Integer

The set of natural numbers ${\displaystyle \mathbb {N} }$ is a subset of ${\displaystyle \mathbb {Z} }$, which in turn is a subset of the set of all rational numbers ${\displaystyle \mathbb {Q} }$, itself a subset of the real numbers ${\displaystyle \mathbb {R} }$.^[a] Like the set of natural numbers, the set of integers ${\displaystyle \mathbb {Z} }$ 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, 5+1/2, 5/4 and √2 are not.^[8]


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.^[9] Historically the term was used for a number that was a multiple of 1,^[10][11] or to the whole part of a mixed number.^[12][13] 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.^[14] For example Leonhard Euler in his 1765 Elements of Algebra defined integers to include both positive and negative numbers.^[15]


The phrase the set of the integers was not used before the end of the 19th century, when Georg Cantor introduced the concept of infinite sets and set theory. The use of the letter Z to denote the set of integers comes from the German word Zahlen ("numbers")^[3][4] and has been attributed to David Hilbert.^[16] The earliest known use of the notation in a textbook occurs in Algèbre written by the collective Nicolas Bourbaki, dating to 1947.^[3][17] The notation was not adopted immediately, for example another textbook used the letter J^[18] and a 1960 paper used Z to denote the non-negative integers.^[19] But by 1961, Z was generally used by modern algebra texts to denote the positive and negative integers.^[20]


The symbol ${\displaystyle \mathbb {Z} }$ is often annotated to denote various sets, with varying usage amongst different authors: ${\displaystyle \mathbb {Z} ^{+}}$,${\displaystyle \mathbb {Z} _{+}}$ or ${\displaystyle \mathbb {Z} ^{>}}$ for the positive integers, ${\displaystyle \mathbb {Z} ^{0+}}$ or ${\displaystyle \mathbb {Z} ^{\geq }}$ for non-negative integers, and ${\displaystyle \mathbb {Z} ^{\neq }}$ for non-zero integers. Some authors use ${\displaystyle \mathbb {Z} ^{*}}$ for non-zero integers, while others use it for non-negative integers, or for {–1, 1} (the group of units of ${\displaystyle \mathbb {Z} }$). Additionally, ${\displaystyle \mathbb {Z} _{p}}$ is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p-adic integers.^[21][22]


The whole numbers were synonymous with the integers up until the early 1950s.^[23][24][25] In the late 1950s, as part of the New Math movement,^[26] American elementary school teachers began teaching that whole numbers referred to the natural numbers, excluding negative numbers, while integer included the negative numbers.^[27][28] The whole numbers remain ambiguous to the present day.^[29]