What each operation does on the Argand diagram

By the end of this lesson

By the end you can look at a rule such as $z \mapsto (1 + i)z$ and say exactly what it does to every point on the Argand diagram: reflect, slide, enlarge, shrink or turn — and by how much.

Key to the symbols

Symbol	Meaning
$z$	the moving point: a general complex number
$w$	a fixed complex number used by the transformation
$\lvert w \rvert$	the modulus of $w$ : the length of its arrow
$\arg w$	the argument of $w$ : its angle from the positive real axis
$z \mapsto f(z)$	the rule that sends each point $z$ to the point $f(z)$
image	the point you land on after the rule is applied

A transformation here is just a rule applied to every point at once. If the rule sends $z$ to $(1 + i)z$ , every point in the plane moves according to that same recipe.

The whole topic in one table

Rule	What happens on the diagram
$z \mapsto z^*$	reflection in the real axis
$z \mapsto z + w$	slide (translate) every point by the arrow $w$
$z \mapsto z - w$	slide every point by the arrow $-w$ . Also, $z - w$ is the arrow from $w$ to $z$ , and its length is the distance between them
$z \mapsto wz$	scale by $\lvert w \rvert$ and turn anticlockwise by $\arg w$
$z \mapsto z/w$	scale by $1/\lvert w \rvert$ and turn clockwise by $\arg w$

The rest of the lesson explains where each row comes from. Nothing here needs memorising: every row follows from adding or multiplying two complex numbers.

Conjugating reflects in the real axis

If $z = x + iy$ , then $z^* = x - iy$ .

The real part $x$ does not change, so the point stays at the same distance across. The imaginary part changes from $y$ to $-y$ , so the point flips to the other side of the horizontal axis.

That is a reflection in the real axis, and it explains three facts immediately.

$\lvert z^* \rvert = \lvert z \rvert$ , because a mirror image is the same distance from the origin.
$\arg z^* = -\arg z$ , because the angle flips sign.
$z + z^* = 2x$ is real, and $z - z^* = 2iy$ is purely imaginary, because the vertical parts either double or cancel.

One small exception: a point on the negative real axis, such as $-3$ , has $\arg z = \pi$ . Its conjugate is itself, so $\arg z^* = \pi$ too, not $-\pi$ . The two angles name the same direction.

Quick check on conjugation

Multiple choiceQuestion 1 · Recall

Answer the question above to continue.