Additive inverse

In mathematics, the additive inverse of a number $a$ (sometimes called the opposite of $a$ )^[1] is the number that, when added to $a$ , yields zero. The operation taking a number to its additive inverse is known as sign change^[2] or negation.^[3] For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.

The additive inverse of $a$ is denoted by unary minus: $- a$ (see also § Relation to subtraction below).^[4] For example, the additive inverse of 7 is −7, because 7 + (−7) = 0, and the additive inverse of −0.3 is 0.3, because −0.3 + 0.3 = 0.

Similarly, the additive inverse of $a - b$ is $-(a - b)$ which can be simplified to $b - a$ . The additive inverse of 2x − 3 is 3 − 2x, because 2x − 3 + 3 − 2x = 0.^[5]

The additive inverse is defined as its inverse element under the binary operation of addition (see also § Formal definition below), which allows a broad generalization to mathematical objects other than numbers. As for any inverse operation, double additive inverse has no net effect: $-(- x) = x$ .

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

^ Tussy, Alan; Gustafson, R. (2012), Elementary Algebra (5th ed.), Cengage Learning, p. 40, ISBN 9781133710790.
^ Brase, Corrinne Pellillo; Brase, Charles Henry (1976). Basic Algebra for College Students. Houghton Mifflin. p. 54. ISBN 978-0-395-20656-0. ...to take the additive inverse of the member, we change the sign of the number.
^ The term "negation" bears a reference to negative numbers, which can be misleading, because the additive inverse of a negative number is positive.
^ Weisstein, Eric W. "Additive Inverse". mathworld.wolfram.com. Retrieved 2020-08-27.
^ "Additive Inverse". www.learnalberta.ca. Retrieved 2020-08-27.

[1] Tussy, Alan; Gustafson, R. (2012), Elementary Algebra (5th ed.), Cengage Learning, p. 40, ISBN 9781133710790.

[2] Brase, Corrinne Pellillo; Brase, Charles Henry (1976). Basic Algebra for College Students. Houghton Mifflin. p. 54. ISBN 978-0-395-20656-0. ...to take the additive inverse of the member, we change the sign of the number.

[3] The term "negation" bears a reference to negative numbers, which can be misleading, because the additive inverse of a negative number is positive.

[4] Weisstein, Eric W. "Additive Inverse". mathworld.wolfram.com. Retrieved 2020-08-27.

[5] "Additive Inverse". www.learnalberta.ca. Retrieved 2020-08-27.

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