Integer Damath Piece Setup

An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold $\mathbb$.

The set of natural numbers $\mathbb$ is a subset of $\mathbb$, which in turn is a subset of the set of all rational numbers $\mathbb$, itself a subset of the real numbers $\mathbb$. Like the natural numbers, $\mathbb$ is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not.

The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers. In fact, (rational) integers are algebraic integers that are also rational numbers.

History

The word integer comes from the Latin ''integer'' meaning "whole" or (literally) "untouched", from ''in'' ("not") plus ''tangere'' ("to touch"). "Entire" derives from the same origin via the French word ''entier'', which means both ''entire'' and ''integer''. Historically the term was used for a number that was a multiple of 1, or to the whole part of a mixed number. Only positive integers were considered, making the term synonymous with the natural numbers. The definition of integer expanded over time to include negative numbers as their usefulness was recognized. For example Leonhard Euler in his 1765 ''Elements of Algebra'' defined integers to include both positive and negative numbers. However, European mathematicians, for the most part, resisted the concept of negative numbers until the middle of the 19th century.

The use of the letter Z to denote the set of integers comes from the German word ''Zahlen'' ("number") and has been attributed to David Hilbert. The earliest known use of the notation in a textbook occurs in Algébre written by the collective Nicolas Bourbaki, dating to 1947. The notation was not adopted immediately, for example another textbook used the letter J and a 1960 paper used Z to denote the non-negative integers. But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.

The symbol $\mathbb$ is often annotated to denote various sets, with varying usage amongst different authors: $\mathbb^+$,$\mathbb_+$ or $\mathbb^$ for the positive integers, $\mathbb^$ or $\mathbb^$ for non-negative integers, and $\mathbb^$ for non-zero integers. Some authors use $\mathbb^$ for non-zero integers, while others use it for non-negative integers, or for (the group of units of $\mathbb$). Additionally, $\mathbb_$ is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of -adic integers.Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008

The whole numbers were synonymous with the integers up until the early 1950s. In the late 1950s, as part of the New Math movement, American elementary school teachers began teaching that "whole numbers" referred to the natural numbers, excluding negative numbers, while "integer" included the negative numbers. "Whole number" remains ambiguous to the present day.

Algebraic properties

Like the natural numbers, $\mathbb$ 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, ), $\mathbb$, 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 $\mathbb$.

$\mathbb$ 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 table lists some of the basic properties of addition and multiplication for any integers , and :

The first five properties listed above for addition say that $\mathbb$, under addition, is an abelian group. It is also a cyclic group, since every non-zero integer can be written as a finite sum or . In fact, $\mathbb$ under addition is the ''only'' infinite cyclic group—in the sense that any infinite cyclic group is isomorphic to $\mathbb$.

The first four properties listed above for multiplication say that $\mathbb$ under multiplication is a commutative monoid. However, not every integer has a multiplicative inverse (as is the case of the number 2), which means that $\mathbb$ under multiplication is not a group.

All the rules from the above property table (except for the last), when taken together, say that $\mathbb$ 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 $\mathbb$ for all values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings.

The lack of zero divisors in the integers (last property in the table) means that the commutative ring $\mathbb$ is an integral domain.

The lack of multiplicative inverses, which is equivalent to the fact that $\mathbb$ is not closed under division, means that $\mathbb$ 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 numbers), its ring of integers can be extracted, which includes $\mathbb$ as its subring.

Although ordinary division is not defined on $\mathbb$, the division "with remainder" is defined on them. It is called Euclidean division, and possesses the following important property: given two integers and with , there exist unique integers and such that and , where denotes the absolute value of . The integer is called the ''quotient'' and is called the ''remainder'' of the division of by . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions.

The above says that $\mathbb$ is a Euclidean domain. This implies that $\mathbb$ 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.

Order-theoretic properties

$\mathbb$ is a totally ordered set without upper or lower bound. The ordering of $\mathbb$ is given by:

An integer is ''positive'' if it is greater than zero, and ''negative'' if it is less than zero. Zero is defined as neither negative nor positive.

The ordering of integers is compatible with the algebraic operations in the following way:
# if and , then
# if and , then .

Thus it follows that $\mathbb$ 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. This is equivalent to the statement that any Noetherian valuation ring is either a field—or a discrete valuation ring.

Construction

Traditional development

In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, zero, and the negations of the natural numbers. This can be formalized as follows. First construct the set of natural numbers according to the Peano axioms, call this $P$. Then construct a set $P^-$ which is disjoint

HOME


Content is Copyleft
Website design, code, and AI is Copyrighted (c) 2014-2017 by Stephen Payne


Consider donating to Wikimedia



As an Amazon Associate I earn from qualifying purchases