Proving the Most Beautiful Equation Bob Ross Style

Hi, I'm Toby and welcome to the joy of mathematics. Today we are going to work on something very

beautiful

. So if you want to work at home, now you have the opportunity to grab your own notebook and pen and you can follow along. Don't be afraid of what's already on the board. This is just a small introduction that we will need to get started. I know that math can sometimes evoke scary emotions in people, but I really wish it didn't. Math can be a lot of fun and I think that's one aspect we'll look at today.

Don't be afraid of this math, I'll be here and we'll work on it together. I think we're going to have a great time. So let's get into it. What we do have here are some Taylor expansions for three common functions. We have our e to the power of X, our exponential function, our cos of X and our sine of X and terms that allow us to write expansions for these functions. To show you what the e in the the maths. Let's move on to the cosine. This is one of our trigonometric functions. So if we were to make a cosine graph on a small axis.

Cosene is someone who is starting out, perhaps born into a rich family. It starts very well in life But then things get constantly worse and there is a downfall. You know, everyone has low moments. But then things get better and, in fact, the cosine shows us that life is a constant cycle of ups and downs, you know? As I like to say these low points. This darkness is really necessary to appreciate the light. We gave the cosine a friend and that is the sine. They are very similar people. The sine, unlike the cosine, had to work for what it had.

He started with nothing, but worked up to being pretty good and then nothing lasts forever. He eventually lost that and he too discovered that life is a constant cycle. of ups and downs. Now you can notice some similarities between our functions, they all have these similar terms that are actually X to the power of N over N factorial. In the case of cos, its terms are the even terms, x squared, e to X All signs are positive. We don't have this positive negative alternation like we had here and we have all the powers of X. So you can see that there may be some relationship between these three people here.

It's al

most

like they are a big family. But it doesn't seem like it. to add because if we added cos and sine, you know, maybe we would get something similar to e in the X, we would get all the right terms, but we would still have these minus signs. here So we can't just do a simple addition to relate these functions to this one here. We need a little help and you know, in those moments when I'm struggling and I need a little help, I'd like you to know that. , it's embarrassing to say this But I resort to asking my imaginary friends.

Some people laugh when I tell them about my imaginary friends, but today I would like to introduce you to one of them and that is my i friend. can be written as the square root of negative 1, now it's a pretty funny little character to have i multiplied by i equal a negative number, but that's exactly what's happening here. Hold on to your calculators, ladies and gentlemen, but it's going to equalize. get a little strange. We'll get to that later. Now we will use our little friend to help us. In fact, let's rewrite this e to the function X as e to i-x.

That means anywhere there's an X here I can replace it with an i-x. Let's go over and do that. Keep the brackets correct so we know the whole term is squared, the same way here and there we go. Now we have our function e raised to i-x. But what do some of these terms do? What can I really do for us? Let's take a look. So i to the power of 1 will simply be i, but i squared will be negative 1, that's by our definition here and if we did i cubed it would be i, or negative i. i to 4 would be 1 and in fact this series repeats between i, minus 1, minus i and 1, and we can use that to rewrite some of these terms.

So if we have an i squared. In this term here we can replace that with negative 1, let's take that i out. Take that bracket out and change this to negative 1. Here we have i cubed, we can replace that with a. i i to four, we can replace it with one. I should quit anyway. There are no errors here. Just happy accidents and our i five, well, we can work on this series to solve for that one and an i to the power of five It would be equal to one multiplied by i, that would be i Okay, that looks pretty good.

We still have our spotted imaginary friend here and one tip I have for you is to always keep your real friends and your imaginary friends separate from each other. So let's write this function, but separating our real and imaginary friends. We can delete this here Clear these little demons And here we go, e Al i-x Let's make the real terms first, so that he is real He is real, he is real and I believe that anything that has even power will be real. So let's put it in parentheses 1 Minus x squared over 2 factorial Plus X to the power of 4 4 factorial And there will be an infinite number of those real terms.

Let's make our imaginaries. Now all of our imaginary terms will have this i in front. So in fact, we can just remove the i in front of our square brackets. What do we have with an X? Minus X cubed over 3 factorial plus i x 5 Over 5 factorial and an infinite number of those terms. So if we take a look at what we're really working on here, the eagle-eyed of you will be able to recognize that this here is really nothing. more than our old friend the cosine of X. This is this function. And you'll also notice that what's in parentheses here is just our old friend the sine of X plus i sine X That's pretty surprising so far, but let's go a little further.

Let me delete this and make a little more space for us, okay, so I've written a backup of our

equation

that we've worked on and we're going to continue building so that e is a pretty interesting function. Like I said, it's exponential and it's also a good number in math. I am our imaginary friend and X is anyone's guess, but what if we add one more curveball here? I'm sure you guys can handle it. I believe in you and our curveball will be Pi. Many of you would have heard of Pi; in fact, it is related to circles.

If you knew the diameter of a circle and you wanted to find the circumference or how far around the outside you looked. make pi multiplied by the diameter. It's a funny little number and it's equal to 3.14159 and it goes on forever. It really has no end. So it's a very pretty number in mathematics and some people would consider it quite

beautiful

. I definitely consider it quite beautiful. That's why I want to replace our X with pi now. If we did that we would get e until i pi is equal to the cosine of Pi plus i sine Pi Now, what is the cosine of Pi?

This is our cosine graph from before, now what I didn't mention is a couple of things first of all, you know, the cosine repeats the cycle over and over again if it starts high with a value of 1, it goes down to the value of 1 negative and the time it takes for it to return to where it started is one period and the other period can be written as 2 pi. Now, if I had to know, one period would be one full rotation. turn that is one full rotation or one period of turning and you could say I spanned 360 degrees.

If, instead, we stopped using the unit degrees and used something called radians, a complete rotation would be expressed with 2 pi radians and also with sine. It will complete a full cycle right here. And that's your 2pi. So what is cosine with the value of 1 pi? That's half a rotation Well, half a rotation will be down here at its minimum at its lowest point unfortunately That will be pi and it will have a value of minus 1 Sine of pi well, where will it be halfway? a rotation? He's going to be in the middle again, back to 0, so this value will be 0 e to the i pi equals negative 1 plus 0.

We don't need to have unnecessary things in our artwork here, e to the i pi equals to minus 1 Now I think it is a truly remarkable

equation

because it not only has three small constants or distinctive, beautiful and curious variables of mathematics. We have pi, our non-ending irrational number, e, a similar number that relates to all kinds of things and comes up in nature all the time, and our i, our little imaginary friend who we ask for help. All equal to quite Good number of minus one. One of the reasons I think it's beautiful is because it's so simple and because it brings together things that seem unrelated to each other into something with a very clear relationship.

Some of the

most

beautiful equations and works of art you can make in mathematics are those. some simple ones with surprising results. I'm going to show some headings at the bottom of the screen and they will be Areas where this type of math and these types of equations can be very useful. These topic areas are also courses on the website. . Brilliant.org. Brilliant is the sponsor of today's episode. So I would like to thank you for sharing the joys of mathematics through your online courses. If you want to dive a little deeper into the world of mathematics, you can visit shiny.org/Tibees. and sign up for free the first 200 of you to follow that link will also get 20% off an annual premium subscription.

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