Number that, when added to the original number, yields the additive identity
In mathematics, the additive inverse of an elementx, denoted -x[1], is the element that when added to x, yields the additive identity, [2]. In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element.
The concept can also be extended to algebraic expressions, which is often used when balancing equations.
Relation to Subtraction
The additive inverse is closely related to subtraction, which can be viewed as an addition using the inverse:
a − b = a + (−b).
Conversely, the additive inverse can be thought of as subtraction from zero:
−a = 0 − a.
This connection lead to the minus sign being used for both opposite magnitudes and subtraction as far back as the 17th century. While this notation is standard today, it was met with opposition at the time, as some mathematicians felt it could be unclear and lead to errors.[8]
Formal Definition
Given an algebraic structure defined under addition with an additive identity , an element has an additive inverse if and only if , , and .[7]
Addition is typically only used to refer to a commutative operation, but it is not necessarily associative. When it is associative, so , the left and right inverses, if they exist, will agree, and the additive inverse will be unique. In non-associative cases, the left and right inverses may disagree, and in these cases, the inverse is not considered to exist.
The definition requires closure, that the additive element be found in . This is why despite addition being defined over the natural numbers, it does not an additive inverse for its members. The associated inverses would be the negative numbers, which is why the integers do have an additive inverse.
Further Examples
In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction.[9]
In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11).[10]
In a Boolean ring, which has elements addition is often defined as the symmetric difference. So , , , and . Our additive identity is 0, and both elements are their own additive inverse as and .[11]
See also
Absolute value (related through the identity |−x| = |x|).
^Gallian, Joseph A. (2017). Contemporary abstract algebra (9th ed.). Boston, MA: Cengage Learning. p. 52. ISBN 978-1-305-65796-0.
^Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 169–170. ISBN 978-1-292-02496-7.
^Mazur, Izabela (March 26, 2021). "2.5 Properties of Real Numbers -- Introductory Algebra". Retrieved August 4, 2024.
^"Standards::Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts". learninglab.si.edu. Retrieved 2024-08-04.
^ a b"2.2.5: Properties of Equality with Decimals". K12 LibreTexts. 2020-07-21. Retrieved 2024-08-04.
^ a bFraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 37–39. ISBN 978-1-292-02496-7.
^Cajori, Florian (2011). A History of Mathematical Notations: two volume in one. New York: Cosimo Classics. pp. 246–247. ISBN 978-1-61640-571-7.
^Axler, Sheldon (2024), Axler, Sheldon (ed.), "Vector Spaces", Linear Algebra Done Right, Cham: Springer International Publishing, pp. 1–26, doi:10.1007/978-3-031-41026-0_1, ISBN 978-3-031-41026-0, retrieved 2024-08-04
^Gupta, Prakash C. (2015). Cryptography and network security. Eastern economy edition. Delhi: PHI Learning Private Limited. p. 15. ISBN 978-81-203-5045-8.
^Martin, Urusula; Nipkow, Tobias (1989-03-01). "Boolean unification — The story so far". Journal of Symbolic Computation. Unification: Part 1. 7 (3): 275–293. doi:10.1016/S0747-7171(89)80013-6. ISSN 0747-7171.