# Positive integers

Order-theoretic properties[ edit ] Z is a totally ordered set without upper or lower bound.

The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered. 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.

This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers.

## Positive integers examples

The integer q is called the quotient and r is called the remainder of the division of a by b. Generalizations[ edit ] Two important generalizations of natural numbers arise from the two uses of counting and ordering: cardinal numbers and ordinal numbers. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. 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, zero , and the negations of the natural numbers. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism more than a bijection! 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. Natural numbers are also used as linguistic ordinal numbers : "first", "second", "third", and so forth. 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 first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. Infinity[ edit ] The set of natural numbers is an infinite set. Other generalizations are discussed in the article on numbers. This concept of "size" relies on maps between sets, such that two sets have the same size , exactly if there exists a bijection between them. 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.

In elementary school teaching, integers are often intuitively defined as the positive natural numbers, zeroand the negations of the natural numbers. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid.

## 10 situations where positive integers are used

It follows that Z together with the above ordering is an ordered ring. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. This concept of "size" relies on maps between sets, such that two sets have the same size , exactly if there exists a bijection between them. 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. Zero is defined as neither negative nor positive. These properties of addition and multiplication make the natural numbers an instance of a commutative semiring. It is the prototype of all objects of such algebraic structure. For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. Linked red points are equivalence classes representing the blue integers at the end of the line. This monoid satisfies the cancellation property and can be embedded in a group in the mathematical sense of the word group. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism more than a bijection! This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. In fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z.

It follows that Z together with the above ordering is an ordered ring. However, not every integer has a multiplicative inverse; e. Linked red points are equivalence classes representing the blue integers at the end of the line.

### Positive integers symbol

This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set. The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions. For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. The hypernatural numbers are an uncountable model that can be constructed from the ordinary natural numbers via the ultrapower construction. This assignment can be generalized to general well-orderings with a cardinality beyond countability, to yield the ordinal numbers. All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. However, not every integer has a multiplicative inverse; e. Here S should be read as " successor ". Zero is defined as neither negative nor positive. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. Natural numbers are also used as linguistic ordinal numbers : "first", "second", "third", and so forth. It follows that Z together with the above ordering is an ordered ring.

All sets that can be put into a bijective relation to the natural numbers are said to have this kind of infinity. 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.

Certain non-zero integers map to zero in certain rings.

### Positive integers

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. Semirings are an algebraic generalization of the natural numbers where multiplication is not necessarily commutative. Natural numbers are also used as linguistic ordinal numbers : "first", "second", "third", and so forth. A countable non-standard model of arithmetic satisfying the Peano Arithmetic i. Construction[ edit ] Red points represent ordered pairs of natural numbers. An important property of the natural numbers is that they are well-ordered : every non-empty set of natural numbers has a least element. A natural number can be used to express the size of a finite set; more precisely, a cardinal number is a measure for the size of a set, which is even suitable for infinite sets. An ordinal number may also be used to describe the notion of "size" for a well-ordered set, in a sense different from cardinality: if there is an order isomorphism more than a bijection! Other generalizations are discussed in the article on numbers. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. Although ordinary division is not defined on Z, the division "with remainder" is defined on them. The integers are the only nontrivial totally ordered abelian group whose positive elements are well-ordered.

For finite well-ordered sets, there is a one-to-one correspondence between ordinal and cardinal numbers; therefore they can both be expressed by the same natural number, the number of elements of the set. These properties of addition and multiplication make the natural numbers an instance of a commutative semiring.

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