Complex Numbers

Topic overview

In complex-number questions you need to switch between algebraic form, modulus–argument form and geometric loci. With each transformation, track the centre, radius, direction and boundary conditions at the same time.

Key concepts

$z^*$ reflects a point in the real axis; adding a constant represents translation; multiplying by a complex number represents scaling and rotation. $|z-a|=r$ is the circle with centre $a$ and radius $r$ .

Key points

◆The argument must be considered together with the quadrant
◆Conjugation changes the sign of the argument
◆The inequality determines which side of the circular arc or half-plane is taken

Worked examples

Try each example yourself before checking the explanation.

1Locus transformation

What are the centre and radius of $|z-2i|=3$ after $w=z^*-1$ ?

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