ACT Preparation / Unit 7 ACT 0617 worked example, 60 to 90 seconds
Dividing Complex Numbers
The idea
To divide complex numbers, work in three steps. First, find the conjugate of the denominator: for c + di it is c − di. Second, multiply the numerator and the denominator by that conjugate; the denominator becomes the real number c² + d². Third, multiply out the numerator, replace i² with −1, and write the result as a + bi by dividing each part by the real denominator.
In practice
Worked example: (5 + 5i) ÷ (1 + 2i). The conjugate is 1 − 2i. The denominator becomes 1² + 2², which is 5. The numerator: (5 + 5i)(1 − 2i) = 5 − 10i + 5i − 10i² = 5 − 5i + 10 = 15 − 5i. Divide each part by 5: 3 − i.
Where people slip
Check by multiplying back: (3 − i)(1 + 2i) = 3 + 6i − i − 2i² = 5 + 5i, the original numerator. The conjugate works because (c + di)(c − di) has no imaginary part.
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