Formulas
- (a + bi) × (c + di) = (ac − bd) + (ad + bc)i
- (a + bi) ÷ (c + di) = [(ac + bd) + (bc − ad)i] ÷ (c² + d²)
- Modulus |z| = √(a² + b²); argument θ = atan2(b, a)
- Polar form: z = |z| (cos θ + i sin θ)
Example
With z₁ = 3 + 4i and z₂ = 1 − 2i: z₁ × z₂ = 11 − 2i, z₁ ÷ z₂ = −1 + 2i, |z₁| = 5 and the argument of z₁ is 53.13°.
Remember
i² = −1. To divide, multiply the top and bottom by the conjugate of the denominator.
Frequently asked questions
What is the conjugate?
The same number with the sign of the imaginary part flipped: the conjugate of 3 + 4i is 3 − 4i. A number times its conjugate gives a real number.
Is the argument in degrees or radians?
Both are shown. The argument goes from −180° to 180°.
See also
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