Hi, I'm Toby and welcome to the joy of mathematics. Today we're going to work up towards something very beautiful
So if you'd like to work along at home, now's your chance to grab your own notepad and pen and you can follow along.
Don't be afraid by what is already on the board
This is just a bit of primer that we're going to need to get started.
I know that mathematics can sometimes evoke emotions of fear in people but I really wish it wasn't that way.
Mathematics can be very fun, and I think that's a side of it that we're going to see today.
Don't be afraid by this math, I'm going to be here and we're going to work through 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 X, our exponential function, our
Cos of X and our sine of X and the terms which allow us to write out expansions for these functions.
To show you what e to the X looks like if you don't remember him
just something that
grows at almost an alarming rate really, something that
Increases with an ever-increasing rate and something that follows this sort of shape would be something like my love for mathematics.
Let's move on to cosine. This is one of our trig functions.
So if we were going to do a plot of cosine on a little axis
Cosine is someone who starts off, maybe he's born into a rich family. He starts off very high in life
But then things steadily get worse and there's a dip
You know, everyone has low moments
But but then things get better and in fact cosine shows us that life is a constant cycle of ups and downs, you know?
Like I like to say these low points. This darkness is really needed to appreciate the light
We gave cosine a friend and that's sine. They're very similar people
Sine unlike cosine had to work for what he had
He's sort of started with nothing, but he worked up to be pretty great and then nothing lasts forever
He eventually lost that and he too found that life is a constant cycle of ups and downs.
Now you might notice a few
Similarities between our functions, they all have these similar terms which are actually X to the N over N factorial
In the case of cos,
His terms are the even terms, x squared, X to the fourth, X to the 6, and then sine
The odd terms, X cubed, x to the 5, X to the 7, and in e to the X
All the signs are positive. We don't have this
Alternating positive negative like we did down here and we have every power of X
So you can see that there may be some relation between these three people here
It's almost like they are one big family
But it doesn't seem to quite add up because if we, we were to add cos and sine
You know, maybe we'd get something similar to e to the X, we'd get all of the right terms
But we'd still have these minus signs in here
So we can't just do a simple addition to relate these functions to this one up here
We need a little bit of help and you know in those times when I struggle and I need a little bit of help
I do like to you know, it's embarrassing to say this
But resort to asking my imaginary friends. Some people laugh when I tell them about my imaginary friends
But I'd like to introduce you to one of them today and that's my friend i.
i can be written as the square root of minus 1, now
It's a pretty funny little character to have i times i equal to a negative number, but that's exactly what's happening here
Hold on to your calculators ladies and gentlemen, but it's going to even get a bit stranger. We'll get to that later
Now we're going to use our little friend for some help. We're going to actually rewrite this e to the X function as e
to the i-x
That means that anywhere there is an X here I can replace it with an i-x. Let's go through and do that
Keep our brackets right so that we know that that whole term is squared, likewise here
And
There we go. We now have our function e to the i-x
But what do some of these terms do what can i really do for us? Let's have a look. So i
To the power of 1 is just going to be i
but i squared
is going to be minus 1, that's from our definition here and
if we were to do i cubed that would be i,
Or minus i. i to the 4
would be
1 and in fact this series repeats itself 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 a minus 1
let's take out that i
Take out that bracket and change this to a minus 1
Here we've got i cubed, we can replace that with a -i
i to the four, we can replace that with a one
Should leave that anyway
No mistakes here. Just happy accidents and
our i five well we can work through this series to work that one out and
an i to the five
Would be equal to
one times i, that would be i
Okay
Now that looks pretty good. We still have though our imaginary friend speckled in here and one piece of advice
I have for you is to always keep your real friends and your imaginary friends separate from each other
So let's write down this function but separating out our real and imaginary friends. We can erase this here
Wipe these little devils off
And here we go, e
To the i-x
Let's do the real terms first, so he's real
He's real he's real and I think anything that has an even power will be real. So let's put it in brackets
1
Minus x squared over 2 factorial
Plus X to the 4
4 factorial
And there's going to be an infinite number of those real terms
Let's do our imaginary ones. Now all of our imaginary terms are going to have this i out front
So, in fact, we can just take the i out in front of our brackets. What have we got 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 little look at what we're actually working with here the eagle-eyed ones amongst you might recognize that this here is
Really nothing more than our old friend
Cosine of X. This is this function
And you also might realize that what's in brackets here is nothing more than our old friend sine of X
so this is i times sine of
X
So e to the i-x can also be written as cosine of X plus i sine X
That's a pretty amazing so far, but we're going to go a little bit further with it
Let me rub this off and make some more space for ourselves
okay, so I've written back up our equation that we've worked towards and let's keep building so
e is a pretty cool function. Like I said, it's an exponential and it's also this nice number in mathematics.
i is our imaginary friend and X is some unknown but what if we threw in one more curveball here?
I'm sure you guys can handle it. I do believe in you and our curveball is going to be Pi
Many of you would have heard of Pi, it's related to circles
in fact
If you knew the diameter of a circle and wanted to find the circumference or how far around the outside
You look do pi times the diameter. It's a curious little number and it's equal to
3.14159 and it keeps going forever. There is really no end to it
So it's a very nice number in mathematics and some people would consider it quite beautiful.
I definitely consider it pretty beautiful. So I want to replace our X with pi now. If we were to do that we would get
e to the i pi is equal to cosine of
Pi
plus i
sine
Pi
Now what is cosine of Pi?
This is our cosine graph from before, now what I didn't mention are a couple of things
first of all, you know
You know cosine repeats the cycle over and over again if he starts up high at a value of 1
He sinks down to the value of negative 1
And the time it takes with him to go all the way back to where he started
Is one a period and one a period can be written as 2 pi
Now you might know a period as a full rotation if I was to spin around
that's a full rotation or one period of spinning and
You could say I spanned
360 degrees. If we were to instead stop using the unit degrees and use something called radians a
Full rotation would be expressed with 2 pi radians
likewise with sine
He will complete a full cycle
Right here. And that's his 2pi
So what is cosine at the value of 1 pi? That's half a rotation
Well half a rotation he will be right down here at his minimum at his lowest point
unfortunately
That's going to be pi and it's going to be a value of minus 1
Sine of pi well, where is he going to be at half a rotation?
He's going to be back at the middle, back at 0 so this value is going to be 0
e to the i pi is equal to minus 1
plus 0
We don't need to have unnecessary things in our artwork here, e to the i pi is equal to minus 1
Now I think that is a truly remarkable equation because not only does it have three
Distinct beautiful and curious little constants or variables from mathematics. We have pi our
Irrational number that doesn't end, e a similar number to that
that relates to all sorts of things and crops up in nature all the time and our i, our
Little imaginary friend who we asked for help
All equalling to quite the nice number of minus one. One of the reasons
I think it's beautiful is because it is so simple and because it brings together things that seem to be unrelated to each other
Into something with a very clear relationship
some of the most beautiful equations and artworks you can make in mathematics are those simple ones with
surprising results
I'm going to run some titles across the bottom of the screen and that's going to be
Areas where this sort of mathematics and this sort of equation can come in quite handy
These subject areas are all also courses on the website. Brilliant.org. Brilliant are the sponsor for today's episode
So I would like to thank them for sharing the joys of mathematics through their online
Courses if you would like to dip your toes a little deeper into the pool of mathematics
then you can go to brilliant.org/Tibees and sign up for free the first
200 of you that follow that link will also get 20% off an annual premium subscription
This is also your personal invitation to subscribe to my channel
If you would like more lessons like this and from the bottom of my heart, I wish you an absolutely mathematical day