Inverses

The additive inverse of a number undoes the effect of adding that number. This means that, for example, the effect of adding a number by subtracting the same number can be undone. So, if 7 is added to 4, the result is 11. If 7 is subtracted from 11, the addition is undone, and the result is 4.

4 + 7 = 11 and 11 −; 7 = 4

Multiplication and division are related in the same way: They are inverse operations that undo each other. If 6 is multiplied by 3, the result is 18. Then 18 can be divided by 3 to undo the multiplication, and the result is 6.

6 × 3 = 18 and 18 ÷ 3 = 6

Most mathematical operations have an inverse operation that undoes or reverses its effect. The squaring of a number, for example, as in 7² = 49, can be undone by taking the square root. The square root of 49 is 7. Thus, squaring and taking the square root are also inverse operations.

Inverse operations can also be used to find the additive inverse of a specific number. For example, −9 is the additive inverse of 9 since the sum of −9 and 9 is 0. Additive inverses come in pairs; 9 is the additive inverse of −9, just as −9 is the additive inverse of 9. Any two numbers are additive inverses if they add up to 0.

Visualize a pair of additive inverses on the number line. The number 9 and its additive inverse −9 are both nine units away from 0 but on opposite sides of 0. For this reason, −9 is called the opposite of 9, and 9 is the opposite of −9. The opposite of a number may be positive or negative. The opposite of −4 is 4, a positive number. The opposite of 8 is −8, a negative number. The number −2 can be read as "the opposite of 2" or as "negative 2."

A negative number is always to the left of 0 on a number line. Every number on the number line has an additive inverse. The additive inverse of 0 is 0 because 0 + 0 = 0.

Multiplicative inverses come in pairs also. Any two numbers are multiplicative inverses if they multiply to 1. For example, because the product of 3 and ⅓ is 1, 3 and ⅓ are multiplicative inverses. In the same way, ⅔ and are multiplicative inverses because their product is 1.

Decimal numbers also have inverses, of course. The decimal number 0.04 can be written as the fraction , so the multiplicative inverse of 0.04 is , which equals 25. This can be checked by multiplying 0.04 and 25 to verify that the product is indeed 1. The additive inverse of 0.04 is - 0.04 because these two numbers add to 0.

A pair of numbers that multiply to 1, such as ⅓ and 3 or 0.04 and 25, are also called reciprocals. To find the reciprocal of any number, write 1 over that number. Thus ⅓ is the reciprocal of 3. One shortcut for finding the reciprocal of a fraction is to "flip" the fraction. The reciprocal of ⅔ is ; the reciprocal of is . Pairs of reciprocals always multiply to give a product of 1.

Every number on the number line, except 0, has a multiplicative inverse. The multiplicative inverse of 0, or the reciprocal of 0, is undefined because division by 0 is undefined.

see also Integers; Mathematics, definition of; Numbers, real.

Lucia McKay

Bibliography

Anderson, Raymond. Romping Through Mathematics. New York: Alfred A. Knopf, 1961.

SUMMARIZING INVERSES

Inverses can be summarized in a few sentences.

Additive inverses add to 0. The additive inverse of a number is the opposite of the number. The additive inverse of a positive number is a negative number.

Multiplicative inverses multiply to 1. A number and its multiplicative inverse are called reciprocals. The reciprocal of a number is 1 divided by that number. The reciprocal of a fraction is the same as the fraction flipped.