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4 (four; ) is a number, numeral, and glyph. It is the natural number following 3 and preceding 5. Four is the only number that has the same number of characters as its value in the English language.^{[1]}
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 (p^{2}) 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.
A number is a multiple of 4 if its last two digits are a multiple of 4. For example, 1092 is a multiple of 4 because 92 = 4×23.
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 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 = 2^{2} = 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. (That is, 2[n]2 = 4 for every positive integer n, where a[n]b is the hyperoperation.)
A four-sided plane figure is a quadrilateral (quadrangle) which include kites, rhombi, rectangles and squares, sometimes also called a tetragon. A circle divided by 4 makes right angles and four quadrants. 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 as well as four vertices 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 only natural integer n for which the (non trivial) alternating group A_n is not simple.
Four is the maximum number of dimensions of a real 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.^{[2]} 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 = y^{2} − z^{2}.
Four is an all-Harshad number and a semi-meandric number.
Four is the highest degree general polynomial equation for which there is a solution in radicals.
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 Shunga 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.^{[3]}
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.
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".
See also 4.
Torah, Kabbalah, Israel, Hebrew language, Mishnah
Martin Luther, Anglicanism, Bible, Lutheranism, Protestantism
4 (number), 0 (number), 2 (number), Taoism, 1 (number)
Venus, Earth, Neptune, Jupiter, Mars
ٔ, A, O, E, C
4 (number), 0 (number), Christianity, V for Vendetta, 2 (number)
3 (number), Ten Commandments, 1 (number), 0 (number), 4 (number)