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