6

🎲 the perfect number

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).

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Cardinal	six
Ordinal	6th (sixth)
Numeral system	senary
Factorization	2 × 3
Divisors	1, 2, 3, 6
Greek numeral	Ϛ´
Roman numeral	VI, vi, ↅ
Greek prefix	hexa-/hex-
Latin prefix	sexa-/sex-
Binary	1102
Ternary	203
Senary	106
Octal	68
Duodecimal	612
Hexadecimal	616
Greek	στ (or ΣΤ or ς)
Arabic, Kurdish, Sindhi, Urdu	٦
Persian	۶
Amharic	፮
Bengali	৬
Chinese numeral	六，陸
Devanāgarī	६
Gujarati	૬
Hebrew	ו
Khmer	៦
Thai	๖
Telugu	౬
Tamil	௬
Saraiki	٦
Malayalam	൬
Armenian	Զ
Babylonian numeral	𒐚
Egyptian hieroglyph	𓏿
Morse code	_ ....

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

In mathematics

A six-sided polygon is a hexagon, 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. It is also the first number that is the sum of its proper divisors, making it the smallest perfect number. 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. 6 is the largest of the four all-Harshad numbers.

6 is the 2nd superior highly composite number, the 2nd colossally abundant number, the 3rd triangular number, the 4th highly composite number, a pronic number, a congruent number, a harmonic divisor number, and a semiprime. 6 is also the first Granville number, or $$

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

${\mathcal {S}}$-perfect number. A Golomb ruler of length 6 is a "perfect ruler".

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

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

$\mathrm {S_{3}}$ which has 3! = 6 elements. 6 the answer to the two-dimensional kissing number problem.

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).

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". Thus:

• "Hexadecimal" combines hexa- with the Latinate decimal to name a number base of 16
• A hexagon is a regular polygon with six sides
  • L'Hexagone is a French nickname for the continental part of Metropolitan France for its resemblance to a regular hexagon
• A hexahedron is a polyhedron with six faces, with a cube being a special case
• 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". Thus:

• Senary is the ordinal adjective meaning "sixth"
• 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.

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. 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.

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 .

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

• Les Six ("The Six" in English) was a group consisting of the French composers Georges Auric, Louis Durey, Arthur Honegger, Darius Milhaud, Francis Poulenc and Germaine Tailleferre in the 1920s
• Bands with the number six in their name include Six Organs of Admittance, 6 O'Clock Saints, Electric Six, Eve 6, Los Xey (sei is Basque for "six"), Out On Blue Six, Six In Six, Sixpence None the Richer, Slant 6, Vanity 6, and You Me At Six
• #6 is the pseudonym of American musician Shawn Crahan, when performing with the band Slipknot

In instruments

• A standard guitar has six strings
• 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.
• There are six semitones in a tritone.

In works

• "Six geese a-laying" were given as a present on the sixth day in the popular Christmas carol, "The Twelve Days of Christmas".
• Divided in six arias, Hexachordum Apollinis is generally regarded as one of the pinnacles of Johann Pachelbel's oeuvre.
• The theme of the sixth album by Dream Theater, Six Degrees of Inner Turbulence, was the number six: the album has six songs, and the sixth song—that is, the complete second disc—explores the stories of six individuals suffering from various mental illnesses.
• Aristotle gave six elements of tragedy, the first of which is Mythos.

In religion

See also: 666

Judaism

• Six points on a Star of David
• Six orders of the Mishnah
• Six symbolic foods placed on the Passover Seder Plate
• God took six days to create the world in the Old Testament Book of Genesis; humankind was created on day 6. In the City of God, Augustine of Hippo suggested (book 11, chapter 30) that God's creation of the world took six days because 6 is a perfect number.
• The Jewish holiday of Shavuot starts on the sixth day of the Hebrew month of Sivan
• Seraphs have six wings.

Islam

• There are Six articles of faith
• Six points on a Seal of Solomon
• Fasting six days of Shawwal, together with the month of Ramadan, is equivalent to fasting the whole year
• The heavens and earth were created in six days according to the Quran:

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 Hindu theology, a trasarenu is the combination of six celestial paramānus (atoms).
• In Taoism:
  • Six Lines of a Hexagram
  • Six Ministries of Huang Di

In science

Astronomy

• Messier object M6, a magnitude 4.5 open cluster in the constellation Scorpius, also known as the Butterfly Cluster
• The New General Catalogue object NGC 6, a spiral galaxy in the constellation Andromeda
• The Roman numeral VI:
  • Stands for subdwarfs in the Yerkes spectral classification scheme
  • (Usually) stands for the sixth-discovered satellite of a planet or minor planet (e.g. Jupiter VI)
• 6 Hebe, a large main-belt asteroid.
• A six-sided cloud pattern persists around the north pole on the planet Saturn.

Biology

The cells of a beehive are six-sided.

• The cells of a beehive are six-sided.
• Insects have six legs (see Hexapoda).
• Six kingdoms in the taxonomic rank below domain (biology); Animalia, Plantae, Fungi, Protista, Archaea/Archaeabacteria, and Bacteria/Eubacteria. See Kingdom (biology).
• The six elements most common in biomolecules are called the CHNOPS elements; the letters stand for the chemical abbreviations of carbon, hydrogen, nitrogen, oxygen, phosphorus, and sulfur. See CHON.

Chemistry

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

• A benzene molecule has a ring of six carbon atoms.
• 6 is the atomic number of carbon. Carbon-12, its most stable isotope, has six protons, six neutrons, and six electrons.
• The sixfold symmetry of snowflakes arises from the hexagonal crystal structure of ordinary ice.
• A hexamer is an oligomer made of six subunits.
• The p-orbital allows an element to bind up to a maximum of six electrons.

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.
• Phase 6 is one of six pandemic influenza phases.

Physics

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

• In the Standard Model of particle physics, there are six types of quarks and six types of leptons.
• In statistical mechanics, the six-vertex model has six possible configurations of arrows at each vertex
• There are six colors in the RGB color wheel: (primary) red, blue, green, (secondary) cyan, magenta, and yellow. (See Tertiary color)
• In three-dimensional Euclidean space, there are six unknown support reactions for a statically determinate structure: one force in each of the three dimensions, and one moment through each of three possible orthogonal planes.

In technology

6 as a resin identification code, used in recycling.

• On most phones, the 6 key is associated with the letters M, N, and O, but on the BlackBerry Pearl it is the key for J and K, and on the BlackBerry 8700 series and Curve 8900 with full keyboard, it is the key for F
• The "6-meter band" in amateur radio includes the frequencies from 50 to 54 MHz
• 6 is the resin identification code used in recycling to identify polystyrene

In calendars

• In the ancient Roman calendar, Sextilis was the sixth month. After the Julian reform, June became the sixth month and Sextilis was renamed August
• Sextidi was the sixth day of the décade in the French Revolutionary calendar

Anthropology

• The name of the smallest group of Cub Scouts and Guiding's equivalent Brownies, traditionally consisting of six people and is led by a "sixer".
• A coffin is traditionally buried six feet under the ground; thus, the phrase "six feet under" means that a person (or thing, or concept) is dead
• There are said to be no more than six degrees of separation between any two people on Earth.
• In Western astrology, Virgo is the 6th astrological sign of the Zodiac
• The Six Dynasties form part of Chinese history
• Six is a lucky number in Chinese culture.
• The Birmingham Six were a British miscarriage of justice, held in prison for 16 years.
• "Six" is used as an informal slang term for the British Secret Intelligence Service, MI6.

See also

• List of highways numbered 6
• Six degrees (disambiguation).

References

• Todd, J. A. (1945). "The 'odd' number six". Mathematical Proceedings of the Cambridge Philosophical Society. 41 (1): 66–68. doi:10.1017/S0305004100022374.
• A Property of the Number Six, Chapter 6, P Cameron, JH v. Lint, Designs, Graphs, Codes and their Links ISBN 0-521-42385-6
• Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 67 - 69

External links

• The Number 6
• The Positive Integer 6
• Prime curiosities: 6