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Let omega be a complex cube
root of unity with omega not being equal to 1. A fair die is
thrown three times. If r1, r2, r3 are the
numbers obtained on the die, then the probability
that omega to the r1 plus omega to the r2 plus
omega to the r3 equals 0 is-- it's a fascinating problem. So let's just start
with the first part and then see if some patterns
fall into place for us to be able to tackle
the second part. So omega is a complex
cube root of unity, which is just a fancy way of
saying it's a complex cube root of 1, with omega
not being equal to 1. So let's just set
up an equation here. Let's just think about all
of the cube roots of 1. So if we say-- that's analogous
to figuring out all of the x's that satisfy x to the
third is equal to 1. Or we can subtract 1 from
both sides of this equation and write this as x to the
third minus 1 is equal to zero. All of the roots of
this equation right here will be cube roots of 1. They will satisfy
this over here. This is the same thing as--
I could write it over here-- x is equal to the
cube root of 1. These are all
equivalent statements. So how do we do that? Well, we know what
one of the roots are. We know that x
equals 1 is a root. Or another way we
could think about it, we know that x minus
1 is a factor here. We know that this
can be factored as x minus 1 times something,
something something squared plus something x. Some quadratic,
some-- let me write here-- some quadratic right
here is equal to zero. We can factor this. And to figure out
what the quadratic is, we just have to do a
little bit of polynomial, or I guess we could call
it algebraic long division. We just have to divide
x minus 1 into this, and then we'll get
what this quadratic is. So let's do that. Let us divide x minus 1
into x to the third minus 1. And I'll do it in place. So this is the third degree
place. x to the third, I'll leave space for
the x squared place, leave space for the
x to the first place, and then we have x to the
zero, which is the ones place. So that's negative 1. So I just wrote x to
the third minus 1, but I wrote it
out this so I have space to keep our
counting pretty clean. So we just do algebraic
long division here. So x minus 1 goes into
x to the third minus 1, you just look at the
highest degree terms. x goes into x to the
third x squared times. It goes x squared times. x squared times x
is x to the third. x squared times negative
1 is minus x squared. Now, we want to subtract
the magenta stuff from the orange stuff, or we
could add the negative of it. So let's just multiply
this times negative 1. So positive-- sorry, we're
multiplying times negative 1. So this is a negative, and then
this will become a positive. We're subtracting
what I originally had written from that, but
I just made it negative. And now we'll add. And so this will become--
these cancel out. We get x squared minus 1. Just algebraic long division. Once again, how many
times does x minus 1 go into x squared minus 1? Look at the highest
degree terms. x goes into x squared--
I'll just arbitrarily switch colors-- x goes into
x squared x times. So this is plus x. x times x is x squared. x times negative
1 is negative x. Same drill. Subtract this from up there,
or we could add the negative. So this becomes negative,
this becomes positive. These guys cancel out. You get x-- bring
this down-- x minus 1. x minus 1 goes into x
minus 1 exactly one time. It goes exactly one time. 1 times x minus 1 is x minus 1. And you get a remainder of zero. So it divided evenly,
which makes sense. Because this is one of
the factors of that. We know that this is a root. So if we wanted to
completely factor this, we could write this as x
minus 1 times this thing over here, times the
quadratic, times x squared plus x plus
1 is equal to 0. So if we wanted to find
the non-one roots of this over here, we just have to find
the roots of this thing right over here. Whatever the roots
are of this are going to make this expression
equal to 0, which would make this entire
expression equal to 0, make this entire
expression equal zero. So we just have to
find this guy's roots. And I can already guess that
they're going to be complex. So don't even
attempt to factor it. You could attempt to
factor it if you like just by looking at it,
but we're going to use the quadratic formula. So the roots here
for this part-- we say x is equal to negative b. b is 1, so negative 1 plus
or minus the square root of b squared, b is 1, so 1
minus 4 times a times c. Well, a and c are both 1. So it's just minus 4. All of that over 2
times a. a is just 1. So all of that over 2. So x is equal to
negative 1 plus or minus the square root of
negative 3-- we were right, this is going to be a
complex root-- over 2. Now, what's the square
root of negative 3? We could write this-- the
square root of negative 3, this is the same thing as
the square root of negative 1 times the square root of 3. And this we know from
studying imaginary numbers and complex numbers. This is i. That right there is i. So the other roots--
and there are two here, because we're plus
square root of negative 3 and subtracting the
square root of negative 3. So the other roots are x
could be equal to negative 1 plus or minus the square
root of 3 times i. And I'll put i here, just
so it's nice and clear. That's not a vector. Just like that. i. All of that over 2. So what we just did
so far is we were able-- we now know
all three roots of 1. Especially if we're thinking
about the complex roots. 1 is one of the roots. The other roots are negative
1 plus the square root of 3i over 2, and negative 1
minus the square root of 3i over 2. And so they're saying that
omega is one of these two roots. We either take the
positive or the negative. And it looks like
it doesn't matter. It says let omega be a
complex cube root of unity. So it could either one of these. So let's just pick one of them. So let's just say omega
is equal to negative 1 minus the square root
of 3i-- and I'll just distribute the divided by 2. So we could say we
could call this-- let me write it this way. It's negative 1/2 minus
the square root of 3/2i. I'm just picking-- once
again, not a vector. My brain keeps wanting me to
put a little hat on top of it. It's not a vector. It's just an imaginary number. It's i. This is actually
a complex number, because we have a real
part and an imaginary part. So I'm just going to
pick that to be my omega. It's one of the non-one
complex roots, third roots, or I guess we could say-- what
was the actual wording they said-- one of the
complex cube roots of 1, of unity, that is not 1. So let's just let
omega be equal to that. So this first statement
can translate into this. And then a fair die
is thrown three times. If r1, r2, and r3 are the
numbers obtained on the die, then the probability
that omega to the r-- So let's think about
this a little bit. Because we're going to be taking
omega to different powers. So let's think
about what happens when we take this omega
to different powers. So this right here is clearly
omega to the first power, the way we've defined it. Let's think about
what omega squared is. So omega squared is
equal to negative 1/2 minus square root of 3 over
2i-- and this will give us good practice multiplying
complex numbers-- times negative 1/2 minus times--
well, it's the same thing. We're squaring it. And so this is going to be
equal to negative 1/2 times negative 1/2 is positive 1/4. And then you have negative
1/2 times negative 3/2i. The negatives cancel out. The square root of
3/2 I should say. So negative 1/2
times this is going to be positive
square root of 3/4i. Then we could multiply
this times this. It's going to give us
the same thing as that. So the negatives cancel out. And so we have plus the
square root of 3/4i. And then we're going to
want to multiply the two imaginary terms. So the negatives cancel out. And so you have square root
of 3 times square root of 3, which is just 3. And then you have 2
times 2, which is 4. And then you have i times
i, which is negative 1. So let me just put a negative 1. So times negative 1 over here. So let's simplify this. This gives us 1/4 plus
square root of 3/4i plus square root of 3/4i. That's 2 square roots
of 3/4i or square root of-- so that's the same
thing as square root of 3/2i. And then minus 3/4. And we can simplify
this even more. The real parts, we can add
1/4 minus 3/4 is negative 2/4, or negative 1/2. So this is negative 1/2 plus
the square root of 3/2i. So that's interesting. When I squared omega,
I got another one of the complex cube roots of 1. I actually got its conjugate. And if any of y'all who've
studied complex analysis and think about
the complex plane, it should be pretty
clear why this happened. But if you don't, don't
worry too much about it. Now what I want to do
is I want to take omega to the third power, just
to really understand what's happening with omega. And if you've studied
complex analysis, this part would be
quicker for you. You would know it. So what's omega to
the third power? Omega to the third
power is going to be omega squared times omega. Omega squared is just
that right over there. We just calculated it. It's negative 1/2 plus
square root of 3/2i. And we're going to have to
multiply that times omega. Omega was the conjugate
of this thing. So it's negative 1/2
minus-- conjugate just means opposite side
on the complex part. So minus square root of 3/2i. That's what we defined
omega to be right over here. This right over here is omega. That's omega. This right here
is omega squared. So let's multiply these. So first, negative
1/2 times negative 1/2 gives us negative-- sorry,
gives us positive 1/4. Have to be careful. This gives us 1/4. And then if we
take negative 1/2-- I shouldn't do it that way. Let's do negative 1/2 times
negative square root of 3/2i. Negatives cancel out, so we
get plus the square root of 3-- let me make sure I have--
square root of 3/4i. And then we can multiply
this times this. It gives us the same thing. So it gives us plus-- let me
make sure I'm doing this right. Oh, no, it's not going to
give us the same thing. Now one is negative,
one is positive. So it's going to give
us-- be very careful. So negative 1/2 times
positive square root of 3/2i is negative square root of 3/4i. And then finally, multiply
the two imaginary parts. Multiply that-- I want to use
a different color-- multiply that times that. That gives us positive times a
negative gives us a negative. Square root of 3 times
square root of 3 is 3. 2 times 2 is 4. i times i is negative 1. Or we can multiply
the negative 1 times this negative over here,
becomes a positive. So what do we get? These guys cancel out. We have 1/4 fourth plus 3/4. It is equal to 1. So this is interesting. This is interesting over here. So omega is equal to this
thing right over here. Omega squared is equal
to its conjugate. Omega squared is equal to its
conjugate right over here. We just swapped the
sign over there. And then omega to
the third is going to be equal to this thing. It's going to be equal to unity. Actually, we knew that. I shouldn't even have
to worry about this. I don't know why I even
went through that process. Well, it was good multiplying. We knew it was the
cube root of unity. So we knew if we took
it to the third power. So this was to some degree
just a waste of time. But what does this tell us? So we now kind of
understand what the different powers of omega. What will omega to
the fourth power be? What's going to be
this times omega? So it's just going
to be omega again. What's omega to the fifth power? It's going to be this. What's omega to the sixth power? It's going to be this again. So let's just use
that, and we're going to tackle that in the
next part of the problem in the next video.