[Music] okay as we all know when we have the principal square root of negative one we get
the Imaginary unit i but have you ever thought about what if we have the i-th root of -1?
oh well yes we can use this shirt to solve this right away but let me give you guys
some detail first we are going to take negative 1 to the complex world and here's a quick review
here is the real axis and here is the imaginary axis and let's say we have a complex number right
here and let's write the as a+bi a and b are real and a means the distance from here to here
and b means the distance from here to here and this is the standard form of a complex number
but we can actually represent a complex number by using another coordinate system the polar
coordinate if we know the distance from the origin to this point that's called R and if
we know the angle from here to here that's called the theta then we can write this as
re^(iθ) and to make sense of e^(iθ) we will have to utilize the Euler's formula
so here is the deal note e to the I Theta is equal to cosine theta plus I sine Theta
so this right here should also remind you of the coordinate that you guys have done back in pre-cal
or cal2 yeah and now it's just a complex version of that pretty much okay now let's look at the
negative one so if we look at negative one let's say it's right here it's a real number which is
also a complex number we need two things the first thing is the distance from the origin
to this point and because we're talking about the distance R is just equal to one yeah and the end
goal well it's just this rotation which is 180 degrees that we are adults now so let's use Pi
angle Theta is equal to Pi cool so this means negative 1 is equal to R is 1 and then e is e i
is I and the pi for the Theta so we have the PI right here that's pretty cool now let's go back
E1 that's ready yes e to the I Pi which is exactly what this show is right if we move
the one to the other side yeah e to the I pi but what do we do with the radical though just
as usual because this right here is the index so we can write it as this part of polar it's a power
form and then the one over I [Music] yeah yeah you know why I have this space but I'll tell you
but anyway if you look at this and that we can just cancel the ini right multiply the powers
so in the hand we see that this right here is just nasty equal to e to the pi and guess what
this right here it's a real number earlier when you just take an innocent square root yeah of
negative one okay complex number I but you take the ice root you get real number e to the pi
yeah and in fact this is not the only answer though because if you look at this picture again
Pi is not the only angle that can give you the rotation from here to here because we could have
gone the other direction in that case it would be negative pi or if you want to keep rotating then
you know once you get to Pi you can rotate 2 pi again right and then keep on going what the other
way keep on going so the whole angle for all this is not just Pi but we also have to add to
um Pi where n is an integer that means you can do whatever however many notations that you want
so we'll come here instead of Pi we already as plus 2 um PI right here for the angle and the
multiply by I right here and hold out so that's why I left a space right here because I will have
to add to um Pi here and of course this is still really nice because the I and one over I they
cancel so we get e to the pi plus two and pi and let's make a note that n is an integer so in fact
we have infinitely many answers and when we have the e to the PI right here this is the so-called
principle value and you can just think about this as like the first answer for all these answers
and of course if you want more answers you can plug in and is equal to 1 into here then you
can see we get e to the 3 pi and then the next one will be e to the 5 pi and so on so on so on
but don't forget we can also plug in negative integers so if you have negative one we will
get e to the negative pi and the next one will be e to the negative 3 Pi e to a negative 5 pi
and so on so and so on you can see when we have a crazy ice root of negative one we actually end up
with infinitely many real numbers but earlier we just get a not real number right how crazy
is this right and I know a lot of videos are interested in learning complex numbers but
unfortunately you might not have the resources so in this case I would highly recommend you guys to
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really cool and now here's a question for you guys to try what's (-1)^pi comment your answer
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