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4 (four; ) is a number, numeral, and glyph. It is the natural number following and preceding . Four is the only number that has the same number of characters as its value in the English language.〔(【引用サイトリンク】title=Four facts about 4 )〕 ==In mathematics== Four is the smallest composite number, its proper divisors being and . Four is also a highly composite number. The next highly composite number is . 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, , 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 . 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 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, , and so on, for any number of up arrows. (That is, 2()2 = 4 for every positive integer ''n'', where ''a''()''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: * Two-dimensional: infinitely many convex regular polygons. * Three-dimensional: five convex regular polyhedra (the five Platonic Solids). * Four-dimensional: six convex regular polychora. * Five-dimensional and every higher-dimensional: three regular convex polytopes (regular simplexes, hypercubes, cross-polytopes). Four-dimensional differential manifolds have some unique properties. There is only one differential structure on 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 for which the (non trivial) alternating group is 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.〔Bryan Bunch, ''The Kingdom of Infinite Number''. New York: W. H. Freeman & Company (2000): 48〕 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; 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. 4''x'' = ''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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「4 (number)」の詳細全文を読む スポンサード リンク
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