Chapter 5

Spinning Numbers

What kind of number could square to −1, and why would it wiggle?

Numbers are arrows

Mika: Last time the spring needed a number with $a^2 = -4$ . Theo said no such number exists. But what would one even look like?

Picture every number as an arrow from 0 along the number line. Multiplying by 2 stretches the arrow. Multiplying by ½ shrinks it. And multiplying by −1 flips it to point the other way. That's a half turn.

June: Ooh. If ×(−1) is a half turn… what's half of a half turn? A quarter turn! The arrow would point straight up, off the number line.

Let's give the quarter turn a name: i. Do it twice and you've done a half turn, which is ×(−1). So $i \cdot i = -1$ . There's our number that squares to a negative!

Chapter 5: Spinning Numbers ↗

i^2 = -1 \qquad\text{(two quarter turns = a half turn)}

Theo: Hold on. These are the "imaginary numbers". Everyone knows they're made up. They can't describe a real, physical spring.

June: They're not imaginary, they're sideways! It's just a 2-D arrow. Turning left is as real as walking forward.

A number like $3 + 2i$ just means "3 steps along, 2 steps sideways". The whole plane of arrows is the stage for everything that follows.

Measuring turns

June: If i is a quarter turn, we're going to be spinning arrows all day. So how do we say how much an arrow has turned?

Degrees are an old habit: 360 of them in a turn, because an ancient calendar liked 360. Mathematicians measure a turn with the circle itself. Roll a wheel of radius 1 and see how far its rim travels. A radian is the angle whose arc is exactly one radius long.

Predict first! How many radians make one full turn?

Chapter 5: Spinning Numbers ↗

Follow the arrow's tip as it turns. Its shadow on the real axis is called the cosine of the angle, $\cos\theta$ ; its shadow on the sideways axis is the sine, $\sin\theta$ . Turn steadily and both shadows trace waves, the sine a quarter turn behind the cosine.

Mika: Got it. So a spinner at 2 radians per second goes round twice a second!

Theo: Twice a second would be 2 turns, $4\pi$ radians. Two radians per second is one full turn every $2\pi/2 \approx 3.14$ seconds.

New words

Radian: The angle whose arc is one radius long. A full turn is $2\pi$ radians, half a turn is $\pi$ .
Cosine and sine: The shadows of a length-1 arrow at angle θ: $\cos\theta$ along the real axis, $\sin\theta$ sideways.

A number that spins

Speeds of turning are measured in radians per second: a spinner at 2 radians per second makes one full turn in $2\pi/2 \approx 3.14$ seconds. The symbol ω tells us that turning speed.

Now picture a point running around a circle at a steady speed, ω radians per second. Where does its velocity point? (Velocity is Chapter 3's speed with a direction, so a moving arrow's velocity is an arrow too: which way its tip moves, and how fast.) Always along the circle, which is a quarter turn ahead of the arrow to the point.

A quarter turn is ×i, and the speed is ω times the radius. So the velocity is the position turned a quarter turn, times ω: velocity = iω × position.

Theo: Wait. "The slope is a fixed number times itself" is exactly the exponential property from Chapter 4! With a = iω.

Mika: But hang on. In Chapter 4 a power meant "how much multiplying has happened". What would $e$ to the power $i\theta$ even mean? Multiply by $e$ , $i\theta$ times?

June: Build it the Chapter 4 way. $e$ was $n$ tiny steps of growing by $\frac1n$ . Make each tiny step a little nudge sideways instead.

e^{i\theta} = \lim_{n\to\infty}\left(1 + \frac{i\theta}{n}\right)^{n}

Chapter 5: Spinning Numbers ↗

Each step turns the arrow by a tiny angle, about $\theta/n$ radians, and stretches it a tiny bit. With more steps the stretching fades away and the path lands on the circle, exactly θ radians round. That's why radians: the tiny sideways nudges add up along the rim. So $e^{i\theta}$ is simply the point on the circle at angle θ.

So $e^{i\omega t}$ is not scary at all. It's a point spinning around the circle of radius 1, ω radians every second. Watch it, and watch its shadow on the real axis:

Chapter 5: Spinning Numbers ↗

\begin{aligned}\frac{d}{dt}\,e^{i\omega t} &= i\omega\,e^{i\omega t} \\ \text{shadow on the real axis} &= \cos(\omega t)\end{aligned}

The shadow slides back and forth as a perfect cosine wave. A spinning arrow, seen side-on, is a wiggle. That's where the wiggle has been hiding.

What we know so far

What we figured out

Numbers are arrows. Multiplying rotates and stretches them. i is a quarter turn, so $i^2 = -1$ .
A radian is the angle whose arc is one radius long: a full turn is $2\pi$ . cos θ and sin θ are the shadows of a turned arrow.
$e^{i\theta}$ , built from tiny sideways steps $\left(1 + \frac{i\theta}{n}\right)^n$ , is the point on the circle at angle θ.
$e^{i\omega t}$ is a point spinning around a circle. Its shadow is a cosine wave.
A spinner's velocity is always a quarter turn ahead of its position: velocity $= i\omega \times$ position. That's the exponential rule with $a = i\omega$ .

Mika: "So the spring's mystery number exists after all: it's 2i. Does that mean a real spring swings sideways?"

Chapter 5: Spinning Numbers ↗