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# Integer  ( Elementary Mathematics )

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An integer is the number zero (), a positive (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.).

(2000). 9780226742670, University of Chicago Press. .
The are the of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the or $\mathbb\left\{Z\right\}$.
(1998). 9780198501954, Oxford University Press. .

The set of natural numbers $\mathbb\left\{N\right\}$ is a of $\mathbb\left\{Z\right\}$, which in turn is a subset of the set of all $\mathbb\left\{Q\right\}$, itself a subset of the $\mathbb\left\{R\right\}$. Like the natural numbers, $\mathbb\left\{Z\right\}$ is . An integer may be regarded as a real number that can be written without a . For example, 21, 4, 0, and −2048 are integers, while 9.75, , and  are not.

(2019). 9781506248448, Simon and Schuster. .

The integers form the smallest group and the smallest ring containing the . 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 .

History
The word integer comes from the meaning "whole" or (literally) "untouched", from in ("not") plus tangere ("to touch"). "" derives from the same origin via the word , which means both entire and integer.
(1995). 9780792333524, Kluwer Academic Publishers. .
Historically the term was used for a that was a multiple of 1, or to the whole part of a . Only positive integers were considered, making the term synonymous with the . The definition of integer expanded over time to include as their usefulness was recognized. For example 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 word ("number")

(1998). 9780198501954, Oxford University Press. .
and has been attributed to . The earliest known use of the notation in a textbook occurs in Algébre written by the collective , 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 $\mathbb\left\{Z\right\}$ is often annotated to denote various sets, with varying usage amongst different authors: $\mathbb\left\{Z\right\}^+$,$\mathbb\left\{Z\right\}_+$ or $\mathbb\left\{Z\right\}^\left\{>\right\}$ for the positive integers, $\mathbb\left\{Z\right\}^\left\{0+\right\}$ or $\mathbb\left\{Z\right\}^\left\{\geq\right\}$ for non-negative integers, and $\mathbb\left\{Z\right\}^\left\{\neq\right\}$ for non-zero integers. Some authors use $\mathbb\left\{Z\right\}^\left\{*\right\}$ for non-zero integers, while others use it for non-negative integers, or for } (the group of units of $\mathbb\left\{Z\right\}$). Additionally, $\mathbb\left\{Z\right\}_\left\{p\right\}$ is used to denote either the set of integers modulo (i.e., the set of congruence classes of integers), or the set of .Keith Pledger and Dave Wilkins, "Edexcel AS and A Level Modular Mathematics: Core Mathematics 1" Pearson 2008LK Turner, FJ BUdden, D Knighton, "Advanced Mathematics", Book 2, Longman 1975.

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

Algebraic properties
Like the , $\mathbb\left\{Z\right\}$ is closed under the of addition and , that is, the sum and product of any two integers is an integer. However, with the inclusion of the negative natural numbers (and importantly, ), $\mathbb\left\{Z\right\}$, unlike the natural numbers, is also closed under .

The integers form a 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 in the category of rings, characterizes the ring $\mathbb\left\{Z\right\}$.

$\mathbb\left\{Z\right\}$ 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 , 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 :

 +Properties of addition and multiplication on integers ! !scope="col" Addition !scope="col" Multiplication

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

The first four properties listed above for multiplication say that $\mathbb\left\{Z\right\}$ 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 $\mathbb\left\{Z\right\}$ under multiplication is not a group.

All the rules from the above property table (except for the last), when taken together, say that $\mathbb\left\{Z\right\}$ together with addition and multiplication is a with unity. It is the prototype of all objects of such algebraic structure. Only those equalities of expressions are true in $\mathbb\left\{Z\right\}$ values of variables, which are true in any unital commutative ring. Certain non-zero integers map to zero in certain rings.

The lack of in the integers (last property in the table) means that the commutative ring $\mathbb\left\{Z\right\}$ is an .

The lack of multiplicative inverses, which is equivalent to the fact that $\mathbb\left\{Z\right\}$ is not closed under division, means that $\mathbb\left\{Z\right\}$ is not a field. The smallest field containing the integers as a is the field of . 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 $\mathbb\left\{Z\right\}$ as its .

Although ordinary division is not defined on $\mathbb\left\{Z\right\}$, 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 of . The integer is called the quotient and is called the of the division of by . The Euclidean algorithm for computing greatest common divisors works by a sequence of Euclidean divisions.

The above says that $\mathbb\left\{Z\right\}$ is a . This implies that $\mathbb\left\{Z\right\}$ is a principal ideal domain, and any positive integer can be written as the products of in an essentially unique way.

This is the fundamental theorem of arithmetic.

Order-theoretic properties
$\mathbb\left\{Z\right\}$ is a without upper or lower bound. The ordering of $\mathbb\left\{Z\right\}$ 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:

1. if and , then
2. if and , then .

Thus it follows that $\mathbb\left\{Z\right\}$ together with the above ordering is an .

The integers are the only nontrivial whose positive elements are .

(2022). 9780486137094, Courier Corporation. .
. This is equivalent to the statement that any is either a field—or a discrete valuation ring.

Construction

In elementary school teaching, integers are often intuitively defined as the union of the (positive) natural numbers, , and the negations of the natural numbers. This can be formalized as follows.
(1985). 9780898748185, Malabar, Fla. : R.E. Krieger Pub. Co.. .
First construct the set of natural numbers according to the , call this $P$. Then construct a set $P^-$ which is from $P$ and in one-to-one correspondence with $P$ via a function $\psi$. For example, take $P^-$ to be the $\left(1,n\right)$ with the mapping $\psi = n \mapsto \left(1,n\right)$. Finally let 0 be some object not in $P$ or $P^-$, for example the ordered pair $\left(0,0\right)$. Then the integers are defined to be the union $P \cup P^- \cup \\left\{0\\right\}$.

The traditional arithmetic operations can then be defined on the integers in a fashion, for each of positive numbers, negative numbers, and zero. For example is defined as follows: $-x = \begin{cases}$

 \psi(x), & \text{if } x \in P \\
\psi^{-1}(x), & \text{if } x \in P^- \\
0, & \text{if } x = 0

\end{cases}

The traditional 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 integers obey the various laws of arithmetic.

(2022). 9780486457925, Courier Dover Publications. .
.

Equivalence classes of ordered pairs
[[File:Relative numbers representation.svg|thumb|alt=Representation of equivalence classes for the numbers −5 to 5 |Red points represent ordered pairs of . Linked red points are equivalence classes representing the blue integers at the end of the line.|upright=1.5]]

In modern set-theoretic mathematics, a more abstract constructionIvorra Castillo: Álgebra allowing one to define arithmetical operations without any case distinction is often used instead.

(1999). 9780748735150, Nelson Thornes. .
. The integers can thus be formally constructed as the equivalence classes of of .
(1970). 9780390168955, Appleton-Century-Crofts. .

The intuition is that stands for the result of subtracting from . To confirm our expectation that and denote the same number, we define an equivalence relation on these pairs with the following rule:

$\left(a,b\right) \sim \left(c,d\right)$
precisely when
$a + d = b + c.$

Addition and multiplication of integers can be defined in terms of the equivalent operations on the natural numbers; by using to denote the equivalence class having as a member, one has:

$\left(a,b\right) + \left(c,d\right) := \left(a+c,b+d\right).$
$\left(a,b\right)\cdot\left(c,d\right) := \left(ac+bd,ad+bc\right).$

The negation (or additive inverse) of an integer is obtained by reversing the order of the pair:

$-\left(a,b\right) := \left(b,a\right).$

Hence subtraction can be defined as the addition of the additive inverse:

$\left(a,b\right) - \left(c,d\right) := \left(a+d,b+c\right).$

The standard ordering on the integers is given by:

$\left(a,b\right) < \left(c,d\right)$ if and only if $a+d < b+c.$

It is easily verified that these definitions are independent of the choice of representatives of the equivalence classes.

Every equivalence class has a unique member that is of the form or (or both at once). The natural number is identified with the class (i.e., the natural numbers are into the integers by map sending to ), and the class is denoted (this covers all remaining classes, and gives the class a second time since

Thus, is denoted by

$\begin\left\{cases\right\} a - b, & \mbox\left\{if \right\} a \ge b \\ -\left(b - a\right), & \mbox\left\{if \right\} a < b. \end\left\{cases\right\}$

If the natural numbers are identified with the corresponding integers (using the embedding mentioned above), this convention creates no ambiguity.

This notation recovers the familiar representation of the integers as .

Some examples are:

\begin\left\{align\right\}
0 &= [(0,0)] &= [(1,1)] &= \cdots & &= [(k,k)] \\
1 &= [(1,0)] &= [(2,1)] &= \cdots & &= [(k+1,k)] \\

-1 &= (0,1) &= (1,2) &= \cdots & &= (k,k+1) \\
2 &= [(2,0)] &= [(3,1)] &= \cdots & &= [(k+2,k)] \\

-2 &= (0,2) &= (1,3) &= \cdots & &= (k,k+2). \end{align}

Other approaches
In theoretical computer science, other approaches for the construction of integers are used by automated theorem provers and . Integers are represented as built using a few basic operations (e.g., zero, succ, pred) and, possibly, using , which are assumed to be already constructed (using, say, the ).

There exist at least ten such constructions of signed integers. These constructions differ in several ways: the number of basic operations used for the construction, the number (usually, between 0 and 2) and the types of arguments accepted by these operations; the presence or absence of natural numbers as arguments of some of these operations, and the fact that these operations are free constructors or not, i.e., that the same integer can be represented using only one or many algebraic terms.

The technique for the construction of integers presented in the previous section corresponds to the particular case where there is a single basic operation pair$\left(x,y\right)$ that takes as arguments two natural numbers $x$ and $y$, and returns an integer (equal to $x-y$). This operation is not free since the integer 0 can be written pair(0,0), or pair(1,1), or pair(2,2), etc. This technique of construction is used by the Isabelle; however, many other tools use alternative construction techniques, notable those based upon free constructors, which are simpler and can be implemented more efficiently in computers.

Computer science
An integer is often a primitive in computer languages. However, integer data types can only represent a of all integers, since practical computers are of finite capacity. Also, in the common two's complement representation, the inherent definition of sign distinguishes between "negative" and "non-negative" rather than "negative, positive, and 0". (It is, however, certainly possible for a computer to determine whether an integer value is truly positive.) Fixed length integer approximation data types (or subsets) are denoted int or Integer in several programming languages (such as Algol68, C, Java, , etc.).

Variable-length representations of integers, such as , can store any integer that fits in the computer's memory. Other integer data types are implemented with a fixed size, usually a number of bits which is a power of 2 (4, 8, 16, etc.) or a memorable number of decimal digits (e.g., 9 or 10).

Cardinality
The of the set of integers is equal to (). This is readily demonstrated by the construction of a , that is, a function that is and from $\mathbb\left\{Z\right\}$ to $\mathbb\left\{N\right\}= \\left\{0, 1, 2, ...\\right\}.$ Such a function may be defined as
$f\left(x\right) = \begin\left\{cases\right\} -2x, & \mbox\left\{if \right\} x \leq 0\\ 2x-1, & \mbox\left\{if \right\} x > 0, \end\left\{cases\right\}$
with graph (set of the pairs $\left(x, f\left(x\right)\right)$ is

.
Its is defined by
$\begin\left\{cases\right\}g\left(2x\right) = -x\\g\left(2x-1\right)=x, \end\left\{cases\right\}$
with graph
.

Footnotes

Sources

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