Integer Number

References

2018b

• (Mathworld, 2018) ⇒ Weisstein, Eric W. “Integer." From MathWorld -- A Wolfram Web Resource. http://mathworld.wolfram.com/Integer.html Retrieved:2018-7-7.
• QUOTE: One of the numbers $\displaystyle{ \cdots, -2, -1, 0, 1, 2, \cdots }$ The set of integers forms a ring that is denoted $\displaystyle{ \mathbb{Z} }$. A given integer $\displaystyle{ n }$ may be negative ($\displaystyle{ n \in \mathbb{Z}^{-} }$), nonnegative ($\displaystyle{ n \in \mathbb{Z}^{*} }$), zero ($\displaystyle{ n=0 }$), or positive ($\displaystyle{ n in \mathbb{Z}^{+}=\mathbb{N} }$). The set of integers is, not surprisingly, called Integers in the Wolfram Language, and a number $\displaystyle{ x }$ can be tested to see if it is a member of the integers using the command Element[x, Integers]. The command IntegerQ[x] returns True if $\displaystyle{ x }$ has function head Integer in the Wolfram Language.

﻿ ﻿Numbers that are integers are sometimes described as “integral” (instead of integer-valued), but this practice may lead to unnecessary confusions with the integrals of integral calculus.

The ring $\displaystyle{ \mathbb{Z} }$ of integers has cardinal number of aleph0. The generating function for the nonnegative integers is

$\displaystyle{ f(x)=\frac{x}{(1-x)^2}=x+2x^2+3x^3+4x^4+\cdots }$

There are several symbols used to perform operations having to do with conversion between real numbers and integers. The symbol $\displaystyle{ \lfloor x\rfloor }$ ("floor x") means "the largest integer not greater than x", i.e., int(x) in computer parlance. The symbol $\displaystyle{ [x] }$ means "the nearest integer to x" (nearest integer function), i.e., nint(x) in computer parlance. The symbol $\displaystyle{ \lceil x \rceil }$ ("ceiling x") means "the smallest integer not smaller than x", or -int(-x), where int(x) is the integer part of x.

The German mathematician and logician Kronecker vociferously opposed the work of Georg Cantor on infinite sets and summarized his view that arithmetic and analysis should be based on whole numbers only by saying, "God made the natural numbers; all else is the work of man"(Bell 1986, p. 477[5]).

2009a

• (Wordnet, 2009) ⇒ wordnet.princeton.edu/perl/webwn
• any of the natural numbers (positive or negative) or zero; "an integer is a number that is not a fraction"

2009b

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Integer
• The integers are natural numbers including 0 (0, 1, 2, 3, ...) and their negatives (0, −1, −2, −3, ...). They are numbers that can be written without a fractional or decimal component, and fall within the set {... −2, −1, 0, 1, 2, ...}. For example, 65, 7, and −756 are integers; 1.6 and 1½ are not integers. In other terms, integers are the numbers one can count with items such as apples or fingers, and their negatives, as well as 0.
• More formally, the integers are the only integral domain whose positive elements are well-ordered, and in which order is preserved by addition. [citation needed] Like the natural numbers, the integers form a countably infinite set. The set of all integers is often denoted by a boldface Z (or blackboard bold \mathbb{Z}, Unicode U+2124 ℤ), which stands for Zahlen (German for numbers, pronounced /ˈtsaːlən/ "tsAH-len"). [2]
• In algebraic number theory, these commonly understood integers, embedded in the field of rational numbers, are referred to as rational integers to distinguish them from the more broadly defined algebraic integers.
• http://en.wiktionary.org/wiki/Integer
• An element of the infinite and numerable set [...,-3,-2,-1,0,1,2,3,...]
• http://en.wiktionary.org/wiki/integers
• integers - (plural only; not used in singular form) The smallest ring containing the natural numbers; the set {... -3, -2, -1, 0, 1, 2, 3 ...}

2000

1. Note: Integer 's first literal meaning in Latin is "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
2. Weisstein, Eric W. "Counting Number". MathWorld.
3. Weisstein, Eric W. "Whole Number". MathWorld.
4. Peter Jephson Cameron (1998). Introduction to Algebra. Oxford University Press. p. 4. ISBN 978-0-19-850195-4.
5. Bell, E. T. Men of Mathematics. New York: Simon and Schuster, 1986.