Let's Walk Round a Circle

How a bank, a turning arrow and a piece of string explain the strangest-looking equation on any T-shirt.

My cousin Riya came to lunch wearing a black T-shirt with white writing. It said e^iπ + 1 = 0, and nothing else. "What does it mean?" I asked. She said, "It's the most beautiful equation in maths." "But what does it mean?" She chewed her roti and said, "It links the five most important numbers." That is true, and it is also the least useful sentence anyone has ever said about it. It's like saying a bicycle links a saddle and two wheels.

This page is the answer I wish Riya had given me. We are going to build the whole thing out of four ordinary ideas: multiplying, a number line, turning, and doing the same small thing many, many times. There is no step where you have to "just trust me". Where a step is too hard for a ten-year-old to carry alone, I say so and mark it For older readers or Parent, carry here.

It takes two or three sittings. There are two "good place to stop" signs along the way. Every picture is a real calculation running live in your browser, so drag, press and slide everything.

Section 1
A power is a multiplication done again and again

Suppose you have a photocopier that doubles things. You put in 1 sweet, you get 2. Put the 2 back in: 4. Again: 8. Again: 16. Doubling four times in a row takes 1 to 16.

Writing "2 × 2 × 2 × 2" is long. Mathematicians write it with a tiny raised number: 2⁴. It just means "multiply by 2, four times". 2³ means 2 × 2 × 2 = 8. 10² means 10 × 10 = 100. The raised number is called a power (or an exponent), and the whole page is about one of them, so let's make sure it feels like a machine you can run.

The little raised number was made popular by René Descartes in his book La Géométrie in 1637. Before that, people wrote things like "the cube of the number", in words.

Side remark for older readers: when a grown-up writes 2ⁿ, the letter n is just a space where a number like 4 or 12 will go. "Multiply by 2, n times."

Section 2
Multiplying stretches an arrow

Draw a number line on the floor: 0 in the middle, 1, 2, 3 going right, −1, −2, −3 going left. Now instead of thinking of a number as a dot on the line, think of it as an arrow from 0 to the dot. The number 3 is an arrow three steps long pointing right.

What does "multiply by 2" do to the arrow? It stretches it to twice the length. Multiply by ½ and it shrinks to half. The arrow never turns; it only gets longer or shorter.

Section 3
Multiplying by −1 is a half-turn

Now press × (−1) below. The arrow 3 becomes −3. It could have got there by jumping across. But watch: the sim turns the arrow, like the hand of a clock, through half a turn, from pointing right to pointing left. Length stays the same. Only the direction changes.

Is that cheating? Here's the test. Everybody learns the rule "negative times negative is positive" and nobody is told why. If × (−1) is a half-turn, then × (−1) twice is two half-turns, which is one whole turn, which is back where you started. (−1) × (−1) = 1. The rule isn't a rule at all; it's what two half-turns do.

Section 4
A quarter-turn, done twice, is a half-turn. We call it i

Here is a question. × (−1) is a half-turn. Is there a number which, when you multiply by it twice, gives a half-turn? Something that, done twice, is × (−1)?

Think about turning. What turn, done twice, is a half-turn? A quarter-turn, obviously: turn a quarter, turn another quarter, and you've turned half. So the number we want is "multiply = quarter-turn". There's one problem: a quarter-turn makes the arrow point up, and there is no "up" on a number line.

So we give ourselves an up. Draw a second number line through 0, going up and down. Now the arrow has a whole flat plane to point in. The quarter-turn number is so useful it gets its own letter: i. An arrow of length 1 pointing straight up is called i. An arrow of length 2 pointing up is 2i. Multiplying by i means "turn a quarter-turn, don't stretch".

Now the first puzzle. "How can a number be imaginary?" It isn't, really. i is no more imaginary than −1. −1 means "one step, but turned round", and i means "one step, but turned a quarter". Nobody panics about −1. The word "imaginary" is a 400-year-old insult that stuck.

Descartes used the word "imaginary" in 1637 as a put-down: he meant these numbers were made-up and useless. For 150 years nobody drew them. Then a Norwegian surveyor, Caspar Wessel, drew the number plane in a paper presented in 1797; a Paris bookkeeper, Jean-Robert Argand, drew it again in 1806; and Carl Friedrich Gauss used it and gave the numbers the name "complex" in 1831. Once people could see i as a direction, the insult stopped making sense.

Section 5
Every multiplication is a stretch and a turn

So far we can stretch (× 2), half-turn (× −1) and quarter-turn (× i). What about a small turn, say an eighth of a turn? There must be a number for that too, and there is. Here is how to find it with a rule you already know.

You know that 4 × (10 + 2) is 40 + 8. Multiplying by "10 plus 2" means "multiply by 10, multiply by 2, add the results". The same works for arrows. Multiplying an arrow by "1 + i" means: take one copy of the arrow (that's × 1), take one copy turned a quarter (that's × i), and add them tip-to-tail. One step forward plus one step sideways is a diagonal: longer by about 1.41×, turned by 45°, an eighth of a turn. The sim draws both copies so you can see them join.

Section 6
Growing in tiny slices lands on e

Now a completely different story, which is true. Imagine a bank that pays 100% interest a year. You put in ₹100; after a year the bank adds 100% of it, and you have ₹200. Doubling. Fine.

A friendlier bank says: "We'll pay the same 100%, but in two halves. After six months we add 50%, and after another six months we add 50% of whatever you have then." ₹100 → ₹150 → ₹225. More! Because the second 50% was paid on the first 50% too.

So you ask for monthly: twelve slices of 1/12 each. Then daily: 365 slices of 1/365. Then every second. Surely with enough slices you get rich without limit?

You don't. Slide and watch. The money creeps up and then stops creeping, at about ₹271.83. More slices don't help. That stopping number, 2.71828…, is called e. It is what "growing by 100% in tiny, tiny slices" adds up to.

That last bullet matters. Feed the machine growth 2 and you get e × e, which is e². Feed it 3 and you get e³. So the machine's answer for growth of "some amount" is always "e to the power of that amount". That is why the T-shirt has an e with a power on it. The power is the amount you fed the machine.

Jacob Bernoulli asked exactly this bank question in 1683 and found the answer was stuck between 2 and 3, but he never found its digits. Leonhard Euler worked out those digits (2.71828182845…) and began calling the number e in a manuscript around 1727; the letter first appeared in print in his book Mechanica in 1736. Nobody knows for sure why he picked e. It is probably not for "Euler": he was a modest man and the letters a, b, c, d were already busy.

Section 7
Turning in tiny slices walks you round a circle

Section 5 showed that adding a small sideways slice to an arrow turns it a little. Section 6 showed a machine that adds a small slice many times. Put them together.

Start with the arrow 1. Instead of adding "a slice of the arrow" (which stretches it, like the bank), add "a slice of the arrow turned a quarter", which is a sideways slice. Let's make the total of all the slices add up to 3.14 arrow-lengths, which is π. Why π? Patience: it's a guess for now, and the next section explains it. With 1 slice, you add one big sideways step of length 3.14. With 10 slices, ten sideways steps of 0.314 each, and each one is sideways to the arrow as it is then, which has already turned a bit. Run it.

This is the moment the whole page was built for. We never said "circle" anywhere. We said "add a sideways slice, again and again", and a circle appeared on its own, and the arrow walked exactly halfway round it, onto −1.

Why does sideways make a circle? Whirl a stone on a string. At every instant the stone is pulled sideways to the way it's going, and it goes round in a circle. A sideways push never makes you go faster or slower; it only changes which way you're facing. Our slices are sideways pushes on the arrow's tip. So the tip can't get much longer; it can only turn.

Section 8
π radius-lengths of string reach exactly halfway round

We fed the machine a total of 3.14 arrow-lengths of sideways slices, and the arrow ended up halfway round. Why 3.14?

Because the tip of the arrow was walking along the circle, and the slices were its steps. So "how far did it turn" is really "how much string did it walk". And there's a fact about circles that every ten-year-old has heard but few have felt: the distance all the way round a circle is 2π times the radius, about 6.28 radius-lengths. Halfway round is 3.14 radius-lengths. Our arrow has length 1, so the circle it walks has radius 1, so halfway round is exactly π of string.

Side remark for older readers: grown-ups write the amount of string as θ (theta, a Greek letter) and say "an angle of θ radians". It's just the slider position above.

Roger Cotes, a young Cambridge mathematician, was measuring angles by string length around 1714, three decades before Euler's famous book. The word "radian" is much younger: James Thomson, a professor in Belfast, coined it in 1873.

Section 9
Run the machine with a turn of π and you land on −1

Now we can say the whole thing in words, with no symbols at all.

The bank machine keeps your money and adds a slice of it. Run it with a growth of 1 in tiny slices and your money becomes e times bigger. The turn machine keeps your arrow and adds a sideways slice of it. Run it with a turn of π in tiny slices and your arrow walks halfway round a circle, from 1 to −1.

It's the same machine. The only difference is the slice. A straight slice is "more of the same direction" (growth). A sideways slice is "a quarter-turn of the arrow". That answers the third puzzle. Why does e show up in bank accounts and in circles? Because it is one machine. Fed a straight slice, it makes e. Fed a sideways slice, it makes a circle.

Now the naming. In Section 6 the machine fed growth 2 gave e², and fed growth 3 gave e³. The machine's answer is always "e to the power of what you fed it". A sideways slice of π is written iπ. (That's π turned a quarter, which is what multiplying by i does.) So when the machine is fed iπ, mathematicians write its answer as e^iπ, to match. And we've watched where it lands:

Then add 1 to both sides. The arrow at −1, plus one step to the right, is at 0, home:

Read it as a sentence: "Walk halfway round the circle, take one step right, and you're back at the start." That is what Riya's T-shirt says.

Here is the machine with the turn under your control. Check that it lands on the circle for every turn, not just π.

Roger Cotes wrote down the link between "string length" and the machine in 1714. His form is hard to recognise today, because he wrote it with logarithms. He died two years later, at 33. Newton said "if he had lived we might have known something". Leonhard Euler stated the general fact in his 1748 book Introductio in analysin infinitorum: the machine fed a turn lands on the circle. Euler never wrote the line e^iπ + 1 = 0 as such. It is one special case of his formula. The habit of writing it that way, and calling it "Euler's identity", came in the 19th and 20th centuries. The physicist Richard Feynman, in his 1964 lectures, called the general formula "our jewel".

Interlude
Stranger things

• A full lap: e^2πi = 1. Feed the machine a turn of 2π. The arrow walks all the way round and lands back on 1. In Section 9's sim, press 2π. The equation looks like it should be exciting. It just says "a full turn gets you home".
• i to the power i is an ordinary number: about 0.208. Here's why. i is where the machine lands when fed a turn of π/2 (press π/2 above). Raising to a power multiplies the thing you fed in. So i to the power i means feeding i × iπ/2. That is −π/2, because i × i = −1. A feed of −π/2 is a shrink, not a turn: the bank machine with growth −π/2. The result is e^−π/2 = 0.2079. For older readers: it isn't the only value. You could also reach i by walking π/2 plus any number of full laps. 0.208 is the one everyone means by default.
• Your house electricity is a spinning arrow. The voltage in a wall socket swings back and forth 50 times a second (60 in some countries). Engineers picture it as an arrow spinning round the number plane. They read off its up-and-down shadow. The multiplication-is-turning idea makes the sums easy. Charles Steinmetz pushed this method in 1893. It is still how electrical engineers work.
• A pure musical note is a spinning arrow too. A tuning fork spins 440 times a second. Its shadow traces the wave you'd see on a sound app. Every sound you hear is a pile of spinning arrows, at different speeds, added together.
• The insult that stuck. "Imaginary" (Descartes, 1637) was meant to be dismissive, like calling something "so-called". Gauss later said it would have been better to call them "lateral" numbers: sideways numbers. He was right, and it's too late.
• Bernoulli's problem (1683) was about interest paid in ever-smaller slices. It is the first place e appears in history. He never knew its value.
• Three people drew the plane. Wessel (1797), Argand (1806), Gauss (1831, though he had the idea by 1799). Wessel's paper was in Danish and was ignored for a century. The picture is sometimes still called an "Argand diagram".
• The turn machine is in your phone. When a game rotates a character, the code multiplies by a number like the orange dot in Section 5. A GPS working out a heading does the same. One multiplication that stretches and turns is simply cheaper than doing the two separately.

Section 10
Make your own

You've seen the machine fed a straight slice. That's the bank, and it lands on e. You've seen it fed a sideways slice. That's the turn, and it lands on the circle. The capstone is to feed it both at once: a bit of growth and a bit of turn. Growth makes the arrow longer as it goes. Turn makes it curl. Together they make a spiral. Make your own, and predict where it lands before you look.

Next time someone says "it links the five most important constants", you can say: "Walk halfway round a circle, step right once, and you're home. The five constants are just the names of the things in that sentence."