Sal's old Maclaurin and Taylor series tutorial
Polynomial approximation of functions (part 7) The most amazing conclusion in mathematics!
Polynomial approximation of functions (part 7)
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- Welcome back.
- So let's continue where we left off.
- 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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At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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