Complex arithmetic and the conjugate
Add, subtract and multiply as if i were an algebraic letter, then replace every by −1. Division uses the conjugate to make the denominator real. Finish every answer in form.
Collect into a plus bi after every operation
- Add or subtract real parts together and imaginary parts together.
- For multiplication, expand both brackets fully.
- Finish in Cartesian form with no powers of i above 1.
Worked example
Multiply in Cartesian form
Answer
The conjugate clears a complex denominator
Conjugation reflects in the real axis
ReIm
z = 2 + 3i
z̄ = 2 − 3i
same real part and modulus · opposite Im and argument
Definition
What is meant by complex conjugate?
Model answer: If , then its conjugate is . Only the sign of the imaginary part changes.
Division by zero is not defined, so the complex denominator must be non-zero.
Worked example
Divide by multiplying by the denominator's conjugate
Answer
Key idea
The conjugate is not chosen at random. The cross terms cancel, leaving the real number .
Use a structural check before trusting the answer
If , then must return z. This catches a wrong conjugate or a sign error using only the arithmetic you already know.
Your turnIndependent arithmetic check [4 marks]
Express in Cartesian form.
Show worked answer
Answer
Check:
Common mistake
Use the conjugate of the denominator, not the numerator, and multiply both numerator and denominator by it.