eiπ + 1 = 0
In The Most Beautiful Ideas in Math this equation was the last exhibit in the gallery: five fundamental numbers, from five different corners of mathematics, locked into one line. That piece said you don't need to understand why it is true to feel that it shouldn't be. This piece is for when you do want the why.
The secret is that the equation isn't really about numbers at all. It is about turning. Once you see multiplication as a kind of motion, the equation stops being a magic trick and becomes something close to obvious.
Picture the numbers on a line, with 0 in the middle. Multiplying by 2 stretches everything away from 0. Multiplying by −1 flips everything to the other side: 3 goes to −3, −5 goes to 5. You can think of that flip as a rotation: swing the whole line half a turn around 0.
Now the old puzzle: what number, multiplied by itself, gives −1? If −1 is a half-turn, the answer is easy: a quarter-turn. Do it twice and you've done a half-turn. That quarter-turn is i. It doesn't live on the number line. It lives straight "up" from 0, which turns the line into a whole plane of numbers. Every point in that plane is a number like 3 + 2i: three steps right, two steps up.
i isn't imaginary in the sense of fake. It's a direction the number line didn't know it had.
In this plane, every multiplication does two things at once: lengths multiply and angles add. Multiplying by i keeps the length and adds 90°. Multiplying by 1 + i (which points at 45° and has length √2 ≈ 1.414) stretches by 1.414 and turns by 45°. Try it.
Press × i four times and you come back to 1: four quarter-turns make a full circle. Press it twice and you land on −1, which is the whole content of i2 = −1. Press × (1 + i) eight times and you also come back to the positive side, but much further out: 8 turns of 45° make 360°, and the length grows to √28 = 16.
Now the other ingredient. Put $1 in an account paying 100% interest a year. Paid once, you end with $2. Paid as two half-year instalments of 50%, you get 1.5 × 1.5 = $2.25. Split the year into more and more pieces and the total creeps up, but never past a ceiling:
| Interest paid… | Formula | After one year |
|---|---|---|
| once | (1 + 1/1)1 | 2 |
| twice | (1 + 1/2)2 | 2.25 |
| 10 times | (1 + 1/10)10 | 2.5937… |
| 100 times | (1 + 1/100)100 | 2.7048… |
| 1,000 times | (1 + 1/1000)1000 | 2.7169… |
| continuously | the limit | e = 2.71828… |
That ceiling is e. More generally, ex is what you get from (1 + x/n)n when n is huge: growth at rate x, compounded in countless tiny steps. (For why this kind of growth runs the world, see Why Exponential Growth Rules Everything.)
Here is the leap. What happens if the "interest rate" is iθ instead of an ordinary number? Then every one of the tiny steps multiplies by 1 + iθ/n. That is a number pointing almost straight ahead with a tiny sideways nudge. By Step 3, multiplying by it barely stretches, and it turns by a tiny angle of about θ/n. Do that n times and you have turned by θ in total, while the stretching fades away as the steps get smaller.
So compounding at an imaginary rate doesn't grow. It rotates. Watch it happen:
With one step and θ = π, (1 + πi) lands far away, at 1 + 3.14i. With 10 steps the path already curls around towards the left, ending at about −1.59 + 0.16i. With 100 steps it ends at −1.051, with 1,000 at −1.005, and the path hugs the circle of radius 1 more and more tightly. In the limit, it lands exactly on the point of the circle at angle θ, measured as distance travelled around the circle.
That is Euler's formula: eiθ = cos θ + i sin θ, the point on the unit circle at angle θ. Halfway around a circle of radius 1 is a distance of π. So go halfway round from 1 and you arrive at −1:
eiπ = −1, so eiπ + 1 = 0
One more way to feel it. Growth at an ordinary rate means your speed always points straight away from 0, so you rush outward. Growth at rate i means your speed always points at right angles to where you are. Anything that always moves at right angles to the line back to the centre goes round in a circle, like a stone on a string. Run that motion for π units of time and you are on the opposite side.
Strictly speaking, "e to an imaginary power" had no meaning until somebody gave it one. The formula is the only definition that keeps the familiar rules of exponents working, and it turns out to be consistent with everything else. That is a discovery of a kind, but a subtler one than "nature hid this equation".
The credit is also less tidy than the legend. Euler published eix = cos x + i sin x in 1748, in his Introductio in analysin infinitorum, and Roger Cotes had found an equivalent result in 1714. Whether Euler ever wrote the famous five-constant version himself is doubtful. Its fame is more recent. In a 1988 poll of readers of The Mathematical Intelligencer it was voted the most beautiful theorem in mathematics, and in a 2004 Physics World poll of great equations physicists ranked it second only to Maxwell's equations. Richard Feynman, in his Lectures on Physics, called the general formula "our jewel" and "the most remarkable formula in mathematics".
Because rotation and oscillation are everywhere. An alternating current, a sound wave, a swinging pendulum and a quantum wavefunction all go round and round, and eiθ lets you handle "going round" with the simple algebra of exponents instead of the messy algebra of sines and cosines. The Fourier transform (see The Fourier Transform in this library) is built entirely out of it.
Growth and rotation turn out to be the same idea, pointed in different directions.
Multiplying by −1 is a half-turn and multiplying by i is a quarter-turn. In the plane of complex numbers, multiplication stretches and rotates. e is what compounding converges to, and compounding at an imaginary rate turns instead of grows, so eiθ is the point at angle θ on the unit circle. Halfway round, at θ = π, sits −1. That's Euler's identity: not a coincidence of five constants, but a statement that exponential growth, pointed sideways, is rotation.
Try this: in the second interactive, set θ = π and drag n slowly from 1 to 1,000. Watch the end of the path close in on −1. Then set θ to about 6.28 (2π) and see where you land.
Sources. Euler's formula in the Introductio in analysin infinitorum (1748) and Cotes's 1714 precursor; doubts that Euler wrote the identity itself (Wikipedia, "Euler's identity", citing Sandifer and R. Wilson). The 1988 Mathematical Intelligencer poll and 2004 Physics World poll: R. Wilson, Euler's Pioneering Equation (OUP, 2018) and reports of both polls. Feynman's phrases: The Feynman Lectures on Physics, Vol. I, ch. 22 ("Algebra"). Every number in the tables, text and interactives was computed in node and Python: for θ = π, (1 + πi/n)n = 1 + 3.1416i (n = 1), −1.5934 + 0.1561i (10), −1.0506 + 0.0011i (100), −1.0049 (1,000).