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An integer (from the Latin ''integer'' meaning "whole")〔''Integer'' 's first, literal meaning in Latin is "untouched", from ''in'' ("not") plus ''tangere'' ("to touch"). "Entire" derives from the same origin via French (see: )〕 is a number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, 5½, and are not. The set of integers consists of zero (), the natural numbers (, , , …), also called ''whole numbers'' or ''counting numbers'', and their additive inverses (the negative integers, i.e. −1, −2, −3, …). This is often denoted by a boldface Z ("Z") or blackboard bold (Unicode U+2124 (unicode:ℤ)) standing for the German word ''Zahlen'' ((:ˈtsaːlən), "numbers"). (unicode:ℤ) is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the (rational) integers are the algebraic integers that are also rational numbers. == Algebraic properties == Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers, and, importantly, , Z (unlike the natural numbers) is also closed under subtraction. The integers form a unital ring which is the most basic one, in the following sense: for any unital ring, there is a unique ring homomorphism from the integers into this ring. This universal property, namely to be an initial object in the category of rings, characterizes the ring Z. Z is not closed under division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer. Although the natural numbers are closed under exponentiation, the integers are not (since the result can be a fraction when the exponent is negative). The following lists some of the basic properties of addition and multiplication for any integers ''a'', ''b'' and ''c''. In the language of abstract algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, since every non-zero integer can be written as a finite sum or . In fact, Z under addition is the ''only'' infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse; e.g. there is no integer ''x'' such that because the left hand side is even, while the right hand side is odd. This means that Z under multiplication is not a group. All the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in Z for all values of variables, which are true in any unital commutative ring. Note that certain non-zero integers map to zero in certain rings. At last, the property ( *) says that the commutative ring Z is an integral domain. In fact, Z provides the motivation for defining such a structure. The lack of multiplicative inverses, which is equivalent to the fact that Z is not closed under division, means that Z is ''not'' a field. The smallest field with the usual operations containing the integers is the field of rational numbers. The process of constructing the rationals from the integers can be mimicked to form the field of fractions of any integral domain. And back, starting from an algebraic number field (an extension of rational numbers), its ring of integers can be extracted, which includes Z as its subring. Although ordinary division is not defined on Z, the division "with remainder" is defined on them. It is called Euclidean division and possesses the following important property: that is, given two integers ''a'' and ''b'' with , there exist unique integers ''q'' and ''r'' such that and , where | ''b'' | denotes the absolute value of ''b''. The integer ''q'' is called the ''quotient'' and ''r'' is called the ''remainder'' of the division of ''a'' by ''b''. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. Again, in the language of abstract algebra, the above says that Z is a Euclidean domain. This implies that Z is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way. This is the fundamental theorem of arithmetic. 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「Integer」の詳細全文を読む スポンサード リンク
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