How doing one small multiplication over and over walks you halfway round a circle, and why that makes eiπ + 1 = 0. Every diagram here is a live simulation you can poke. Each one is worked out with real arithmetic, step by step, as you watch.
Your cousin comes home from college wearing a new T-shirt. Printed across the front is eiπ + 1 = 0. "Most beautiful equation in all of maths," they say. You ask why. They shrug and mumble that it's famous.
That isn't an answer. A number called e, raised to the power of an "imaginary" number, times π? How do you even multiply something by itself an imaginary number of times? And why would the answer come out as plain old −1?
By the end of this page you'll be able to see it happen, with nothing more than multiplying, a number line, turning, and doing the same multiplication again and again. Along the way we'll answer:
Picture every number as an arrow that starts at zero. The number 3 is an arrow three steps long, pointing right along the number line. The number 1 is an arrow one step long.
Multiplying by 2 doubles the arrow. Multiplying by ½ halves it. It's like zooming a photo in or out. The picture gets bigger or smaller, but it still faces the same way.
A few things to notice:
So what does ×(−1) do? Press it, and watch closely.
Multiplying 3 by −1 gives −3. Same length, opposite direction. There are two ways you could picture that. You could squash the arrow down through zero and out the other side. Or you could swing it round, like the hand of a clock, until it points the other way.
The simulation swings it. A swing that ends up facing the opposite way is a half-turn. So multiplying by −1 means "turn halfway round".
Now press ×(−1) a second time. Another half-turn. Two half-turns make a whole turn, and a whole turn leaves you facing where you started. That's all that "a negative times a negative is a positive" means.
Try (−2) × (−3). The arrow for −2 is "2, turned halfway". Multiplying by −3 means "stretch 3 times, and turn halfway". Put it together: stretched to 6, and turned halfway twice. That's 6, facing right. So (−2) × (−3) = 6.
About 1,400 years ago, in the year 628, the Indian mathematician Brahmagupta wrote down rules for negative numbers. He thought of them as debts. One of his rules says that a debt times a debt is a fortune.
Hold on to this idea: multiplying can turn an arrow, not just stretch it.
Here's a puzzle. What number, multiplied by itself, gives −1?
On the number line, nothing works. 1 × 1 = 1. (−1) × (−1) = 1 too, because that's two half-turns. Any number times itself is two copies of the same turn, and on a line the only turns are "none" and "halfway". Two "nones" face right. Two "halfways" also face right. You can never end up facing left.
But think about turning instead of numbers. Multiplying by −1 is a half-turn. What could you do twice to make a half-turn? A quarter-turn! Turn a quarter of the way round, then another quarter, and you're facing the opposite way. (A whole turn is 360°, so a half-turn is 180° and a quarter-turn is 90°.)
There's one problem. A quarter-turn makes the arrow point straight up, off the number line. So let's give it somewhere to go. Draw a second line going up through zero. Now the arrow can point anywhere on a flat sheet.
Mathematicians call "multiply by a quarter-turn" multiplying by i. Turn 1 a quarter of the way round and you get an arrow one step long pointing straight up. That arrow is the number i itself. And i × i = −1, because two quarter-turns are a half-turn.
A few things to notice:
In 1637 the French thinker René Descartes called numbers like this "imaginary". He meant it as a put-down: you can imagine them, but they aren't really there. The name stuck, even after people found they were perfectly useful. Leonhard Euler started using the letter i for the quarter-turn number in 1777.
Draw a big plus sign on a sheet of paper. Write 1 at the right end, i at the top, −1 at the left and −i at the bottom. Cut a small arrow out of card, put its tail on the middle, and point it at 1.
Now turn it a quarter-turn anticlockwise. It points at i. Another quarter-turn: −1. You've just done i × i = −1 with your hands.
It works because "times i" never meant anything mysterious. It only ever meant "turn a quarter". The paper arrow is the same as pressing ×i in the simulation above.
You now own the number i. That's the hardest new idea on this page. The next sections build the number e and then put the two together. Come back with fresh eyes.
Every point on the sheet is now a number. The point two steps right and one step up is called 2 + 1i, or just 2 + i. Numbers like this are called complex numbers. "Complex" here means "made of two parts", the way a sports complex has several buildings. It doesn't mean "hard".
Drawing these numbers as points on a flat sheet took a surprisingly long time. A Norwegian-Danish surveyor called Caspar Wessel published the idea in 1799. A Swiss bookkeeper in Paris, Jean-Robert Argand, came up with it again in 1806. Neither got much attention until the great German mathematician Carl Friedrich Gauss backed the picture in 1831.
How do you multiply two of these numbers? The computer does it with ordinary multiplication plus the one new fact that i × i = −1. (The working is in the box for older readers below.) What matters is what comes out. Take the arrow 1 + i, one step right and one step up. It points 45° up, and it's about 1.41 long. Multiply it by itself and you get 2i, an arrow pointing 90° up that is 2 long.
Look at what happened. The angles added: 45° + 45° = 90°. The lengths multiplied: 1.41 × 1.41 = 2.
Is that a fluke? Try it for any two arrows. Drag the two handles, and the simulation multiplies them using i × i = −1, nothing else.
A few things to notice:
Hold on to that last one. A number that's 1 plus a tiny bit of i is almost a pure turn. The smaller the tiny bit, the less it stretches.
First, the multiplication the page skipped. Multiply out the brackets and use i × i = −1:
(1 + i)(1 + i) = 1 + i + i + i×i = 1 + 2i − 1 = 2i
The simulation checks the rule for whatever arrows you choose, but checking isn't proving. The proof needs trigonometry. Write a = r(cos α + i sin α) and b = s(cos β + i sin β). Multiply out using i² = −1, and the real part is rs(cos α cos β − sin α sin β) = rs cos(α + β). The i part is rs sin(α + β). Those are the angle-addition formulas, so the product has length rs and angle α + β.
Let's leave arrows for a moment and go to the bank. This is a true story, or at least a true question.
Imagine a very generous bank that pays 100% interest a year. Put in ₹100, and after a year they add another ₹100. You have ₹200.
Now a clever customer asks: "Could you pay me half the interest every six months instead?" After six months they add 50%, so ₹100 becomes ₹150. After another six months they add 50% of ₹150, which is ₹75. Now you have ₹225. Better! The interest from the first half-year starts earning interest of its own.
Every month? Each month adds 1/12 of what's there, twelve times. You end up with ₹261.30. Every day? ₹271.46. So does paying more often make you endlessly rich? In 1683 the Swiss mathematician Jacob Bernoulli asked exactly this question. Find out for yourself.
A few things to notice:
That ceiling is 2.71828… times your money. The number 2.71828… goes on forever without repeating, and it's called e. Euler started calling it e in the late 1720s, and the name stuck.
Look at what the bank was really doing. It took your money and multiplied it by "one plus a small slice", over and over. With 12 payments the slice is 1/12, done 12 times. With 365 payments the slice is 1/365, done 365 times. The more payments, the thinner the slice, and the closer the answer creeps to 2.718.
Hold on to this idea. Think of the bank as a machine. You feed it an interest rate. It chops the rate into tiny slices and multiplies 1 by "one plus a slice" over and over, as many times as you like. Feed it 100% and out comes 2.718, which is e. Feed it 200% and out comes e × e, about 7.39. Feed it 20% and out comes about 1.22.
Mathematicians write the rate as a plain number, so 100% is 1 and 200% is 2, and they call the machine's answer e to the power of the rate, written ex when the rate is x. The grown-up way to write the machine itself is (1 + x/n)n, where n is how many slices. You don't need the letters to use the machine. You only need to remember what it does.
On a calculator (a phone works), work out 1.01 multiplied by itself 100 times. If it has an x^y key, type 1.01, x^y, 100. If not, type 1.01 × 1.01 and keep pressing = until you've done it 100 times. Then race a friend: they do 1.001 multiplied by itself 1,000 times.
You'll get 2.7048. They'll get 2.7169. Both are closing in on 2.71828.
Why? 1.01100 is a bank paying 1% interest, 100 times a year. That's the bank simulation with n = 100. Your friend's calculator is the same bank with n = 1,000.
You now own e as well: it's what the bank machine gives when you feed it 100%. The next three sections feed that same machine a turn, and that's where the T-shirt comes from. Do them in one go, and it's fine to do them with a grown-up.
Now for the big move. Remember the machine from section 5. Remember, too, the almost-forward arrow from section 4. What happens if we feed the machine an imaginary number?
Let's feed it x = iπ. That's the number π (about 3.14) turned a quarter-turn, so it points straight up. The machine splits it into n pieces, so each step multiplies by 1 + iπ/n.
With 8 steps, that's 1 + 0.39i. That's an arrow pointing mostly forward, with a little nudge up, just like b = 1 + 0.1i in section 4. So each step turns our arrow a bit and stretches it a bit. Let's do it 8 times and see where we land.
A few things to notice:
Why does it bend into a circle? Look at any one step. Multiplying the arrow by 1 + iπ/n keeps the whole arrow (that's the "1") and adds a small piece that is the arrow turned a quarter-turn and shrunk to π/n of its length (that's the "iπ/n"). So every step is a small push at right angles to the arrow. The small squares in the simulation show it. This is the bank all over again. The bank kept your money and added a small slice of it. This machine keeps your arrow and adds a small sideways slice of it.
Think of whirling a ball on a string round your head. The string keeps it the same distance from your hand, and the push you give it is always sideways. The ball goes round in a circle. Each step of our chain is a sideways push too. The pushes are a tiny bit too long to keep the arrow exactly the same length, which is why it ends up a bit longer than 1. But the more steps you take, the smaller each push is. Then that tiny extra stretch fades away, and the walk becomes a perfect circle.
Each step multiplies by m = 1 + iπ/n. By section 4, after n steps the length is |m|n = (1 + π²/n²)n/2 and the angle is n·arctan(π/n). Since 1 + u ≤ eu, the length is at most eπ²/(2n), which goes to 1. Since u − u³/3 ≤ arctan u ≤ u, the angle is within π³/(3n²) of π. So the landing point goes to length 1, angle π: that's −1. This is a statement about limits, which is where "exactly" really lives. The simulation can only show you it getting closer and closer.
We've seen that the chain lands near −1. But why does a turn amount of π get you exactly halfway round? To see that, we need to measure how far you've walked round a circle.
Take a circle whose radius is 1 step. Imagine laying a piece of string along its edge, starting at the point 1 and going anticlockwise. Measure the string in radii: one radius of string, two radii, and so on.
A few things to notice:
That's what π is. All the way round a circle is π times its width, which is 2π radii. So halfway round is π radii. People have known this for thousands of years. In the year 499 the Indian astronomer Aryabhata gave the distance round a circle as 3.1416 times its width, which is correct to four decimal places. A walk of one radius along a circle is called a radian.
Take a round roti, or a round steel plate. Cut a piece of string exactly as long as the radius, from the very middle to the edge.
Now lay the string along the edge, mark where it ends, and lay it again from there. Count how many lengths it takes to get halfway round. You'll get 3 and a bit, the bit being about a seventh of the string. Going the whole way round takes 6 and a bit more.
It works for any circle, big or small, because π is the same for every circle. Your string is doing what the "1 radius" button does in the simulation above.
Now go back to the chain in section 6. With 8 steps, each push was 0.39 long, and 8 × 0.39 = 3.14. With 100 steps, each push was 0.0314 long, and 100 × 0.0314 = 3.14. However many steps you take, the pushes add up to π, because each one is π shared out over the steps. Add them up and you've walked about π. And because each push is sideways, you've walked it round a circle of radius about 1. Walking π radii round a circle takes you halfway. Halfway round from 1 is −1. The more steps you take, the more those "abouts" melt away.
Go back to the chain simulation in section 6 and set n to 1,000. Now try the turn amount buttons.
Here is the whole thing in words first. "e to the power of something" means: run the bank machine with that something. Run it with a plain rate and your money grows. Run it with a turn and your arrow walks round the circle, by that many radii. Run it with a turn of π and you walk halfway round, to −1.
Now the symbols. Let's call the turn amount θ (the Greek letter theta). In section 5 we called the machine's answer ex. So when we feed it a turn of θ, which is x = iθ, the answer is eiθ. And we've just seen what that answer is.
eiθ is the point you reach by walking θ radii round a circle of radius 1.
Put θ = π. Walking π radii takes you halfway round, to −1. So eiπ = −1. Add 1 to both sides and you get the T-shirt: eiπ + 1 = 0.
The story behind it took a while. In 1714 the English mathematician Roger Cotes found a link between circles and e, written in a different form. In 1748 Euler published a book called Introductio in which he wrote the rule for any angle. In today's letters it says eiθ = cos θ + i sin θ, which is a mathematician's way of naming the point θ radii round the circle. As far as anyone has found, Euler never actually wrote eiπ + 1 = 0 in that tidy T-shirt form. It follows from his rule in one line, and other people wrote it down later.
You can't multiply e by itself iπ times, so "eiπ" needs a definition. This page uses the machine: ex is the limit of (1 + x/n)n. For ordinary numbers this agrees with the usual powers of e. It also keeps the rule that makes powers powers: ex+y = ex × ey, for complex x and y too. That's why the name is earned and not just borrowed. Many textbooks define it with an infinite sum instead. Both definitions give the same answer, but showing that takes more than this page has room for.
Walk 2π radii and you're back where you started, so e2πi = 1. Walk 4π and you've gone round twice: still 1. Set the chain simulation to 2π to see it close up into a full circle.
Feed the machine −iπ and each step pushes the other way. The chain walks clockwise, under the line, and still lands on −1. A half-turn is a half-turn, whichever way you go.
This one uses the rules for powers, which this page hasn't taught, so skip it if you like. Walking π/2 radii round gets you to i, so i = eiπ/2. Raise both sides to the power i. The power rules say you multiply the powers, and i × iπ/2 = −π/2, because i × i = −1. So ii = e−π/2, about 0.2079. An imaginary number to an imaginary power gives a plain real number. (For older readers: i is also e5iπ/2, one extra lap round, so ii has other values too. 0.2079 is the usual one.)
Spin an arrow round the circle at a steady speed and watch only its height, the way a shadow on a wall would show it. The height swings up and down in a smooth wave.
That wave is the shape of the electricity in a wall socket. In India the mains supply swings back and forth 50 times every second, so engineers picture it as an arrow spinning round 50 times a second. Sound works the same way. The note A that orchestras tune to is a wave that swings 440 times a second. Engineers and physicists use eiθ every day for exactly this reason: spinning arrows are much easier to multiply than wiggly waves.
Feed the machine a number that has both a growing part and a turning part. The arrow stretches as it turns, and the chain curls into a spiral, like a snail shell. The next section lets you build your own.
Here is the machine with both dials. "Grow" is the ordinary part of x, the bank's interest. "Turn" is the i part. The machine multiplies 1 by (1 + x/n), n times.
A few things to notice:
You need a ruler, a pencil and something with a square corner (a set square, or the corner of a book). Mark a dot O in the middle of a page. Draw a line 5 cm to the right and call its end A.
At A, draw a short line at right angles going up, 0.39 times as long as OA. That's about 2 cm. Call its end B. Measure OB. At B, draw a new line at right angles to OB, 0.39 times as long as OB. Keep going. Four short lines are enough to see the chain bend. Eight short lines take you to the landing point below.
Your last point should land about 8.9 cm from O, a little above the line on the left. That matches the chain simulation at n = 8. It works because each short line is exactly what multiplying by 1 + 0.39i does. The "1" keeps the old arrow, and the "0.39i" adds a piece turned a quarter and 0.39 times as long. (0.39 is π/8.) Try 16 lines that are each 0.196 as long, and you'll land closer to the 5 cm mark on the left.
So the next time you see that T-shirt, you'll know what it says: multiply 1 by "one plus a tiny sideways push" again and again, and the pushes walk you round a circle, so a total push of π walks you halfway round, to −1.