Integer Number

An Integer Number is a number from The Integer Number Sequence.

• AKA: Discrete Whole Number.
• Context:
  • It can range from being a Negative Integer to being a Non-Positive Integer to being a Non-Negative Integer to being a Positive Integer.
• Example(s):
  • any number in [math]\displaystyle{ ..., -1, 0, 1, 2, ... }[/math], such as a binary number.
  • …
• Counter-Example(s):
  • a Rational Number, such as ⅛, ¼, ⅓, ½, ⅜ ⅝, ⅔, ¾, ⅞.
  • an Irrational Number, such as [math]\displaystyle{ \pi }[/math]
• See: Real Number, Natural Number.

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References

2018a

• (Wikipedia, 2018) ⇒ https://en.wikipedia.org/wiki/Integer Retrieved:2018-7-7.
  • An integer (from the Latin integer meaning "whole") is a 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 set of integers consists of zero the positive natural numbers ( …), also called whole numbers or counting numbers, and their additive inverses (the negative integers, i.e., −1, −2, −3, …). The set of integers is often denoted by a boldface Z ("Z") or blackboard bold [math]\displaystyle{ \mathbb{Z} }[/math] (Unicode U+2124 ℤ) standing for the German word Zahlen ("numbers").

    Z is a subset of the set of all rational numbers Q, in turn a subset of the real numbers R. Like the natural numbers, Z is countably infinite.

    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, the (rational) integers are the algebraic integers that are also rational numbers.

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 [math]\displaystyle{ \cdots, -2, -1, 0, 1, 2, \cdots }[/math] The set of integers forms a ring that is denoted [math]\displaystyle{ \mathbb{Z} }[/math]. A given integer [math]\displaystyle{ n }[/math] may be negative ([math]\displaystyle{ n \in \mathbb{Z}^{-} }[/math]), nonnegative ([math]\displaystyle{ n \in \mathbb{Z}^{*} }[/math]), zero ([math]\displaystyle{ n=0 }[/math]), or positive ([math]\displaystyle{ n in \mathbb{Z}^{+}=\mathbb{N} }[/math]). The set of integers is, not surprisingly, called Integers in the Wolfram Language, and a number [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ \mathbb{Z} }[/math] of integers has cardinal number of aleph0. The generating function for the nonnegative integers is

    [math]\displaystyle{ f(x)=\frac{x}{(1-x)^2}=x+2x^2+3x^3+4x^4+\cdots }[/math]

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

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

• (Witten & Frank, 2000) ⇒ Ian H. Witten, and Eibe Frank. (2000). “Data Mining: Practical Machine Learning Tools and Techniques with Java implementations." Morgan Kaufmann.
  • QUOTE: The value of an attribute for a particular instance is a measurement of the quantity that the attribute refers to. There is a broad distinction between quantities that are numeric and ones that are nominal. Numeric attributes, sometimes called continuous attributes, measure numbers — either real- or integer-valued. Note that the term continuous is routinely abused in this context: integer-valued attributes are certainly not continuous in the mathematical sense. Nominal attributes take on values in a prespecified, finite set of possibilities and are sometimes called categorical. But there are other possibilities. Statistics texts often introduce "levels of measurement" such as nominal, ordinal, interval and ratio.

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