College Algebra / Unit 4 Alg 188 worked example, 60 to 90 seconds
Dividing Complex Numbers
The idea
Dividing complex numbers means multiplying numerator and denominator by the complex conjugate of the denominator, turning the denominator real.
In practice
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.
Where people slip
Conjugate the denominator only; the numerator rides along unconverted.
Your place on the trail
Every lesson rests on earlier ones and opens later ones. Tap any to jump.
Builds on
- Nothing. This is a starting point.
Unlocks
- Nothing yet depends on this.
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