Dividing Complex Numbers Using Conjugates
Simple Explanation
To divide by a complex number, multiply both the numerator and denominator by the denominator's complex conjugate (same real part, opposite sign on the imaginary part). This clears the imaginary part out of the denominator entirely, since a number times its own conjugate is always a real number.
Why Do We Need It?
Division would otherwise leave an "i" sitting in the denominator, which isn't standard form — the conjugate trick is the standard, reliable way to fix that.
Formula
Dividing Complex Numbers Using Conjugates
(a+bi) / (c+di) = [(a+bi)(c−di)] / (c²+d²)
Multiply both the numerator and denominator by the denominator's conjugate (c−di) — this makes the denominator a real number (c²+d²), clearing i from it entirely.
- c−di
- — the complex conjugate of the denominator c+di — same real part, opposite sign on the imaginary part
- c²+d²
- — the result of (c+di)(c−di), always a real, non-negative number
When to use it: Whenever dividing by a complex number.
Worked Example
Divide two complex numbers using the conjugate
Divide (4+2i) by (1−i).
Why Does This Work?
For any complex number c+di, its conjugate is c−di, and (c+di)(c−di) = c²+d² — always a real, non-negative number, since it's a sum of two squares. Multiplying both the top and bottom of a fraction by the same quantity never changes its value, so this trick legally clears i from the denominator.
Real-Life Example
Finding current from voltage and impedance
In an AC circuit, current equals voltage divided by impedance — both are complex numbers.
The conjugate technique is exactly how engineers compute this division to find the actual current, in proper standard form.
Practice
Divide (3+i) by (1+i).
MediumCommon mistake
Multiplying only the numerator by the conjugate and forgetting the denominator — both must be multiplied by the same conjugate to keep the fraction's value unchanged.
Quick Review
- (a+bi)/(c+di) = [(a+bi)(c−di)] / (c²+d²).
- Multiply top and bottom by the denominator's conjugate.
- A number times its own conjugate is always a real number.