith
  46. ^ Algül, Hüseyin (2005). The Blessed Days and Nights of the Islamic Year. Tughra Books. p. 65. ISBN 978-1-932099-93-5. ...it was blessed to fast for six days in the month of Shawwal...
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  49. ^ Rhoads, Samuel E. (1996). The Sky Tonight: A Guided Tour of the Stars Over Hawaiʻi. Bishop Museum Press. ISBN 978-0-930897-93-2. Three Messier objects are visible in this part of the sky : M6, M7 and M8 .
  50. ^ Sedgwick, Marcus (2011-07-05). White Crow. Roaring Brook Press. p. 145. ISBN 978-1-4299-7634-3. The cells of honeycombs are six-sided because a hexagon is the most material-efficient tessellation
  51. ^ Parker, Steve (2005). Ant Lions, Wasps & Other Insects. Capstone. p. 16. ISBN 978-0-7565-1250-7. Insects have six legs each...
  52. ^ Pendarvis, Murray P.; Crawley, John L. (2019-02-01). Exploring Biology in the Laboratory: Core Concepts. Morton Publishing Company. p. 10. ISBN 978-1-61731-899-3. ...presently at least six kingdoms are recognized;
  53. ^ Mader, Sylvia S. (2004). Biology. McGraw-Hill. p. 20. ISBN 978-0-07-291934-9. The acronym CHNOPS helps us remember these six elements
  54. ^ Dufour, Fritz (2018-09-19). The Realities of Reality - Part II: Making Sense of Why Modern Science Advances. Vol. 1. Fritz Dufour. p. 100. The benzene molecule has its six carbon atoms in a ring[self-published source?]
  55. ^ Starr, Cecie; Evers, Christine (2012-05-10). Biology Today and Tomorrow without Physiology. Cengage Learning. p. 25. ISBN 978-1-133-36536-5. For example, the atomic number of carbon is 6,
  56. ^ Webb, Stephen; Webb, Professor of Australian Studies Stephen (2004-05-25). Out of this World: Colliding Universes, Branes, Strings, and Other Wild Ideas of Modern Physics. Springer Science & Business Media. p. 16. ISBN 978-0-387-02930-6. snowflake, with its familiar sixfold rotational symmetry
  57. ^ Woo, Teri Moser; Robinson, Marylou V. (2015-08-03). Pharmacotherapeutics For Advanced Practice Nurse Prescribers. F.A. Davis. p. 145. ISBN 978-0-8036-4581-3. Ayurvedic herbology is based on the tridoshic theory that there exist six basic tastes
  58. ^ Pandemic Influenza Preparedness and Response Guidance for Healthcare Workers and Healthcare Employers. OSHA, U.S. Department of Labor. 2007. p. 8. The WHO Plan describes six phases of increasing public health risk associated with the emergence of a new influenza
  59. ^ Sanghera, Paul (2011-03-08). Quantum Physics for Scientists and Technologists: Fundamental Principles and Applications for Biologists, Chemists, Computer Scientists, and Nanotechnologists. John Wiley & Sons. p. 64. ISBN 978-0-470-92269-9. ...there are six types of quarks and six types of leptons.
  60. ^ Jimbo, M.; Jimbo, Michio; Miwa, Tetsuji; Tsuchiya, Akihiro (1989). Integrable Systems in Quantum Field Theory and Statistical Mechanics. Academic Press. p. 588. ISBN 978-0-12-385342-4. Allowed configurations in the six-vertex model and their statistical weights
  61. ^ Sloan, Robin James Stuart (2015-05-07). Virtual Character Design for Games and Interactive Media. CRC Press. p. 34. ISBN 978-1-4665-9820-1. placing six primaries around the wheel in the following order: red, yellow, green, cyan, blue, magenta.
  62. ^ Stevens, E. S. (2002). Green Plastics: An Introduction to the New Science of Biodegradable Plastics. Princeton University Press. p. 45. ISBN 978-0-691-04967-0.
  63. ^ Bunson, Matthew (2014-05-14). Encyclopedia of the Roman Empire. Infobase Publishing. p. 90. ISBN 978-1-4381-1027-1. Augustus was also originally called Sextilis, the sixth month.
  64. ^ Nicolas, Sir Nicholas Harris (1833). The Chronology of History: Containing Tables, Calculations and Statements, Indispensable for Ascertaining the Dates of Historical Events and of Public and Private Documents from the Earliest Period to the Present Time. Longham, Rees, Orme, Brown, Green, & Longman and John Taylor. p. 172. SEXTIDI, or " Jour de la Révolution, "
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  67. ^ "Virgo | constellation and astrological sign". Encyclopedia Britannica. Retrieved 2020-08-06.
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  • The Odd Number 6, JA Todd, Math. Proc. Camb. Phil. Soc. 41 (1945) 66–68
  • A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0-521-42385-6
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