6 (six) is the natural number following 5 and preceding 7.
It is a composite number and the smallest perfect number.
| ||||
---|---|---|---|---|
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 | _ .... |
Six is the smallest positive integer which is neither a square number nor a prime number. It is the second smallest composite number after four, equal to the sum and the product of its three proper divisors (1, 2 and 3). As such, 6 is the only number that is both the sum and product of three consecutive positive numbers. It is the smallest perfect number, which are numbers that are equal to their aliquot sum, or sum of their proper divisors. It is also the largest of the four all-Harshad numbers (1, 2, 4, and 6).
6 is a pronic number and the only semiprime to be. It is the first discrete biprime (2 × 3) which makes it the first member of the (2 × q) discrete biprime family, where q is a higher prime. All primes above 3 are of the form 6n ± 1 for n ≥ 1.
As a perfect number:
Six 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; sixty (10 × 6) and ninety (15 × 6) are the next two.
It is the first primitive pseudoperfect number, and all integers that are multiples of 6 are pseudoperfect (all multiples of a perfect number are pseudoperfect); six is also the smallest Granville number, or -perfect number.
Unrelated to 6's being a perfect number, a Golomb ruler of length 6 is a "perfect ruler". Six is a congruent number.
6 is the second primary pseudoperfect number, and harmonic divisor number. It is also the second superior highly composite number, and the last to also be a primorial.
There are 6 non-equivalent ways in which 100 can be expressed as the sum of two prime numbers: (3 + 97), (11 + 89), (17 + 83), (29 + 71), (41 + 59) and (47 + 53).
There is not a prime such that the multiplicative order of 2 modulo is 6, that is, By Zsigmondy's theorem, if is a natural number that is not 1 or 6, then there is a prime such that . See A112927 for such .
The ring of integer of the sixth cyclotomic field Q(ζ6), which is called Eisenstein integer, has 6 units: ±1, ±ω, ±ω2, where .
The six exponentials theorem guarantees (given the right conditions on the exponents) the transcendence of at least one of a set of exponentials.
There are six basic trigonometric functions: sin, cos, sec, csc, tan, and cot.
The smallest non-abelian group is the symmetric group which has 3! = 6 elements.
Six is a triangular number and so is its square (36). It is the first octahedral number, preceding 19.
A six-sided polygon is a hexagon, one of the three regular polygons capable of tiling the plane. Figurate numbers representing hexagons (including six) are called hexagonal numbers. Because 6 is the product of a power of 2 (namely 21) with nothing but distinct Fermat primes (specifically 3), a regular hexagon is a constructible polygon with a compass and straightedge alone. A hexagram is a six-pointed geometric star figure (with the Schläfli symbol {6/2}, 2{3}, or {{3}}).
Six similar coins can be arranged around a central coin of the same radius so that each coin makes contact with the central one (and touches both its neighbors without a gap), but seven cannot be so arranged. This makes 6 the answer to the two-dimensional kissing number problem. The densest sphere packing of the plane is obtained by extending this pattern to the hexagonal lattice in which each circle touches just six others.
There is only one non-trivial magic hexagon: it is of order-3 and made of nineteen cells, with a magic constant of 38. All rows and columns in a 6 × 6 magic square collectively generate a magic sum of 666 (which is doubly triangular). On the other hand, Graeco-Latin squares with order 6 do not exist; if is a natural number that is not 2 or 6, then there is a Graeco-Latin square of order .
The cube is one of five Platonic solids, with a total of six squares as faces. It is the only regular polyhedron that can generate a uniform honeycomb on its own, which is also self-dual. The cuboctahedron, which is an Archimedean solid that is one of two quasiregular polyhedra, has eight triangles and six squares as faces. Inside, its vertex arrangement can be interpreted as three hexagons that intersect to form an equatorial hexagonal hemi-face, by-which the cuboctahedron is dissected into triangular cupolas. This solid is also the only polyhedron with radial equilateral symmetry, where its edges and long radii are of equal length; its one of only four polytopes with this property — the others are the hexagon, the tesseract (as the four-dimensional analogue of the cube), and the 24-cell. Only six polygons are faces of non-prismatic uniform polyhedra such as the Platonic solids or the Archimedean solids: the triangle, the square, the pentagon, the hexagon, the octagon, and the decagon. If self-dual images of the tetrahedron are considered distinct, then there are a total of six regular polyhedra that are formed by three different Weyl groups in the third dimension (based on tetrahedral, octahedral and icosahedral symmetries).
How closely the shape of an object resembles that of a perfect sphere is called its sphericity, calculated by:
In four dimensions, there are a total of six convex regular polytopes: the 5-cell, 8-cell, 16-cell, 24-cell, 120-cell, and 600-cell.
, with 720 = 6! elements, is the only finite symmetric group which has an outer automorphism. This automorphism allows us to construct a number of exceptional mathematical objects such as the S(5,6,12) Steiner system, the projective plane of order 4, the four-dimensional 5-cell, and the Hoffman-Singleton graph. A closely related result is the following theorem: 6 is the only natural number for which there is a construction of isomorphic objects on an -set , invariant under all permutations of , but not naturally in one-to-one correspondence with the elements of . This can also be expressed category theoretically: consider the category whose objects are the element sets and whose arrows are the bijections between the sets. This category has a non-trivial functor to itself only for .
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).
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 |
Hexa is classical Greek for "six". Thus:
Sex- is a Latin prefix meaning "six". Thus:
The SI prefix for 10006 is exa- (E), and for its reciprocal atto- (a).
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.
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
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."
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