Students meeting complex numbers for the first time usually find addition and multiplication easy enough. Division is where confidence drops, because the answer has to be written in the form a + bi, and the denominator contains an imaginary part that has no obvious way of disappearing. The standard method, multiplying by the conjugate, is easy to memorise and easy to misapply. This guide explains why it works and gives a routine that keeps the signs straight.
The problem with an imaginary denominator
Consider (3 + 2i) ÷ (1 − 4i). We would like the result as a single number with a real part and an imaginary part, but the denominator 1 − 4i contains i. In ordinary arithmetic we never leave a square root of a negative number sitting in a denominator, and the same spirit applies here: we want a real denominator.
The conjugate
The conjugate of a + bi is a − bi. Only the sign of the imaginary part changes. The conjugate of 1 − 4i is 1 + 4i. The key property is that a number times its conjugate is always real:
(a + bi)(a − bi) = a² − (bi)² = a² + b²
because i² = −1. So (1 − 4i)(1 + 4i) = 1 + 16 = 17, a plain positive number.
Why multiplying top and bottom is allowed
Multiplying the numerator and denominator by the same non-zero number does not change the value of a fraction. We are multiplying by (1 + 4i)/(1 + 4i), which equals 1. The trick changes how the number looks, not what it is.
The routine
- Write the division as a fraction.
- Find the conjugate of the denominator.
- Multiply numerator and denominator by it.
- Expand the numerator carefully, replacing i² with −1.
- Simplify the denominator to a real number.
- Split into real and imaginary parts and reduce.
Worked example
Compute (3 + 2i) ÷ (1 − 4i).
Multiply top and bottom by 1 + 4i.
Numerator: (3 + 2i)(1 + 4i) = 3 + 12i + 2i + 8i² = 3 + 14i − 8 = −5 + 14i.
Denominator: (1 − 4i)(1 + 4i) = 1 + 16 = 17.
Result: (−5 + 14i)/17 = −5/17 + (14/17)i.
Checking the answer
Multiply the result by the divisor and confirm that you return to the dividend. (−5/17 + 14i/17)(1 − 4i) = −5/17 + 20i/17 + 14i/17 − 56i²/17 = (−5 + 56)/17 + 34i/17 = 51/17 + 2i = 3 + 2i. It matches.
A second example with a pure imaginary divisor
Compute 6 ÷ (2i). The conjugate of 2i is −2i. Multiply: 6(−2i) / (2i)(−2i) = −12i / 4 = −3i. So 6 ÷ 2i equals −3i. You can check it: −3i × 2i = −6i² = 6.
Common mistakes
- Changing the sign of the real part as well as the imaginary part when forming the conjugate.
- Forgetting that i² = −1 and leaving terms like 8i² unsimplified.
- Multiplying only the denominator by the conjugate.
- Sign errors when multiplying a negative imaginary part, such as 2i × −4i.
- Forgetting to divide both the real and imaginary parts by the denominator.
A geometric picture
Complex numbers can be plotted on a plane, with the real part along the horizontal axis and the imaginary part along the vertical. Multiplying by a complex number scales and rotates; dividing by one undoes that scaling and rotation. The conjugate is the reflection across the real axis, and a number times its conjugate is the square of its distance from the origin. This explains why the denominator becomes a² + b².
Where this appears
Division of complex numbers appears in electrical engineering, where impedances are complex, in signal processing and in solving quadratic equations with negative discriminants. Being fluent with the routine saves time and errors in those settings.
Practice problems
Try (2 + i) ÷ (3 − i), (5 − 2i) ÷ (1 + i), 4i ÷ (2 − 2i) and 1 ÷ (3 + 4i). After working them by hand, a complex number long division calculator can show the real and imaginary parts step by step so you can see exactly where any difference arises.
Summary
Multiply by the conjugate to make the denominator real, expand carefully with i² = −1, split the result into real and imaginary parts and check by multiplying back. The trick is simple once you know why it works.