Chapter exploration

Complex numbers and the Argand plane

Explore

Use the worked example below, then change one input or assumption. Predict which part of the answer must change before calculating.

Explore this related example, then return to the prediction task above.

$4iz$ is $4z$ turned through $90^\circ$ anticlockwise. Slide to apply the turn, and read how far $4iz$ lands from $3w$ .

Turn applied to 4z0°

Try an example

turned 4z

20 + 8i

turned 4z − 3w

29 − 7i

Key idea

Equation (1) says $4iz$ and $3w$ differ by exactly $1+5i$ . The quarter-turn turns $4z$ into $4iz$ and closes that gap.

Ideas to try

Slide to $90^\circ$ . The amber point is $4iz=-8+20i$ , and the red dashed segment from $3w$ is the vector $1+5i$ .
Slide back to $0^\circ$ : the same segment no longer equals $1+5i$ , which is why the rotation matters.