Module 11 · Lesson 11.1

Complex Numbers as 2D Rotations

A complex number is a point on a plane, and multiplying by i is just a 90-degree spin around the origin — nothing more mysterious than that.

Intuition

Plot a complex number $a + bi$ as the point $(a, b)$: the real part is how far right, the imaginary part is how far up. Once you see it that way, multiplication by $i$ stops looking like an algebra trick — it's a rotation. Take any point, spin it a quarter turn counterclockwise around the origin, and you've just multiplied it by $i$.

This is why complex numbers show up anywhere a phase or rotation needs tracking, like the phase shift of an audio signal: rotating the "angle" of a wave is literally multiplying its complex representation by a number on the unit circle.

Core concept

Multiplying $a + bi$ by $i$ swaps the real and imaginary parts and flips a sign: $i(a + bi) = ai + bi^2 = -b + ai$. The point $(a, b)$ becomes $(-b, a)$ — exactly a 90° rotation.

3+2i -2+3i
z = 3 + 2i(a=3, b=2)
i · z = 3i + 2i²3i − 2
Simplify−2 + 3i
Answer: new point(−2, 3)

Interactive Lab

Drag the angle slider to rotate the point $z = 3 + 2i$ around the origin. Watch what happens right at 90° — that's exactly multiplication by $i$.

w = 3.00 + 2.00i Aligned with ×i at θ = 90°

Key Idea

$$ i \cdot (a + bi) = -b + ai $$

More generally, multiplying any complex number by $\cos\theta + i\sin\theta$ rotates it by angle $\theta$ around the origin without changing its distance from the origin. Multiplying by $i$ is just the special case $\theta = 90°$.

$a, b$The real and imaginary parts of the starting point
$i$The imaginary unit, defined so that $i^2 = -1$
$-b + ai$The same point after a 90° counterclockwise turn
$\theta$The rotation angle when multiplying by a point on the unit circle

Think Further

  1. What point do you land on if you multiply by $i$ twice in a row, and how does that match $i^2 = -1$?
  2. If multiplying by $i$ is a 90° rotation, what rotation does multiplying by $-i$ perform?
  3. An audio signal's phase shift is applied by multiplying by $\cos\theta + i\sin\theta$. Why does this leave the signal's amplitude (distance from origin) unchanged?
Show suggested answers
  1. Two 90° rotations land you 180° from where you started, which is the same as flipping the point to its negative — exactly what multiplying by $i^2 = -1$ does.
  2. Multiplying by $-i$ rotates 90° in the opposite direction (clockwise), since $-i$ sits at $-90°$ on the unit circle.
  3. $\cos\theta + i\sin\theta$ always has distance 1 from the origin, and multiplying two complex numbers multiplies their distances — so multiplying by a distance-1 number leaves distance unchanged, only the angle shifts.