AcademyComplex Arithmetic
Academy
Complex Division
Level 1 - Math I (Physics) topic page in Complex Arithmetic.
Dividing Complex Numbers
Complex division requires eliminating the imaginary part from the denominator.
Division Formula
For \(z_1 = a + bi\) and \(z_2 = c + di \neq 0\):
Division
\[\frac{z_1}{z_2} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}\]
Method: Using the Conjugate
Multiply numerator and denominator by the conjugate of the denominator:
ConjugateMethod
\[\frac{z_1}{z_2} = \frac{z_1 \cdot z_2^*}{z_2 \cdot z_2^*}\]
Reciprocal of a Complex Number
Reciprocal
\[\frac{1}{a + bi} = \frac{a - bi}{a^2 + b^2}\]
The denominator \(a^2 + b^2 = |z|^2\) is always real and positive.
Verification
Verify
\[(a + bi)\frac{a - bi}{a^2 + b^2} = \frac{a^2 + b^2}{a^2 + b^2} = 1\]
Example
Example
\[\frac{1 + i}{2 - i} = \frac{(1 + i)(2 + i)}{(2 - i)(2 + i)} = \frac{2 + i + 2i + i^2}{4 + 1} = \frac{1 + 3i}{5} = \frac{1}{5} + \frac{3}{5}i\]