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IIT JEE complex root probability (part 1)

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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.