Opposite (mathematics)

In mathematics, the additive inverse of an element x, denoted -x, is the element that when added to x, yields the additive identity, 0 (zero). In the most familiar cases, this is the number 0, but it can also refer to a more generalized zero element.

In elementary mathematics, the additive inverse is often referred to as the opposite number, or its negative. The unary operation of arithmetic negation is closely related to subtraction and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers.

Common examples

When working with integers, rational numbers, real numbers, and complex numbers, the additive inverse of any number can be found by multiplying it by −1.

These complex numbers, two of eight values of ⁸√1, are mutually opposite

Simple cases of additive inverses

${\displaystyle n}$	${\displaystyle -n}$
${\displaystyle 7}$	${\displaystyle -7}$
${\displaystyle 0.35}$	${\displaystyle -0.35}$
${\displaystyle {\frac {1}{4}}}$	${\displaystyle -{\frac {1}{4}}}$
${\displaystyle \pi }$	${\displaystyle -\pi }$
${\displaystyle 1+2i}$	${\displaystyle -1-2i}$

The concept can also be extended to algebraic expressions, which is often used when balancing equations.

Additive inverses of algebraic expressions

${\displaystyle n}$	${\displaystyle -n}$
${\displaystyle a-b}$	${\displaystyle -(a-b)=-a+b}$
${\displaystyle 2x^{2}+5}$	${\displaystyle -(2x^{2}+5)=-2x^{2}-5}$
${\displaystyle {\frac {1}{x+2}}}$	${\displaystyle -{\frac {1}{x+2}}}$
${\displaystyle {\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta }}$	${\displaystyle -({\sqrt {2}}\sin {\theta }-{\sqrt {3}}\cos {2\theta })=-{\sqrt {2}}\sin {\theta }+{\sqrt {3}}\cos {2\theta }}$

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.

Formal definition

Given an algebraic structure defined under addition ${\displaystyle (S,+)}$ with an additive identity ${\displaystyle e\in S}$, an element ${\displaystyle x\in S}$ has an additive inverse ${\displaystyle y}$ if and only if ${\displaystyle y\in S}$, ${\displaystyle x+y=e}$, and ${\displaystyle y+x=e}$.

Addition is typically only used to refer to a commutative operation, but it is not necessarily associative. When it is associative, so ${\displaystyle (a+b)+c=a+(b+c)}$, 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 ${\displaystyle y}$ be found in ${\displaystyle S}$. 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 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.
• 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).
• In a Boolean ring, which has elements ${\displaystyle \{0,1\}}$ addition is often defined as the symmetric difference. So ${\displaystyle 0+0=0}$, ${\displaystyle 0+1=1}$, ${\displaystyle 1+0=1}$, and ${\displaystyle 1+1=0}$. Our additive identity is 0, and both elements are their own additive inverse as ${\displaystyle 0+0=0}$ and ${\displaystyle 1+1=0}$.

See also

• Absolute value (related through the identity |−x| = |x|).
• Monoid
• Inverse function
• Involution (mathematics)
• Multiplicative inverse
• Reflection (mathematics)
• Reflection symmetry
• Semigroup

Notes and references

1. Gallian, Joseph A. (2017). Contemporary abstract algebra (9th ed.). Boston, MA: Cengage Learning. p. 52. ISBN 978-1-305-65796-0.
2. Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 169–170. ISBN 978-1-292-02496-7.
3. Mazur, Izabela (March 26, 2021). "2.5 Properties of Real Numbers -- Introductory Algebra". Retrieved August 4, 2024.
4. "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.
5. Weisstein, Eric W. "Negative". MathWorld. Retrieved 2025-01-04.
6. Kinard, James T.; Kozulin, Alex (2008-06-02). Rigorous Mathematical Thinking: Conceptual Formation in the Mathematics Classroom. Cambridge University Press. ISBN 978-1-139-47239-5.
7. Brown, Christopher. "SI242: divisibility". www.usna.edu. Retrieved 2024-08-04.
8. "2.2.5: Properties of Equality with Decimals". K12 LibreTexts. 2020-07-21. Retrieved 2024-08-04.
9. Fraleigh, John B. (2014). A first course in abstract algebra (7th ed.). Harlow: Pearson. pp. 37–39. ISBN 978-1-292-02496-7.
10. 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.
11. Axler, Sheldon (2024), Axler, Sheldon (ed.), "Vector Spaces", Linear Algebra Done Right, Undergraduate Texts in Mathematics, Cham: Springer International Publishing, pp. 1–26, doi:10.1007/978-3-031-41026-0_1, ISBN 978-3-031-41026-0
12. Gupta, Prakash C. (2015). Cryptography and network security. Eastern economy edition. Delhi: PHI Learning Private Limited. p. 15. ISBN 978-81-203-5045-8.
13. 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.