The complex conjugate of a complex number is of the form (a + bi)^(ast) = a - bi(a+bi)∗=a−bi.
We have: frac(3 + 2i)(1 - i) = frac(3 + 2i)(1 + (- i))3+2i1−i=3+2i1+(−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 + bia+bi.
= frac(3 + 2i)(1 - i) cdot frac(1 - (- i))(1 - (- i))=3+2i1−i⋅1−(−i)1−(−i)
= frac(3 + 2i)(1 - i) cdot frac(1 + i)(1 + i)=3+2i1−i⋅1+i1+i
= frac(3 + 3 i + 2i - 2i^(2))(1^(2) - i^(2))=3+3i+2i−2i212−i2
= frac(3 - 2 (- 1) + (3 + 2)i)(1 - (- 1))=3−2(−1)+(3+2)i1−(−1)
= frac(3 + 2 + 5i)(1 + 1)=3+2+5i1+1
= frac(5 + 5i)(2)=5+5i2
= frac(5)(2) + frac(5)(2)i=52+52i
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.
