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 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.
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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
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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