Find the complex conjugate of #(3+2i)/(1-i)# ?

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Answer 1 · 2018-03-29T12:45:40

#frac(5)(2) - frac(5)(2)i#

The complex conjugate of a complex number is of the form #(a + bi)^(ast) = a - bi#.

We have: #frac(3 + 2i)(1 - i) = frac(3 + 2i)(1 + (- i))#

To perform division on complex numbers we multiply the numerator and denominator of the fraction by the denominator's complex conjugate.

This turns the denominator into a real number, allowing the fraction to be expressed in the form #a + bi#.

#= frac(3 + 2i)(1 - i) cdot frac(1 - (- i))(1 - (- i))#

#= frac(3 + 2i)(1 - i) cdot frac(1 + i)(1 + i)#

#= frac(3 + 3 i + 2i - 2i^(2))(1^(2) - i^(2))#

#= frac(3 - 2 (- 1) + (3 + 2)i)(1 - (- 1))#

#= frac(3 + 2 + 5i)(1 + 1)#

#= frac(5 + 5i)(2)#

#= frac(5)(2) + frac(5)(2)i#

#therefore (frac(5)(2) + frac(5)(2)i)^(ast) = frac(5)(2) - frac(5)(2)i#

Therefore, the complex conjugate of #frac(3 + 2i)(1 - i)# is #frac(5)(2) - frac(5)(2)i#.