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Polynomial approximation of functions (part 7)

So we had this intuition that i must have something to do with these sign changes, right? The pattern of the sign changes of i are very similar to the pattern of the sign changes in the Maclaurin representation of cosine of x plus sine of x. And then we also saw that the i’s, whether they’re positive i’s or negative i’s, correspond to the sine terms. So let’s do a little experiment.

And it’s not an experiment because I know where this leads to, but it could have been an experiment. What is e to the i x? Well, raising anything to the i power really isn’t defined. I mean, i, itself, was created by a definition.

We said, “i squared is equal to negative 1 by definition.” So i is a bit of a definition. So if we haven’t defined what something to the i power is yet, we really don’t know what to do with it. But let’s just say that we can treat i just like any other number.

And we do know what happens with i when you put it into a polynomial. That’s one thing we do know. In fact, that’s one of the reasons why i was defined in first place was so that people could take roots of all polynomials, even ones that didn’t have real roots. So what happens if we take e to the i x? Well, I don’t know what that is but we know we could put that into the Maclaurin representation of e to the x and actually, since you’re taking my leap of faith, that that is equal to e of x and all of its derivatives are equal to e to the x’s derivatives at x equals 0, it’s not that hard to imagine.

And actually, you could plot the graph of this and you’ll see that they’re identical. So if we take the Maclaurin representation of this, everywhere we see an x we just replace it with an i x, right? So that will be 1 plus i x plus — let me just write it — plus i squared x squared over 2 factorial. Oops. i squared x squared plus i to the third x to the third over 3 factorial plus i to the fourth x to the fourth over 4 factorial plus i to the fifth x to the fifth over 5 factorial.

I don’t have to keep going. Plus, and it just keeps going, right? So what happens when you simplify that? So that equals 1 plus i x — What’s i squared? That’s negative 1, right? — minus x squared over 2 factorial. What’s i to the third? That’s minus i. So it’s minus i x to the third over 3 factorial plus i to the fourth.

So what’s i to the fourth? That’s just 1 again. So we get plus x to the fourth over 4 factorial. And then we have — what’s i to the fifth? Plus i times x to the fifth over 5 factorial. It just keeps going.

We have something interesting here. Now, all of a sudden, we have something extremely similar to this except for only one difference. Compare that to e to the i x. The dots on my i’s always get merged.

Compare these 2 things that I’m circling. What’s the difference? Let’s see the 1, 1. Well, here, I have an x, I have an i x here. Then minus x squared over 2 fact — so these terms are the same.

Then on the x to the third, the signs are right but have an i. And then, x to the fourth over 4 factorial — that’s identical — but then on x to the fifth, I have an i. So the only difference between this and this is on the terms that involve sin of x, right? So what are the terms that involve sin of x? This term corresponds to that term, right? This term corresponds to that term. These are the terms that correspond to sin of x in this representation.

That term corresponds to that term. And the only difference is — so this has all of the terms that the sin of x would have but they all have an i in front of them, right? Even the sign is right. This is negative, that’s negative.

But this just has an i in front of it. So it turns out, that you could rewrite this, right? You could rewrite this representation. Well, it doesn’t turn out.

It’s pretty obvious you could rewrite it. Let me clear this just so we get a — So we could actually rewrite that e to the i x. And we could write it — we could separate out the imaginary terms and we could separate out the real terms.

What were the real terms? Well, the real terms were 1 minus x squared over 2 factorial plus x to the fourth over 4 factorial minus x to the sixth over 6 factorial. And it just kept going to infinity, right? Those were the real terms. That’s to infinity dot dot dot. This pen tool looks like minus signs.

I don’t want to do that. Oh, I can’t undo it. So this is just dot dot dot. So those are the real terms, essentially.

And then, the imaginary terms — it was plus — well, all of these terms are going to have i on them, right? So let me just take the i out. So, plus i times — and we figured out that those terms were x minus — well, I don’t want to give it away too fast — x to the third over 3 factorial. Plus x to the fifth over 5 factorial minus x to the seventh over 7 factorial and it just kept going on, on, and on to infinity, right? Well isn’t this the Maclaurin representation of cosine of x? And similarly, isn’t this the Maclaurin representation of sin of x? Well yeah, sure. And you probably realized it in the previous screen where I showed that all of the imaginary terms corresponded to the sin of x terms.

And all the real ones, likewise, were the cosine of x when we we compared it to sin of x plus cosine of x. So if you believe me, that the Maclaurin representation of e to the x is equal to e to the x and the Maclaurin representation of cosine and sin of x are equal to those functions, then all of a sudden, we come up with this bizarre and amazing and mystical idea that e to the i x is equal to cosin of x plus i times the sin of x. And this is called Euler’s formula.

And actually e stands for Euler. That’s where it comes from. Euler starts with an E.

E U L E R. But this is amazing. Not only have we found a relationship between this bizarre, mystical, magical number, e, and these trigonometric functions that we defined as a ratio of the sides of right triangles, but now we’re involving this other mystical, magical number that we invented just so that all of our polynomials would have some root, whether or not they’re real or not. We have this number, i, all of a sudden showing up.

This by itself is amazing. But now we can take it one step further and this should blow your mind. If it doesn’t, then you have no emotion. I will just judge you.

So if we take this and, essentially, we’re taking it that when you take something to the i power, that you can just substitute it into this Maclaurin repres — but I won’t go into the details. But I think you can say that this is a pretty reasonable proposition. But what happens if we take something to the pi power? If e to the i pi power? Before, we didn’t have any way of saying, “Well, what does that mean? Taking something to the i pi power?” But now we do because we’re saying that these 2 sides of this are equal to each other.

So what happens? Let me do this in a bold color because it deserves to be bold. e to the i pi is equal to well, where x is pi, is equal to cosine of pi plus i sin of pi. Well what’s cosine of pi? This is equal to negative 1.

And sin of pi, well that’s just equal to 0. We get e to the i pi is equal to negative 1. This is amazing. Or you could also write e to the i pi plus 1 is equal to 0.

Once again, amazing. Either of these should make you question your take on reality because we have the number pi, which is a ratio of a circumference of a circle to its diameter. We have the number, e, that comes from a continuous compound interest. And then we have the number, i, which you can say the square root of negative 1 or it squared is negative 1.

And they all come together. This formula right here involves all the fundamental numbers in mathematics but they come from completely different directions. Completely different directions. And although we can prove this and we can say this is true, I’ll tell you no one — no one — probably in the history of mankind, fully understands why this is.

This is just a glimpse on some type of order in the universe.

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