4 (number)

4

−1 0 1 2 3 4 5 6 7 8 9

Cardinal 4
four
Ordinal 4th
fourth
Numeral system quaternary
Factorization 2^2
Divisors 1, 2, 4
Roman numeral IV or IIII
Roman numeral (Unicode) Ⅳ, ⅳ
Greek δ (or Δ)
Arabic ٤,4
Arabic (Persian, Urdu) ۴
Ge'ez
Bengali
Chinese numeral 四,亖,肆
Devanagari
Telugu
Malayalam
Tamil
Hebrew ארבע (Arba, pronounced AR-bah) or ד (Dalet, 4th letter of the Hebrew alphabet)
Khmer
Thai
prefixes tetra- (from Greek)

quadri-/quadr- (from Latin)

Binary 100
Octal 4
Duodecimal 4
Hexadecimal 4
Vigesimal 4

4 (four; /ˈfɔər/) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5.

Contents

In mathematics

Four is the smallest composite number, its proper divisors being 1 and 2. Four is also a highly composite number. The next highly composite number is 6.

Four is the second square number, the second centered triangular number.

4 is the smallest squared prime (p2) and the only even number in this form. It has an aliquot sum of 3 which is itself prime. The aliquot sequence of 4 has 4 members (4, 3, 1, 0) and is accordingly the first member of the 3-aliquot tree.

Only one number has an aliquot sum of 4 and that is squared prime 9.

The prime factorization of four is two times two.

Four is the smallest composite number that is equal to the sum of its prime factors. (As a consequence of this, it is the smallest Smith number). However, it is the largest (and only) composite number n for which (n - 1)!\ \equiv\ 0 \ ({\rm mod}\ n) is false.

It is also a Motzkin number.

In bases 6 and 12, 4 is a 1-automorphic number.

In addition, 2 + 2 = 2 × 2 = 22 = 4. Continuing the pattern in Knuth's up-arrow notation, 2 \uparrow\uparrow 2 = 2 \uparrow\uparrow\uparrow 2 = 4, and so on, for any number of up arrows.

A four-sided plane figure is a quadrilateral (quadrangle) or square, sometimes also called a tetragon. A circle divided by 4 makes right angles. Because of it, four (4) is the base number of plane (mathematics). Four cardinal directions, four seasons, duodecimal system, and vigesimal system are based on four.

A solid figure with four faces is a tetrahedron. The regular tetrahedron is the simplest Platonic solid. A tetrahedron, which can also be called a 3-simplex, has four triangular faces and four vertices. It is the only self-dual regular polyhedron.

Four-dimensional space is the highest-dimensional space featuring more than three convex regular figures:

Four-dimensional differential manifolds have some unique properties. There is only one differential structure on \mathbb{R}^{n} except when n = 4, in which case there are uncountably many.

The smallest non-cyclic group has four elements; it is the Klein four-group. Four is also the order of the smallest non-trivial groups that are not simple.

Four is the maximum number of dimensions of a real division algebra (the quaternions), by a theorem of Ferdinand Georg Frobenius.

The four-color theorem states that a planar graph (or, equivalently, a flat map of two-dimensional regions such as countries) can be colored using four colors, so that adjacent vertices (or regions) are always different colors.[1] Three colors are not, in general, sufficient to guarantee this. The largest planar complete graph has four vertices.

Lagrange's four-square theorem states that every positive integer can be written as the sum of at most four square numbers. Three are not always sufficient; 7 for instance cannot be written as the sum of three squares.

Four is the first positive non-Fibonacci number.

Each natural number divisible by 4 is a difference of squares of two natural numbers, i.e. 4x = y2z2.

Four is an all-Harshad number and a semi-meandric number.

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 21 22 23 24 25 50 100 1000
4 \times x 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 100 200 400 4000
Division 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
4 \div x 4 2 1.\overline{3} 1 0.8 0.\overline{6} 0.\overline{571428} 0.5 0.\overline{4} 0.4 0.\overline{36} 0.\overline{3} 0.\overline{307692} 0.\overline{285714} 0.2\overline{6} 0.25
x \div 4 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4
Exponentiation 1 2 3 4 5 6 7 8 9 10 11 12 13
4 ^ x\, 4 16 64 256 1024 4096 16384 65536 262144 1048576 4194304 16777216 67108864
x ^ 4\, 1 16 81 256 625 1296 2401 4096 6561 10000 14641 20736 28561

Evolution of the glyph

Representing 1, 2 and 3 in as many lines as the number represented worked well. The Brahmin Indians simplified 4 by joining its four lines into a cross that looks like our modern plus sign. The Sunga would add a horizontal line on top of the numeral, and the Kshatrapa and Pallava evolved the numeral to a point where speed of writing was a secondary concern. The Arabs' 4 still had the early concept of the cross, but for the sake of efficiency, was made in one stroke by connecting the "western" end to the "northern" end; the "eastern" end was finished off with a curve. The Europeans dropped the finishing curve and gradually made the numeral less cursive, ending up with a glyph very close to the original Brahmin cross.[2]

While the shape of the 4 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in .

On the seven-segment displays of pocket calculators and digital watches, as well as certain optical character recognition fonts, 4 is seen with an open top. This is because this was the original way the glyph was drawn (before the illustration above came into usage).

Television stations that operate on channel 4 have occasionally made use of another variation of the "open 4", with the open portion being on the side, rather than the top. This version resembles the Canadian Aboriginal syllabics letter ᔦ. The magnetic ink character recognition "CMC-7" font also uses this variety of "4". Another form of the 4 glyph that was invented for television was the arrow 4, which combines the 4 glyph with an arrow.

In religion

Buddhism
Judeo-Christian symbolism
Hinduism
Islam
Other

In science

In astronomy

In biology

In chemistry

In physics

In logic and philosophy

In technology

In transport

In sports

In other fields

See also 4 (disambiguation).

In music

Groups of four

References

  1. ^ Bryan Bunch, The Kingdom of Infinite Number. New York: W. H. Freeman & Company (2000): 48
  2. ^ 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. 24.64
  3. ^ Chevalier, Jean and Gheerbrant, Alain (1994), The Dictionary of Symbols. The quote beginning "Almost from prehistoric times..." is on p. 402.
  4. ^ http://windowsteamblog.com/blogs/windowsvista/archive/2008/10/14/why-7.aspx

External links