# Inverse of a number: find the multiplicative inverse of the number 3 + 2i.

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Inna Shpiro | Certified Educator

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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...

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justaguide | Certified Educator

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giorgiana1976 | Student

We'll get the multiplicative inverse when multiplying the given complex number by the inverse number we'll get the result 1.

We'll note the inverse as x:

(3 + 2i)*x = 1

We'll divide by (3 + 2i) both sides:

x = 1/(3 + 2i)

Since, it is not allowed to keep a complex number to denominator, we'll multiply the entire fraction by the conjugate of the complex number:

x = (3 - 2i)/(3 + 2i)(3 - 2i)

We'll re-write the denominator as a difference of squares:

(3 + 2i)(3 - 2i) = 3^2 - (2i)^2

(3 + 2i)(3 - 2i) = 9 - 4i^2

But i^2 = -1:

(3 + 2i)(3 - 2i) = 9 + 4

(3 + 2i)(3 - 2i) = 13

We'll re-write x:

x = (3 - 2i)/13

x = 3/13 - 2i/13

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

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