Flip the sign, get a real product: The conjugate of z = a + bi is z* = a − bi — just flip the sign of the imaginary part.
The useful bit: z × z* = a² + b², which is always a real number (the i's cancel).
IB-style question — conjugate and product
Let z = 3 + 2i.
Write down z and find z × z.
Step by step
- Flip the sign of the imaginary part for the conjugate.
- Multiply z by z* — the cross terms cancel and i² → −1.
- A real answer.
Final answer
z = 3 − 2i and z × z = 13.
z* is the mirror image of z in the real axis — same real part, opposite imaginary part.
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Make the bottom real: You can't leave an i on the bottom. Multiply top and bottom by the conjugate of the bottom — that turns the denominator into a real number, and the rest is easy.
IB-style question — divide
Express (5 + i)/(2 − 3i) in the form a + bi.
Step by step
- Multiply top and bottom by the conjugate of the bottom, (2 + 3i).
- Top: expand. Bottom: a² + b² = 4 + 9 = 13 (real).
- Split into a + bi form.
Final answer
7/13 + (17/13)i.