Quaternions· Visually
Home / Part A / 3

Warm-up: multiplying complex numbers is rotating in 2D

Lesson 3 · Interactive

In 2D, multiplying by a complex number of length 1 is a rotation. Quaternions grow from this one fact: they do the same in 3D. So we make sure it is completely clear in 2D first.

3.1A point is a number

A complex number \(p = a + b\,i\) is just a point in the plane, or the arrow from 0 to that point: a steps along the real axis, b steps along the imaginary axis. The only new rule is \(i^2 = -1\). The length of the arrow is \(\sqrt{a^2 + b^2}\).

Visual 1 of 4 · A point is a number
Try it: drag the point. A complex number is nothing more than this: two numbers, drawn as one point.

3.2Multiplying is rotating

We start with the simplest multiplication: by the number i.

Visual 2 of 4 · Times i is a quarter turn
Try it: press × i four times and read the numbers. Each press swaps the two parts and flips one sign, and the arrow makes a quarter turn.

So i is a turn by 90°. For other angles we take another number, r, that sits on the circle of radius 1, at the angle θ from the real axis:

\[r = \cos\theta + i\,\sin\theta\]

Its length is 1, because cos² θ + sin² θ = 1.

The rule
  • Multiplying any point p by r turns p by the angle θ around 0. The length of p does not change.
  • So the number r is a rotation. Its angle on the circle is the angle of the turn.
  • Do it to every point of a shape, and the whole shape turns.
Visual 3 of 4 · Times r is a turn by any angle
Try it: move the slider. The purple arrow r moves along the unit circle to the angle θ, and the orange house, which is the green house multiplied by r, turns by exactly the same θ. Press × i: the number i sits at 90° on the circle, so each press is a quarter turn. Drag the green point to check that it works for any point.

3.3Why it works

With numbers first. Take the point p = 2 + 1i, which is the arrow (2, 1), and multiply it by i:

\[(2 + 1i)\cdot i = 2i + i^2 = -1 + 2i\]

The arrow (2, 1) became (−1, 2): same length, turned by a quarter turn.

For any angle we multiply by \(r = \cos\theta + i\sin\theta\) the same way, and use \(i^2 = -1\):

\[(a + b\,i)(\cos\theta + i\sin\theta) = (a\cos\theta - b\sin\theta) + (a\sin\theta + b\cos\theta)\,i\]

The new point is x′ = a cos θ − b sin θ and y′ = a sin θ + b cos θ. This is exactly the formula for turning a point by θ in 2D.

So in 2D, a rotation is a number on the unit circle, and turning a vector is one multiplication. Two turns one after another are two multiplications, and the angles add up.

Visual 4 of 4 · Two turns: the angles add
Try it: r₁ and r₂ are two rotations. Their product is the orange arrow. Its angle is always the sum of the two.
Aha

What does \(i^2=-1\) mean? Multiplying by i turns by 90°. Doing it twice turns by 180°, and a half turn makes every vector point the other way, which is multiplying by −1. So \(i \cdot i = -1\) is just "two quarter turns are a half turn".

3.4From 2D to 3D

In 2D there is only one plane to turn in, so one imaginary unit i is enough. In 3D we can turn in three different planes, so we need three: i, j and k. In 1843 William Rowan Hamilton found the rules that make it work, and carved them into a bridge in Dublin:

\[i^2 = j^2 = k^2 = ijk = -1\]

A quaternion is one real part plus these three imaginary parts. We meet it properly in the next lesson.

✓Check yourself

What does multiplying by −1 do to a 2D vector?

It turns the vector by 180°.

What is \((\cos 30° + i\sin 30°)^3\)?

Three turns of 30°, so one turn of 90°: \(\cos 90° + i\sin 90° = i\).

+Sources and further reading

Next: Meet the quaternion: four numbers for one turn.