Integer

The integers arranged on a number line

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 $\mathbb {Z}$ .^[3]^[4]

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 $\mathbb {R}$ .^[a] 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 √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.

^ Science and Technology Encyclopedia. University of Chicago Press. September 2000. p. 280. ISBN 978-0-226-74267-0.
^ Hillman, Abraham P.; Alexanderson, Gerald L. (1963). Algebra and trigonometry;. Boston: Allyn and Bacon.
^ Cite error: The named reference earliest was invoked but never defined (see the help page).
^ 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.
^ Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990). Mathematical Methods in Linguistics. Springer Science & Business Media. pp. 78–82. ISBN 978-90-277-2245-4. The natural numbers are not themselves a subset of this set-theoretic representation of the integers. Rather, the set of all integers contains a subset consisting of the positive integers and zero which is isomorphic to the set of natural numbers.
^ Wohlgemuth, Andrew (10 June 2014). Introduction to Proof in Abstract Mathematics. Courier Corporation. p. 237. ISBN 978-0-486-14168-8.
^ Polkinghorne, John (19 May 2011). Meaning in Mathematics. OUP Oxford. p. 68. ISBN 978-0-19-162189-5.
^ 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.

Cite error: There are <ref group=lower-alpha> tags or {{efn}} templates on this page, but the references will not show without a {{reflist|group=lower-alpha}} template or {{notelist}} template (see the help page).

[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.

[earliest-3] Cite error: The named reference earliest was invoked but never defined (see the help page).

[Cameron1998-4] 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] Partee, Barbara H.; Meulen, Alice ter; Wall, Robert E. (30 April 1990). Mathematical Methods in Linguistics. Springer Science & Business Media. pp. 78–82. ISBN 978-90-277-2245-4. The natural numbers are not themselves a subset of this set-theoretic representation of the integers. Rather, the set of all integers contains a subset consisting of the positive integers and zero which is isomorphic to the set of natural numbers.

[6] Wohlgemuth, Andrew (10 June 2014). Introduction to Proof in Abstract Mathematics. Courier Corporation. p. 237. ISBN 978-0-486-14168-8.

[7] Polkinghorne, John (19 May 2011). Meaning in Mathematics. OUP Oxford. p. 68. ISBN 978-0-19-162189-5.

[9] 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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