Complex arithmetic
Use i² = −1 and collect real and imaginary parts.
Read the lesson: Multiplying and dividing complex numbers
A complex number a + bi has real part a and imaginary part b, both real numbers. Multiply brackets as in algebra, then replace i² by −1. The conjugate a − bi changes the sign of the imaginary part.
To divide by a nonzero complex number, multiply top and bottom by its conjugate. The denominator becomes a² + b². Keep the modulus distinct from the real part: |a + bi| = √(a² + b²).
Worked example
Calculate (2 + 3i)(1 − 2i).
- Expand to 2 − 4i + 3i − 6i².
- Replace −6i² by +6.
- Collect real and imaginary terms.
Answer: 8 − i
Revise first: Expanding and factorising.
Practise complex arithmetic
Course mapping
These specification references show where the topic occurs. The questions cover only some parts of each topic.
- 9FM0 · core · 2 Complex numbers: Complex number arithmetic
- 7367 · core · B Complex numbers: Complex number arithmetic
- H245 · core · 2 Complex numbers: Complex number arithmetic
- H645 · core · Complex numbers: Complex number arithmetic
- 8FM0 · core · Complex numbers: Complex number arithmetic
- 7366 · core · Complex numbers: Complex number arithmetic
- H235 · core · Complex numbers: Complex number arithmetic
- H635 · core · Complex numbers: Complex number arithmetic
- 9FM0 · core · 2 Complex numbers: Dividing complex numbers
- 7367 · core · B Complex numbers: Dividing complex numbers
- H245 · core · 2 Complex numbers: Dividing complex numbers
- H645 · core · Complex numbers: Dividing complex numbers
- 8FM0 · core · Complex numbers: Dividing complex numbers
- 7366 · core · Complex numbers: Dividing complex numbers
- H235 · core · Complex numbers: Dividing complex numbers
- H635 · core · Complex numbers: Dividing complex numbers
Next practice: Modulus and argument.