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Complex Number Division — Definition, Formula & Examples

Complex number division is the process of dividing one complex number by another by multiplying both the numerator and denominator by the conjugate of the denominator. This eliminates the imaginary part from the denominator, producing a standard a+bia + bi result.

Given two complex numbers z1=a+biz_1 = a + bi and z2=c+diz_2 = c + di where z2≠0z_2 \neq 0, their quotient is defined as z1z2=(a+bi)(c−di)(c+di)(c−di)=(ac+bd)+(bc−ad)ic2+d2\frac{z_1}{z_2} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}, obtained by multiplying numerator and denominator by the complex conjugate z2‾=c−di\overline{z_2} = c - di.

Key Formula

a+bic+di=ac+bdc2+d2+bc−adc2+d2 i\frac{a + bi}{c + di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}\,i
Where:
  • aa = Real part of the numerator
  • bb = Imaginary part of the numerator
  • cc = Real part of the denominator
  • dd = Imaginary part of the denominator

How It Works

You cannot simply divide the real and imaginary parts separately. Instead, multiply both the numerator and denominator by the conjugate of the denominator. The conjugate of c+dic + di is c−dic - di. Multiplying the denominator by its own conjugate gives (c+di)(c−di)=c2+d2(c + di)(c - di) = c^2 + d^2, which is always a real number. Then expand the numerator using FOIL, separate into real and imaginary parts, and divide each by c2+d2c^2 + d^2.

Worked Example

Problem: Divide (3+4i)(3 + 4i) by (1+2i)(1 + 2i).
Identify the conjugate: The conjugate of the denominator 1+2i1 + 2i is 1−2i1 - 2i.
1+2i‾=1−2i\overline{1 + 2i} = 1 - 2i
Multiply numerator and denominator by the conjugate: Multiply both top and bottom by 1−2i1 - 2i.
3+4i1+2i⋅1−2i1−2i\frac{3 + 4i}{1 + 2i} \cdot \frac{1 - 2i}{1 - 2i}
Expand and simplify: FOIL the numerator: (3)(1)+(3)(−2i)+(4i)(1)+(4i)(−2i)=3−6i+4i−8i2=3−2i+8=11−2i(3)(1) + (3)(-2i) + (4i)(1) + (4i)(-2i) = 3 - 6i + 4i - 8i^2 = 3 - 2i + 8 = 11 - 2i. The denominator becomes 12+22=51^2 + 2^2 = 5.
11−2i5=115−25i\frac{11 - 2i}{5} = \frac{11}{5} - \frac{2}{5}i
Answer: 115−25i\dfrac{11}{5} - \dfrac{2}{5}i

Why It Matters

Complex number division appears throughout electrical engineering when working with impedance in AC circuits, where voltage and current are represented as complex numbers. It is also essential in Algebra 2 and precalculus courses when simplifying rational expressions involving ii.

Common Mistakes

Mistake: Multiplying by the conjugate of the numerator instead of the denominator.
Correction: Always use the conjugate of the denominator. The goal is to make the denominator real, which only happens when you multiply (c+di)(c + di) by (c−di)(c - di).