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Welcome back.

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So let's continue where we left off.

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So we had this intuition that i must have something to do with

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these sign changes, right?

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The pattern of the sign changes of i are very similar to the

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pattern of the sign changes in the Maclaurin representation of

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cosine of x plus sine of x.

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And then we also saw that the i's, whether they're positive

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i's or negative i's, correspond to the sine terms.

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So let's do a little experiment.

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And it's not an experiment because I know where this

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leads to, but it could have been an experiment.

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What is e to the i x?

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Well, raising anything to the i power really isn't defined.

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I mean, i, itself, was created by a definition.

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We said, "i squared is equal to negative 1 by definition." So

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i is a bit of a definition.

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So if we haven't defined what something to the i power is

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yet, we really don't know what to do with it.

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But let's just say that we can treat i just

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like any other number.

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And we do know what happens with i when you put

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it into a polynomial.

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That's one thing we do know.

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In fact, that's one of the reasons why i was defined in

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first place was so that people could take roots of all

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polynomials, even ones that didn't have real roots.

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So what happens if we take e to the i x?

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Well, I don't know what that is but we know we could put that

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into the Maclaurin representation of e to the x

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and actually, since you're taking my leap of faith, that

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that is equal to e of x and all of its derivatives are equal to

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e to the x's derivatives at x equals 0, it's not

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that hard to imagine.

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And actually, you could plot the graph of this and you'll

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see that they're identical.

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So if we take the Maclaurin representation of this,

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everywhere we see an x we just replace it with an i x, right?

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So that will be 1 plus i x plus -- let me just write it -- plus

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i squared x squared over 2 factorial.

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Oops. i squared x squared plus i to the third x to the third

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over 3 factorial plus i to the fourth x to the fourth over 4

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factorial plus i to the fifth x to the fifth over 5 factorial.

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I don't have to keep going.

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Plus, and it just keeps going, right?

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So what happens when you simplify that?

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So that equals 1 plus i x -- What's i squared?

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That's negative 1, right? -- minus x squared

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over 2 factorial.

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What's i to the third?

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That's minus i.

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So it's minus i x to the third over 3 factorial

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plus i to the fourth.

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So what's i to the fourth?

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That's just 1 again.

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So we get plus x to the fourth over 4 factorial.

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And then we have -- what's i to the fifth?

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Plus i times x to the fifth over 5 factorial.

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It just keeps going.

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We have something interesting here.

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Now, all of a sudden, we have something extremely similar to

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this except for only one difference.

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Compare that to e to the i x.

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The dots on my i's always get merged.

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Compare these 2 things that I'm circling.

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What's the difference?

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Let's see the 1, 1.

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Well, here, I have an x, I have an i x here.

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Then minus x squared over 2 fact -- so these

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terms are the same.

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Then on the x to the third, the signs are right but have an i.

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And then, x to the fourth over 4 factorial -- that's identical

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-- but then on x to the fifth, I have an i.

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So the only difference between this and this is on the terms

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that involve sin of x, right?

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So what are the terms that involve sin of x?

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This term corresponds to that term, right?

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This term corresponds to that term.

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These are the terms that correspond to sin of x

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in this representation.

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That term corresponds to that term.

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And the only difference is -- so this has all of the terms

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that the sin of x would have but they all have an i in

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front of them, right?

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Even the sign is right.

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This is negative, that's negative.

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But this just has an i in front of it.

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So it turns out, that you could rewrite this, right?

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You could rewrite this representation.

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Well, it doesn't turn out.

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It's pretty obvious you could rewrite it.

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Let me clear this just so we get a --

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So we could actually rewrite that e to the i x.

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And we could write it -- we could separate out the

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imaginary terms and we could separate out the real terms.

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What were the real terms?

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Well, the real terms were 1 minus x squared over 2

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factorial plus x to the fourth over 4 factorial minus x to the

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sixth over 6 factorial.

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And it just kept going to infinity, right?

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Those were the real terms.

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That's to infinity dot dot dot.

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This pen tool looks like minus signs.

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I don't want to do that.

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Oh, I can't undo it.

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So this is just dot dot dot.

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So those are the real terms, essentially.

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And then, the imaginary terms -- it was plus -- well, all of

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these terms are going to have i on them, right?

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So let me just take the i out.

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So, plus i times -- and we figured out that those terms

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were x minus -- well, I don't want to give it away too fast

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-- x to the third over 3 factorial.

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Plus x to the fifth over 5 factorial minus x to the

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seventh over 7 factorial and it just kept going on, on,

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and on to infinity, right?

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Well isn't this the Maclaurin representation of cosine of x?

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And similarly, isn't this the Maclaurin representation

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of sin of x?

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Well yeah, sure.

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And you probably realized it in the previous screen where I

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showed that all of the imaginary terms corresponded

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to the sin of x terms.

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And all the real ones, likewise, were the cosine of x

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when we we compared it to sin of x plus cosine of x.

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So if you believe me, that the Maclaurin representation of e

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to the x is equal to e to the x and the Maclaurin

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representation of cosine and sin of x are equal to those

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functions, then all of a sudden, we come up with this

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bizarre and amazing and mystical idea that e to the i x

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is equal to cosin of x plus i times the sin of x.

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And this is called Euler's formula.

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And actually e stands for Euler.

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That's where it comes from.

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Euler starts with an E.

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E U L E R.

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But this is amazing.

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Not only have we found a relationship between this

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bizarre, mystical, magical number, e, and these

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trigonometric functions that we defined as a ratio of the sides

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of right triangles, but now we're involving this other

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mystical, magical number that we invented just so that all of

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our polynomials would have some root, whether or not

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they're real or not.

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We have this number, i, all of a sudden showing up.

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This by itself is amazing.

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But now we can take it one step further and this

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should blow your mind.

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If it doesn't, then you have no emotion.

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I will just judge you.

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So if we take this and, essentially, we're taking it

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that when you take something to the i power, that you can just

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substitute it into this Maclaurin repres -- but I

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won't go into the details.

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But I think you can say that this is a pretty

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reasonable proposition.

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But what happens if we take something to the pi power?

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If e to the i pi power?

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Before, we didn't have any way of saying, "Well,

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what does that mean?

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Taking something to the i pi power?" But now we do because

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we're saying that these 2 sides of this are

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equal to each other.

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So what happens?

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Let me do this in a bold color because it deserves to be bold.

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e to the i pi is equal to well, where x is pi, is equal to

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cosine of pi plus i sin of pi.

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Well what's cosine of pi?

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This is equal to negative 1.

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And sin of pi, well that's just equal to 0.

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We get e to the i pi is equal to negative 1.

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This is amazing.

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Or you could also write e to the i pi plus 1 is equal to 0.

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Once again, amazing.

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Either of these should make you question your take on reality

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because we have the number pi, which is a ratio of a

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circumference of a circle to its diameter.

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We have the number, e, that comes from a continuous

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compound interest.

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And then we have the number, i, which you can say the square

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root of negative 1 or it squared is negative 1.

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And they all come together.

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This formula right here involves all the fundamental

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numbers in mathematics but they come from completely

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different directions.

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Completely different directions.

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And although we can prove this and we can say this is true,

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I'll tell you no one -- no one -- probably in the history

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of mankind, fully understands why this is.

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

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