Transcription
This is often called the most beautiful equation in the world. Right now, it probably looks like nonsense, and that's completely fine. Most people never understand it, even people who are good at math. But you're going to. By the end of this video, you'll look at this and know exactly what every piece means and why grown mathematicians say it's beautiful.
To get there, we need to understand two numbers. These two numbers were found by different people in different countries hundreds of years apart. The people who found them were not looking for the same thing. And for a long time, nobody thought these two numbers had anything to do with each other. And then people kept finding them together, showing up in the same places and doing the same jobs. And nobody had planned any of it. To this day, nobody fully understands why it keeps happening. So, let's meet them one at a time, and then I'll show you where they meet.
And before we go ahead, I just want to say this video is made for everyone. I imagined explaining it to someone who only knows basic math. So, I'll go step by step. And even if you already know the basics, I'd still love for you to watch from the beginning. Sometimes the things we think we know have a deeper meaning underneath. And in this video, I really want you to feel that meaning, not just memorize the surface.
Let's understand this first and forget the name and the symbol for now. Say you have a coin. If you put a tiny ant on the edge and it walks all the way around and comes back to where it started, how far did it walk? That measurement and that distance ant just traveled is known as circumference. The full trip around.
Now, a second thing, start at that same point on the edge, but this time go straight across the coin through the center to the other side. That straight distance also has a name, a diameter, the trip through. Here is a simple question. How many trips through fit inside one trip around? You can actually test this at home. Take a piece of string, wrap it once around the coin, and mark where it meets, which is circumference. Now lay the string flat next to a ruler and see how many times the coin's width fits into it. It fits three times. And then a little bit is left over. Measure carefully, and you get 3.14 times. So the circumference is a little more than three times the diameter.
And here's the part that makes it special. Let's do it with a bigger circle. Bigger circle, longer circumference, but a longer diameter, too. And it still fits exactly three times and that same little bit. Try it on a plate, a wheel, the moon. Every single time, the circumference is the same 3.14 diameters. The size never matters. Every circle in the world, big or small, gives you the exact same number. That number is what we finally call pi. That's all it is. The circumference of a circle divided by its diameter.
Now the real question. All right. It's cool that every circle gives the same value when you divide circumference by diameter. But why does it matter? Why should I care? You know, what does it actually get us? Here's the answer, and I know you're going to love it.
Take our own planet, the Earth. We want to know how big it is. With today's satellites, we can measure the circumference of the Earth, the distance all the way around the equator. We've done it. It's about 40,000 km. But now try to measure the Earth's diameter straight through the middle from one side through the center out the other side. How would you even do that? You'd have to dig a tunnel straight through the center of the entire planet through molten rock all the way to the far side. Nobody can do that. There's no way to directly measure straight through the Earth.
And here is where this number saves us. We know the circumference. And we know that for any circle, the circumference is always this number times the diameter. So, we just turn it around. Take the circumference, divide by this number, and out comes the diameter, the one we could never reach. We just measured straight through the Earth without digging a single meter. That's the need. This number is the bridge between the part of a circle you can measure and the part you can't because it's always the same. You can know a circle you can't reach. The Earth, a far away planet, anything round and out of your hands. Measure the part you can use this number and you know the rest.
And there's one more thing about this number, the strangest thing of all that we'll come back to later. Pi was born from circles, but it shows up in places that have nothing to do with circles. It appears when you drop a needle on a lined floor and ask how often it lands crossing a line. It sits inside the equations that describe gravity and gravity's pull between things. It runs through the math of waves, which means your phone quietly uses pi every time it plays a sound or loads a video. No circle anywhere in sight. And there's pi. Keep that in the back of your mind. We'll get there.
Now, the second number, and this one isn't about shape at all. It's about growing. And I will start with the example to understand it with the most common example you will find everywhere about E. But you need to understand it. Imagine a very generous bank. You put in $1 and the bank promises to double your money in one year. 100% interest. So after one year, your $1 becomes $2. Simple.
But then you have an idea. You ask the bank instead of paying me everything at the end, can you pay me half in the middle of the year and let that half start earning two? The bank says fine. So at 6 months, your $1 has grown by half. You have $1.50. And now for the second half of the year, all of it grows, not just your first dollar. The new 50 grows, too. So by the end of the year, you have $2.25. Interesting. Same bank, same 100% but you got more just because the growth started growing earlier.
So you push further. Pay me every month. Now the year ends at about $261. Every day $2.71. Every hour, every second, every tiny moment. And here is the surprise. The number stops climbing. It hits a wall. No matter how fast the bank pays you, you can never pass 2.71828 and a bit more. That number is what we call E. That's what E is. The number you land on when something grows as smoothly and constantly as it possibly can.
Okay, so that's what it is. But like before, the real question is why do we need it? What does it actually do for us? Let me show you with a tree because this is the example that made it click for me. Say you walk past a tree and today it has 20 leaves. You're curious, so you decide to count it every day. The next day it has 22, so two new leaves. The day after, still 22. No new leaves that day. Next day, 25, so three more. Then 30, that's five more. Then 31, one more. Then 33, two more. Then 39, six more. So after 7 days, the tree went from 20 leaves to 39. It grew by 19 leaves. But notice, I only know that because I showed up and counted every single day for a week. That's the slow way. That's measuring.
Let's see. On day one, I saw the tree go from 20 leaves to 22. That's two new leaves out of 20, which is a growth of 10% in one day. A tree with more leaves catches more sunlight. So, it grows more leaves, which catch even more sunlight. The growth feeds itself, and it doesn't wait for tomorrow to do it. It happens a little in every moment the sun is up. Each new leaf earning its keep the instant it opens. Growth feeding itself with no pause. And that smooth non-stop feeding on itself is exactly what our number E is built for. We just can't watch every single moment. So we check once a day. Those seven daily counts are only snapshots, little dots sitting on one smooth curve growing for.
So we use it. We take where we started 20 leaves. We take the growth pace 10%. We take the time, 7 days, and E turns that into a prediction. We put it in, and it tells us the tree should reach about 40 leaves, a growth of about 20. And remember what actually happened when we waited the whole week and counted by hand. It grew by 19. We predicted about 20 before any of it happened. From one single day of watching, we saw a week into the future. That is why we need E. Just like pi let us measure a circle we couldn't reach, e lets us measure a future we haven't reached yet. Anything that grows by feeding on itself, money, a tree, a crowd of people, a sickness spreading, even how a hot cup of tea cools down to room temperature. E lets you jump ahead and know roughly where it's going without living through every underneath.
And one honest little detail because it makes E. S came out a touch high, 40, when the real count was 39. That's not a mistake. E is the number for perfectly smooth growth every moment, day and night, never resting. Nothing real is quite that perfect. A real tree slows at night, rests when the sky is gray. So real growth always lands just under E's smooth ideal. 39. A whisker below 40. E gives you the ceiling. Real things lean right up against it from just below. And still from one single day of watching, we called a whole week and missed by one leaf. That's exactly why E is the number the whole world of growth is built on.
All right. Now that we have an understanding of E, before we move forward, I want to teach you one more number I. And to understand it, I need to show you something about the numbers you already know. Numbers live on a line. Picture it as a straight road and 0 is your house. 1 2 3 are steps to the right of your house. Minus one minus two minus three are steps to the left. And these two directions mean something. Going right means having. Three steps right you have three rupees. Going left means owing. Three steps left you owe three. Right is forward, left is backward. Right is gaining, left is losing. Every number on this road is not just an amount. It's an amount pointing in a direction.
Now, on this road, multiplying is an instruction. It tells you where to move. Multiply by two means double your distance from the house. Stand at three, multiply by two, you walk out to six. But there is one number whose instruction is special. Minus one. Stand at three and multiply by minus1. You land at minus3. Same distance from the house, opposite side. Try it from anywhere. Five lands on minus5. Minus2 lands on + two. Every number jumps across the house, keeping its distance. Now watch the whole road while this happens. Every point on the right swings over to the left. Every point on the left swings over to the right. The whole road has spun half a turn around your house. That's the secret about minus one. Multiplying by it is not really making things negative. It's a rotation. A half turn 180° around zero. Keep that in your pocket. We'll need it soon.
Now, a small question. Which number times itself gives 9? Easy. Three. Because 3 * 3 is 9. Actually, there's a second answer hiding here. Minus3 also works because minus3 * minus3, the two minuses are two half turns, a full spin. You end up back on the plus side, nine again. There's a name for this game. Finding the number that times itself gives you what you want. That's called taking the square root. The square root of 9 is 3. The square root of 25 is five. So far so easy.
Now try this one. The square root of - 9. Which number times itself gives - 9? Try three. That gives + 9. Try minus3. Two half turns back to + 9 again. Try any number on the road. A positive times itself stays positive. A negative time itself turns positive. 0 gives zero. Every single number on the road refuses to come out negative. And now you can see exactly why it refuses. Times itself means do the same instruction twice. Numbers on the right do no turn twice. Still no turn. Numbers on the left do a half turn twice. A full spin back to plus. Nothing on the road done twice leaves you facing backward.
So the question, what is the square root of minus1 is really asking what instruction done twice makes a half turn. Say it like that and you can answer it yourself. A quarter turn. Turn 90° then 90° again and you've turned 180. So stand at one, make a quarter turn around your house. You're not on the road anymore. You're standing one step off it in the field directly above your house. That exact spot is the number we call I. Check it the way we've checked everything in this video. I * I means do the quarter turn twice. Start at one. First quarter turn above the house. Second quarter turn you come down onto the road on the other side. One step out. You're standing on minus one. I * I equals minus1. The impossible question had an answer all along. It was just standing one step off the road in a direction nobody thought numbers could go.
For hundreds of years, nobody looked there. And here's how the number forced its way in asking. In the 1500s in Italy, mathematicians held actual public contests in solving equations. And one of them, a man named Bombelly, hit something disturbing. He was solving a normal equation, and he already knew by checking with his hands that its answer was an ordinary number. But right in the middle of the solving steps, the method forced him to take the square root of a negative number, the impossible thing. If he stopped there and said, "This cannot be done," he could never reach the answer he knew existed. So he tried something bold. He pretended. He said, "Fine, let us suppose there is such a number, the square root of minus1. I will not ask what it is. I will just carry it through my steps like a normal number and keep going." And something amazing happened. A few steps later, all the impossible pieces cancelled each other out. And what remained was the correct ordinary answer. He checked it. It was right. Think about how strange that is. a number that cannot exist had walked through the middle of his calculation and quietly handed him the truth. It worked. It gave right answers. Nobody could say what it actually was. Other mathematicians were so uncomfortable that they called it the imaginary number as an insult. And the insult stuck and became its official name. It took 200 more years before someone finally saw where this number had been living the whole time.
Anyway, so numbers can turn. Let me show you why the world cares about a number that turns. Come to the fair with me. Look at this ferris wheel. Pick one cabin and just watch it. It goes up one side, over the top, down the other side, around and around. Now, I want to ask a strange question. At any moment, where is the cabin? Not somewhere on the wheel. I want two honest numbers. Here's how you get them. Wait for the sun. The cabin throws a shadow straight down onto the ground. As the wheel turns, watch only the shadow. It slides right, slows down, stops, slides left, slows, stops, slides right again, back and forth along the ground forever. That sliding shadow, how far left or right of the wheel center the cabin is, has a name. It's called cosine.
Now, forget the shadow and look at the cabin's height. It rises, slows near the top, comes down, sinks below the middle, comes back up. How high or low the cabin is compared to the center, that's called sign. That's all cosine and s are. One turning cabin watched two ways. Cosine is its left right. S is its up down. Neither of them is the cabin. But give me both numbers and I can point at the exact spot on the wheel where the cabin is. Two flat back and forth numbers and together they rebuild the turn. And look at what shape each one makes on its own. Plot the height as time passes and you get a wave. That smooth rising and falling curve. Plot the shadow. The same wave just shifted. Every wave you've ever met is secretly a turning wheel watched from one.
That's why this matters outside the fair. The electricity in your walls doesn't flow one way. It swings forward and backward 50 or 60 times every second, exactly like the cabin's height. It is a wave. And here's the problem nobody tells you about. In a real power system, there is never just one wave. Your fridge makes its own wave. A big motor pushes its wave a little late. A charger pushes its wave a little early. All of these waves are riding the same wire at the same time. Same speed, but out of step with each other. And the engineer has one job. Tell me what all of these waves add up to. Because if you get that answer wrong, wires overheat, machines break, the lights go out.
So try to add just two of them. Two waves, same speed, one slightly behind the other. Take the height of the first wave at every moment. Add the height of the second wave at that same moment. point by point forever. There is a formula for this from the mathematics of triangles and it is a monster. Two waves fill a page, three waves fill a chapter, and a real power grid has thousands. In the 1880s and 1890s, when electricity was brand new, this was engineer's daily life. The calculations were technically possible and practically hopeless. People built machines half by guessing.
Then one engineer, Charles Steinmets, in 1893 said, "Stop carrying the wave around." Remember what a wave really is. You already know it. You saw it at the fair. A wave is a turning wheel watched from the side. So don't drag the whole wavy curve through your mathematics. Carry the wheel instead. A turning arrow. Its length says how strong the wave is. Its angle says how far ahead or behind it is. And an arrow that turns that lives exactly where it lives. One step along the road, one step into the field. Every wave becomes one single number written with I.
And now watch what happens to the impossible job. To add two waves, you don't touch the wavy curves at all. You put the two arrows tiptoail and draw one new arrow. That's it. That's the sum. The page of triangle formulas became one line of arithmetic that a student can do. Not because the problem got easier, because I is the right language for things that turn. Without I, the grid was drowning engineers. With I, it became homework. Steinmet's trick is still how every power grid, every transformer, every electric motor on Earth is designed today. Your phone does the same trick thousands of times a second. The signal reaching it right now is a messy wiggle. Many waves piled up. The phone asks which turning wheels added together make exactly this wiggle. It takes the mess apart into clean spinning arrows, reads each one, and hands you the video. That taking apart is computed in eyes language. The insulted impossible number quietly runs the modern world.
Now you're holding everything. And I'll show you one line that ties it together. The cabin's position needs two numbers, the shadow and the height. But we know a place where two directions live together. The road and the field above it. So write the cabin spot as one number. Cosine along the road plus i * sign up into the field. One turning point, one address. And here is the line Oiler found. E our growth number raised to the power of I * X=cossine of X + I * sin of X growth raised to a turning number doesn't grow bigger it goes around e to the ix is the ferris wheel the exact position of the cabin after turning through X growth turned sideways becomes rotation sit with that for a second because it's the strangest sentence in this video and by now Every word in it is yours. E is growth. I is the quarter turn. Cosine and sign are the wheel watched two ways. Oilers's line says push growth in the turning direction and it stops racing away and starts circling perfectly forever. That's the whole imaginary number.
Next, we take this wheel and walk it exactly halfway around and watch every number we've met land in one line. Remember where we left the ferris wheel? E to the iix is the wheel, the exact position of the cabin after turning through X. Pick any amount of turning and this little expression hands you the spot. So, let me ask the last question of this video. What happens if we turn through exactly half a circle? The cabin starts at one, one step to the right of the house. We want it to land exactly opposite. Same distance other side. How much turning is that? Well, we measured this. It was the first thing we ever did back with the coin and the ant. The distance all the way around a circle is pi * the distance across. Our wheel has radius 1. So, the distance across is 2. And the full trip around is 2 * pi. Halfway around then is exactly pi. Not roughly pi, exactly pi. That's not a coincidence. That is what pi is. It has been the number for halfway around since the ant took its first walk.
So watch. We stand at one. We turn smoothly through pi half the circle up past the top of the wheel and down the other side. And we land one step to the left of the house on minus one. Say it in the language of the wheel. E to the I pi equals minus1. Now look at what's sitting in that little sentence. E the growth number born in a bank grown on a tree. I the quarter turn, the number they called imaginary, the direction off the road. Pi, the circle number born on the edge of a coin. Three numbers from three different centuries found by different people who never met for reasons that had nothing to do with each other. And when you put them in one line, they don't argue. They fit like they were made for each other.
One last touch. Move the minus1 to the other side. E to the i pi + 1 equals 0. The one where the cabin started. The zero, your house, the center of everything. Now the line contains e, i, pi, 1 and zero. The five most important numbers in mathematics once each and nothing else. This is the line I showed you at the start. Back then it looked like nonsense. Look at it now. E to the i pi grow but push the growth sideways so it turns instead of racing away. through pi half the trip around plus one step back from where you landed to where you began zero home growth turned sideways walked halfway around a circle brings you exactly home that's the whole equation that's why mathematicians call it beautiful not because it's complicated but because three stories that never plan to meet all end at the same front door.
All right, I hope you now understand what that equation really means. And this is just one place these three numbers meet. There are so many more equations all across mathematics where pi, e, and i turn up together for reasons just as surprising as this one. If you'd like a part two where we go and find them, just let me know down in the comments and I'll make it. Thank you for watching.