Numbers & Functions · 8 Numbers That Shouldn't Exist

8.6 The harvest: why e knows about circles

Now we spring the surprise that’s been armed since Chapter 2. Hold the two facts we own: • From Chapter 5: ex is the purest growth machine — the one whose rate of change equals its own current value.

• From just now: multiplying by i means rotating.

Now ask the question that has no business having a nice answer: what is eiθ? What does the pure-growth machine do when fed an imaginary input? real imaginary r, α s, β rs, α+β α α+β Figure 8.4. Multiplication: distances multiply, angles add. The product of arrows at angles α and β lands at angle α + β and distance rs — the rigorous form of “multiplication is rotation.”

Ordinary ex grows in the direction it already points — straight outward. But i turns outward-pushing into sideways-pushing. So eiθ doesn’t grow outward — it’s pushed perpendicular to where it points, at every instant. And a point always pushed exactly sideways to its own direction, never outward, does not get bigger and does not get smaller. It goes in a circle. (A force always perpendicular to your motion is exactly what keeps a planet in orbit.)

So eiθ traces the unit circle. As θ increases, the point walks around the circle of radius 1 — and “the point on the unit circle at angle θ” is precisely the walker from Chapter 6, whose coordinates are (cos θ, sin θ). Written as a complex number, that walker is cos θ + i sin θ. Hence: eiθ = cos θ + i sin θ.

There it is — Euler’s formula, the equation I wrote on the page back in Chapter 6 and told you that you couldn’t read yet. Now you can. It says the growth machine, fed an imaginary input, is the going-around machine. Exponentials and trigonometry — which looked like opposite ends of this book — are the same machine, seen from two directions. The bridge between them is i. A second confirmation, pure algebra. We proved two facts from two directions: exponents add, eiα · eiβ = ei(α+β) (Chapter 5’s first law); and angles add, multiplying two unit-circle points adds their angles (just proven). These are the same rule. So whatever function carries angle θ to the unit-circle point at angle θ must behave like an exponential, because it obeys the exponential’s signature law. The walker and the growth machine aren’t equal by coincidence — they’re forced to be the same thing.

Numbers That Shouldn’t Exist real imaginary 1 eiθ cos θ sin θ θ Figure 8.5. Euler’s formula on the unit circle — deliberately the same picture as the Chapter 6 walker, now relabelled. eiθ is the walker; its real part is cos θ and its imaginary part is sin θ. The walker was a complex number all along.

And the most famous special case: walk half the circle, to θ = π. Then cos π = −1 and sin π = 0, so eiπ = −1, usually written eiπ + 1 = 0 — Euler’s identity, tying together e, i, π, 1, and 0, the five most important numbers in mathematics, in a single line.

real imaginary half a turn (θ = π) −1 eiπ = −1 Figure 8.6. Euler’s identity. Walking half the unit circle (θ = π) lands exactly on −1 — which is why eiπ = −1, i.e. eiπ + 1 = 0.

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