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PROFESSOR: Hi.

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Today we'll explore the complex
numbers and Euler formula.

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So the first part
of the problem is

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to write this complex number,
minus 2 plus 3i, in polar form.

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And at this point, it's written
in rectangular coordinate form.

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The second question asks
you to do the reverse,

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to write three exponential to
the i*pi over 6 in rectangular

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coordinate form.

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Question c asks you to
draw and label the triangle

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relating the
rectangular coordinates

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to the polar coordinate form.

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D asks you to compute 1 over
this reverse complex number

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that we already
saw in question a.

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And e asks you to find
the cube root of 1.

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And in all these questions,
you'll be using Euler formula.

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So why don't you
pause the video,

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take a few minutes to
work out the problem,

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and we'll come back.

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Welcome back.

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So we're asked
throughout the problem

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to go back and forth between
coordinates in polar form

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and in rectangular form.

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So a key thing to remember is
Euler formula from the start.

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So we're just going
to write it up here.

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It allows us to express
a complex exponential

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into the sum of its
cosine plus i sine theta.

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So how do we tackle question a?

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Question a gives us a complex
number in rectangular form.

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So in this form, a plus i*b.

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And we're asked to write
it in polar form, which

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introduces the modulus of the
complex number r and its phase

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theta.

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So r, the modulus of
the complex number

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that we can compute when we
know its rectangular form,

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with its real form squared
plus imaginary part squared,

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the whole thing under the root.

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So in this case,
we have 4 plus 9.

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So we end up with root
of 13 for the modulus

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of the complex number z.

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So now for the phase.

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Using Euler formula,
we can see that we

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can relate the rectangular
form to the polar form

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by just introducing--
I'm going to keep r.

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And you can see now
that we can extract

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the sine and the cosine
of the angle theta

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and relate that to a ratio of a.

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And the modulus r
that we just found,

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b modulus r that we just found.

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Or in one move, just express
it as the tangent of the angle

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theta, just sine
over the cosine,

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just becomes basically b
over a, which we have here.

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So we have the modulus r,
which is now root of 13,

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and the angle theta
that we can now

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extract by using the
reverse of the function tan.

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So just before we move
to the next question,

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this is not one of the classical
angles that you learned.

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So just to have an idea
of where this angle lies

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on the trigonometric
unit circle,

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just recall here that
the sine is positive

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and the cosine of this
angle is negative.

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So we're bound to be in this
region, where basically theta

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is between pi and pi over 2.

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And that's the
answer to question a.

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So now for question b, we're
asked to do the reverse,

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expressing the polar number
3 i*pi over 6 in rectangular

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coordinates.

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So now this is just a
straightforward application

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of the Euler formula
that we just saw.

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By just expanding
the exponential,

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as I already wrote there.

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Plus i 3 sine pi over 6.

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And on the same
trigonometric circle

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here, that's roughly
where pi over 6 lie.

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And you can just re-express this
as 3 root of 3 over 2, plus i 3

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over 2.

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So that ends the
solution for question b.

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So now question c.

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Let me just add a line here.

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Question c we're asked
to draw and label

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the triangle relating
rectangular to polar

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coordinates.

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So that's what we already had
a sense with when we wrote this

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formula going from a plus i*b
to r exponential of i*theta.

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So in the complex plane,
we have the real axis,

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an imaginary axis,
and a complex number

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lying on this plane
written in this form

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in a rectangular coordinate.

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You'd have a projection of a
in the real axis, projection

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of value b on the
imaginary axis.

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And in polar form this
would be its modulus

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or distance from the origin,
and its phase theta that

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would come in the polar form.

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So that's the
triangle that allows

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us to go back and forth between
the rectangular and polar

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coordinates.

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So to almost finish,
question d now

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asks us to compute the reverse
of the original complex number

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that we used.

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So 1 over minus 2 plus 3i.

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So to do this, we can
stay in rectangular form

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and basically multiply the
numerator and denominator

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by the complex
conjugate of the number.

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But clearly now
that we learned how

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to use polar coordinate
expressions of this number,

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it's much easier to just write
it directly in this form.

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In one step we basically
arrived to the results,

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where we express that the angle
was the reverse tan of minus 3

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over 2.

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And that's done.

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So now are for
the last question,

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we were asked to compute
the one third root of 1.

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So basically, 1 to the 1/3.

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So here, obviously, we're
treating 1 as a complex number.

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And if we go in
the complex plane

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and I just introduce
here the number 1,

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we see that in polar
form 1 is just basically

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a complex number with modulus
1, and angle 0, modulo 2pi.

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So we can write 1 as
exponential 2n*pi,

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because it's basically
angle 0 modulus 2pi.

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And from here we know that
we're looking at third roots,

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so we're going to
have three roots.

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And these roots are
going to be expressed

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by changing the value of n.

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First one, n equals
to 0 is just going

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to give us back root
of 1, because we're

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going to have exponential
to 0 is just 1.

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Power of 1/3 is just 1.

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n equals to 1.

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We are going to have
exponential of 2pi

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over 3, which we can express,
again using the Euler formula,

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also in coordinate form.

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And then just write
down the values.

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And for the third root we
take the value n equals to 2,

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so we have i*4pi over 3,
which again we can express

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as the cosine plus the
sine of 4pi over 3.

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So where do these roots lie?

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So we have root 1
for n equals to 0.

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The second root, exponential 2pi
over 3, basically in polar form

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would be here, where we would
have the angle 2pi over 3.

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So 1pi over 3 would here.

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2pi over 3 would be here.

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3pi over 3 would be here.

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And 4pi over 3 is our third
root, it would be here.

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So then we can just
write down the values.

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And you can do that when
you know the angles,

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or just keep it in either form
when you don't know directly

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the expression for the angles.

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So this completes the problems.

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In all of these problems
what we kept using

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is Euler formula to go back
and forth between coordinate

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in rectangular
form to expression

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of complex number in polar form.

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And that's the key formula
that we kept using.

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And you'll be using
this repeatedly when

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we will be solving
other ODEs for which we

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can use complex number
as a trick for solutions.

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And this ends this session.