Half a Turn Is e to the iπ

eⁱᵗ describes a motion: the velocity is always the position times i, which means at right angles to the radius and of length 1. Such a point walks round the unit circle at a steady pace, so after time t it has covered an arc t. Half the circle is an arc of π, so e^(iπ) = −1.

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Chapters · 6

01Times i: a Quarter Turn
02Growth at the Rate of Itself
03Velocity at Right Angles
04A Hundred Small Steps
05Rotation as a Motion
06The Bench

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01

Times i: a Quarter Turn

A complex number a + bi is a point in the plane: a across, b up. By the rule for multiplying, i·(a + bi) = −b + ai. For example, 2 + i times i is −1 + 2i.

The new point is as far from the origin as before, and its direction has turned 90° anticlockwise. Every complex number behaves this way under i. Twice is half a turn, so i·i = −1; four times brings it home.

The brass arm on the instrument can extend. It points at z, and each multiplication by i turns it 90° about the centre, its length unchanged.

02

Growth at the Rate of Itself

Start at 1 and let a quantity grow at a rate always equal to its size: at 2 it grows at rate 2, at 3 at rate 3. After one unit of time it has reached e ≈ 2.718. This is eˣ: the function whose rate of change is itself.

Compound interest is the same thing in steps. Lend 1 at 100% a year, paid once, and you get 2; paid twice at 50%, 2.25; monthly, 2.613; daily, 2.7146. The more payments n, the closer (1 + 1/n)ⁿ comes to e.

Each link of the brass rule is one step: 1/n of the size so far, in the direction of the position. So each link is a little longer than the one before.

03

Velocity at Right Angles

The rule of eˣ is that velocity equals position. Replace x with it, and by the rules of differentiation the velocity of eⁱᵗ is i times its position.

Times i is a 90° turn, so the velocity is always at right angles to the radius. The start is 1, of length 1, so the velocity has length 1 too. The point always moves sideways, never away from the centre and never towards it: it walks the unit circle at speed 1.

At speed 1, after time t the arc covered is t. The unit circle is 2π round, so half of it is π. At t = π/2 the point is at i; at t = π it is on the far side, e^(iπ) = −1; at t = 2π it is back at 1.

04

A Hundred Small Steps

eⁱᵗ can also be worked out in steps, like compound interest. For t = π it is (1 + iπ/n)ⁿ: start at 1, and at each step add iπ/n times where you are.

Adding iπ/n times z means moving at right angles to the radius, by π/n of its length. Each step turns by the same small angle and stretches a little. Each link of the brass rule is one such step.

With n = 1 it is one big step, from 1 straight to 1 + πi. With n = 4 the four links already pass halfway round. With n = 100 the rule almost lies on the unit circle and ends just 0.0506 from −1. The gap is about π²/(2n): double n and it halves.

The end always lands a little outside −1. Each step stretches by √(1 + π²/n²), which over n steps comes to about 1 + π²/(2n); the angle turned falls short of π by only about π³/(3n²), far less.

05

Rotation as a Motion

The across of eⁱᵗ is cos t and the up is sin t, so eⁱᵗ = cos t + i sin t. The two brass riders on the real and imaginary axes follow the shadows of the arrow’s tip.

Change the speed and the motion changes with it: e²ⁱᵗ turns twice as fast, e⁻ⁱᵗ turns the other way, and r·eⁱᵗ walks a circle of radius r.

Join many such arrows head to tail, each turning at a whole-number speed, and the last tip can draw any closed line. That is a Fourier series.

06

The Bench

t runs from 0 to 2π and the arrow points at eⁱᵗ; it also turns by hand on the slate. The brass rule folds to (1 + it/n)ⁿ, with n from 1 to 100. The readout is the distance between the two.

Try thisWith t = 2π, where does the arrow stop? And with t = π/2?

What the model leaves out

The instrument is a model of the idea. The two riders follow the shadows of the arrow’s tip; a real instrument would need linkages for that, left out here. Every link of the rule is placed exactly at (1 + w/n)ᵏ, and its width is only for seeing; with n = 100 each link is about 1.6 mm long. Time t has no units; the scene runs at about 0.8 per second. eⁱᵗ is computed from cos t and sin t, and (1 + it/n)ⁿ by repeated complex multiplication, each without the other.

Two common misreadings

e^(iπ) is not e multiplied by itself iπ times: “how many times” only makes sense for whole numbers. eⁱᵗ means the motion that starts at 1 with velocity always i times its position; it is also the limit of (1 + it/n)ⁿ as n grows, and the series 1 + it + (it)²/2! + (it)³/3! + …, and all three give the same number. And a rule with finitely many steps is not on the unit circle: each step leans slightly outwards, so it ends outside −1, closer as n grows.

Sources

Data

Leonhard Euler printed e^(+v√−1) = cos v + √−1 · sin v in 1748, in the Introductio in analysin infinitorum, where he defines eˣ through (1 + x/n)ⁿ with n infinitely large. Jacob Bernoulli met the limit of (1 + 1/n)ⁿ in 1683, in a question about compound interest. Reading eⁱᵗ as a motion whose velocity is i times its position is the approach of Grant Sanderson (3Blue1Brown) in “e^(iπ) in 3.14 minutes, using dynamics”; his videos are under Watch next. The slate instrument and its jointed rule are our own model of the idea.