Relations and Functions

Relations and Functions

Question

Find the multiplicative inverse of the following complex numbers :

space space space space space space space space space space space space space space space space fraction numerator left parenthesis 2 plus 3 straight i right parenthesis space left parenthesis 3 space plus 2 straight i right parenthesis over denominator 5 space plus space straight i end fraction 

Answer
space space space space space space space space space space space space space space space space space space straight z space equals space fraction numerator left parenthesis 2 plus 3 straight i right parenthesis left parenthesis 3 plus 2 straight i right parenthesis over denominator 5 plus straight i end fraction equals fraction numerator 6 plus 4 straight i plus 9 straight i plus 6 straight i squared over denominator 5 plus straight i end fraction equals fraction numerator 13 straight i over denominator 5 plus straight i end fraction     ( i2 = -1)

Multiplicative inverse of space space space space space space space space space space space space space space space space space space space space space space space space space space straight z space equals space fraction numerator 5 plus straight i over denominator 13 straight i end fraction equals fraction numerator 5 plus straight i over denominator 13 straight i end fraction straight x straight i over straight i equals fraction numerator 5 straight i plus straight i squared over denominator negative 13 end fraction

                   space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space minus 5 over 13 straight i plus 1 over 13 equals 1 over 13 minus 5 over 13 straight i
                        

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