Dividing Complex Numbers

Dividing complex numbers means multiplying numerator and denominator by the complex conjugate of the denominator, turning the denominator real.

Divide (2 + 3i)/(1 − i). The denominator's conjugate is 1 + i; multiply both floors by it. Downstairs: (1 − i)(1 + i) = 1 + 1 = 2, real as promised. Upstairs: (2 + 3i)(1 + i) = 2 + 2i + 3i + 3i² = −1 + 5i. The quotient is (−1 + 5i)/2, standard form −1/2 + 5i/2. The move is rationalizing a denominator wearing new clothes: multiply by a disguised 1, chosen so the bottom's troublesome part multiplies itself away. There the radical vanished; here the imaginary part does.