The multiplicative inverse of a negative number must also be a negative number. By definition, the product of a number and its multiplicative inverse is (positive) 1, which cannot be achieved by multiplying a positive and a negative number together.
The multiplicative inverse is also known as the reciprocal of a given number. All real numbers have reciprocals with the exception of zero, since anything multiplied by zero is also zero. The product of a negative number multiplied by its reciprocal (which is also a negative number) is also 1.
The reciprocal of a number expresses the inverse property of multiplication. For the variable a it can be written as:
- a x 1⁄a = 1⁄a x a = 1, a ≠0
- -a x -1⁄a = -1⁄a x -a = 1, a ≠0
Division is defined in terms of multiplication. In fact, division is equivalent to multiplication by the reciprocal or multiplicative inverse of a given number. Division and multiplication are therefore considered inverse operations to each other, just as addition and subtraction are inverse operations. If a number x is multiplied by a number y, then to end back with the result x, the product must be divided again by y - or multiplied by 1≠y. This demonstrates the multiplicative inverse relationship of x and its reciprocal 1≠x.