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Integer ○꠹｜Definition｜1st｜20251119205401-00-⌔

Integer - Wikipedia

Integer

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An integer is the number zero (0), a positive natural number (1, 2, 3,…), or the negation of a positive natural number (−1, −2, −3,…).¹ The negations or additive inverses of the positive natural numbers are referred to as negative integers.² The set of all integers is often denoted by the boldface Z or blackboard bold ⁠ $\mathbb{Z}$ ⁠.³⁴

The set of natural numbers ⁠ $\mathbb{N}$ ⁠ is a subset of ⁠ $\mathbb{Z}$ ⁠, which in turn is a subset of the set of all rational numbers ⁠ $\mathbb{Q}$ ⁠, itself a subset of the real numbers ⁠ $\mathbb{R}$ ⁠.⁵ Like the set of natural numbers, the set of integers ⁠ $\mathbb{Z}$ ⁠ 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 the square root of 2 are not.⁶

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

Printed 2026-06-28.

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Footnotes

1. Science and Technology Encyclopedia. University of Chicago Press. September 2000. p. 280. ISBN 978-0-226-74267-0. ↩

2. Hillman, Abraham P.; Alexanderson, Gerald L. (1963). Algebra and trigonometry;. Boston: Allyn and Bacon. ↩

3. Miller, Jeff (29 August 2010). “Earliest Uses of Symbols of Number Theory”. Archived from the original on 31 January 2010. Retrieved 20 September 2010. ↩

4. Peter Jephson Cameron (1998). Introduction to Algebra. Oxford University Press. p. 4. ISBN 978-0-19-850195-4. Archived from the original on 8 December 2016. Retrieved 15 February 2016. ↩

5. More precisely, each system is embedded in the next, isomorphically mapped to a subset.${}^{[5]}$ The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the reals.${}^{[6]}$ ↩

6. Prep, Kaplan Test (4 June 2019). GMAT Complete 2020: The Ultimate in Comprehensive Self-Study for GMAT. Simon and Schuster. ISBN 978-1-5062-4844-8. ↩

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