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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. 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. Note that certain non-zero integers map to zero in certain rings. 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 as a subring 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 numbersits 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: 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. Order-theoretic properties[ edit ] Z is a totally ordered set without upper or lower bound.
The ordering of Z is given by: Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: It follows that Z together with the above ordering is an ordered ring.
The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.
Construction[ edit ] Red points represent ordered pairs of natural numbers. Linked red points are equivalence classes representing the blue integers at the end of the line. In elementary school teaching, integers are often intuitively defined as the positive natural numbers, zeroand the negations of the natural numbers.
However, this style of definition leads to many different cases each arithmetic operation needs to be defined on each combination of types of integer and makes it tedious to prove that these operations obey the laws of arithmetic.Diploma in Computer Science KPT/JPS(A )04/20 Our Diploma in Computer Science programme has been designed to help the students to acquire knowledge related to computer applications.
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