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The integers consist of the positive natural numbers (1, 2, 3, …) the negative natural numbers (−1, −2, −3, ...) and the number zero. The set of all integers is usually denoted in mathematics by Z (or Z in blackboard bold, ), which stands for Zahlen ( German for "numbers"). They are also known as the whole numbers, although that term is also used to refer only to the positive integers (with or without zero). Like the natural numbers, the integers form a countably infinite set.The branch of mathematics which includes the study of the integers is called number theory.
1 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, zero, Z (unlike the natural numbers) is also closed under subtraction. Z is not closed under the operation of division, since the quotient of two integers (e.g., 1 divided by 2), need not be an integer.
The following table lists some of the basic properties of addition and multiplication for any integers a, b and c.
| | addition | multiplication
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| closure: | a + b is an integer | a × b is an integer
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| associativityAbstract algebra Algebra In mathematics, associativity is a property that a binary operation can have. It means that the order of evaluation is immaterial if the operation appears more than once in an expression. Put another way, no parentheses are requir: | a + (b + c) = (a + b) + c | a × (b × c) = (a × b) × c
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| commutativity: | a + b = b + a | a × b = b × a
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| existence of an identity elementIn mathematics, an identity element (or neutral element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. The term identity element is often shortened to ident: | a + 0 = a | a × 1 = a
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| existence of inverse elementIn mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combinatis: | a + (-a) = 0 |
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| distributivityIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. For example: : 4 · (2 + 3) (4 · 2) + (4 · 3) In the left-hand side of the above equatio: | a × (b + c) = (a × b) + (a × c)
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In the language of abstract algebraAbstract algebra Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. The term "abstract algebra" is used to distinguish the field from " elementary algebra" or "high school algebr, the first five properties listed above for addition, say that Z together with addition is an abelian group (lacking commutativity, such a structure is simply called a group). The lack of inverse elements for multiplication (a × b = 1 implies that a = b = 1) means that Z together with multiplication is not a group. All the properties of the above table taken together say that Z together with addition and multiplication is a ring. 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 containing the integers is the rational numbers. This follows from the fact the that the set of rational numbers (which are essentially just the ratios or "divisions", of integers) is the closure of Z under division.
Although ordinary division is not defined for Z, it does posses an important property called the division algorithm: that is, given two integers a and b with b ≠ 0, there exist unique integers q and r such that a = qb + r and 0 ≤ r < |b|, (where |b| denotes the absolute value of b). The integer q is called the quotient and r is called the remainder, resulting from division of a by b. This is the basis for the Euclidean algorithm for computing greatest common divisors.
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 that whole numbers can be written as products of primes in an essentially unique way. This is the fundamental theorem of arithmetic.
