# Inverse of a number: find the multiplicative inverse of the number 3 + 2i. The multiplicative inverse is another name for the reciprocal. Both numbers are such that, if multiplied by the given number, they result in 1. The number 1 is special with regard to the operation of multiplication because if a number is multiplied by 1, it remains the same.

As an...

The multiplicative inverse is another name for the reciprocal. Both numbers are such that, if multiplied by the given number, they result in 1. The number 1 is special with regard to the operation of multiplication because if a number is multiplied by 1, it remains the same.

As an aside, the operation of addition has a similar such number, which is zero. Adding a zero to a number results in the same number. Additive inverse, also called opposite, is a number such that its addition to the given number results in zero.

In this case, we need to find a number which will result in 1 when multiplied by the complex number 3 + 2i. (Recall that i is the imaginary number such that i^2 = -1.) In other words, we need to find 1/(3 + 2i).

In order to perform the division by a complex number, we need to multiply both the top and bottom of the fraction by its conjugate, 3 - 2i:

1/(3 + 2i) = (3 - 2i)/[(3+2i)(3-2i)].

The product on the bottom can be simplified using the "difference of two squares" formula, (a+b)(a-b) = a^2 - b^2, and the result is

(3+2i)(3-2i) = 9-(2i)^2 = 9 - (-4) = 13.

Thus, 1/(3+2i) = (3-2i)/13 = 3/13 - (2/13)i.

The complex number 3/13 -(2/13)i is the multiplicative inverse of the number 3 + 2i.

Approved by eNotes Editorial Team The multiplicative inverse of a number A is another number B which when multiplied by A gives 1.

Here we have the number 3 + 2i

Let A = 3 + 2i, we need to find B such that A*B = 1

=> B = 1/(3 + 2i)

=> B = (3 - 2i)/(3 + 2i)(3 - 2i)

=> 3 - 2i /( 9 + 4)

=> 3/13 - 2i/13

The multiplicative inverse is 3/13 - 2i/13

Approved by eNotes Editorial Team