Multiplying by i
Take z = 3 + i. Then iz . On the Argand diagram, the point (3, 1) has moved to (−1, 3).
Both points are from O, and the arrows to them are at right angles: multiplying by i turns z a quarter turn, 90°, counterclockwise about O. In general i(a + bi) = −b + ai, so the real and imaginary parts swap and the new real part changes sign.
Multiplying by −i turns the other way: −iz = 1 − 3i, a quarter turn clockwise.
Negating
−z = −3 − i. Both parts change sign, so the point moves to the opposite side of O, the same distance away. That is a half turn about O.
It agrees with the quarter turns: , so multiplying by −1 is two quarter turns.
Conjugating
z* = 3 − i. Only the imaginary part changes sign, so the point moves to the other side of the real axis: a reflection in the real axis.
A reflection is not a turn. −z and z* both change a sign, but −z changes both and z* changes one.
Adding and subtracting
Adding w = 1 + 2i adds 1 to the real part and 2 to the imaginary part, so z + w = 4 + 3i. Every point moves by the same step, the arrow of w: adding is a translation.
Subtracting w translates by the arrow of −w, 1 back and 2 down.
Two moves in a row
The order of two moves can matter. Turning first and then reflecting gives (iz)* = (−1 + 3i)* = −1 − 3i. Reflecting first and then turning gives i(z*) = i(3 − i) = 1 + 3i. These are different points.
The usual slips
Turning the wrong way: multiplying by i turns counterclockwise, and by −i clockwise.
Taking −z for iz. Negating is two quarter turns, not one.
Taking z* for a turn. The conjugate is a reflection in the real axis, and only the imaginary part changes sign.