Complex conjugates and dividing complex numbers
Dividing complex numbers
We're asked to divide. And we're dividing six plus three i by seven minus 5i. And in particular, when I divide this, I want to get another complex number. So I want to get some real number plus some imaginary number, so some multiple of i's. So let's think about how we can do this. Well, division is the same thing -- and we rewrite this as six plus three i over seven minus five i. These are clearly equivalent; dividing by something is the same thing as a rational expression where that something is in the denominator, right over here. And so how do we simplify this? Well, we have a tool in our toolkit that can make sure that we don't have an imaginary or complex number in the denominator. And that's the complex conjugate. If we multiply both the numerator and the denominator of this expression by the complex conjugate of the denominator, then we will have a real number in the denominator. So let's do that. Let's multiply the numerator and the denominator by the conjugate of this. So seven PLUS five i. Seven plus five i is the complex conjegate of seven minus five i. So we're going to multiply it by seven plus five i over seven plus five i. And anything divided by itself is going to be one (assuming you're not dealing with zero; zero over zero is undefined). But seven plus five i over seven plus five i is one. So we're not changing the value of this. But what this does is it allows us to get rid of the imaginary part in the denominator. So let's multiply this out. Our numerator -- we just have to multiply every part of this complex number times every part of this complex number. You can think of it as FOIL if you like; we're really just doing the distributive property twice. We have six times seven, which is forty two. And then we have six times five i, which is thirty i. So plus thirty i. And then we have three i times seven, so that's plus twenty-one i. And then finally we have three i times five i. Three times five is fifteen. But we have i times i, or i squared, which is negative one. So it would be fifteen times negative one, or minus 15. So that's our numerator. And then our denomenator is going to be -- Well, we have a plus b times a minus b. (You could think of it that way. Or we could just do what we did up here. Actually, let's just do what we did up here so you don't have to remember that difference of squares pattern and all that.) Seven times seven is forty-nine. Let's think of it in the FOIL way, if that is helpful for you. So first we did the 7X7. And we can do the outer terms. 7 X 5i is +35i. Then we can do the inner terms. -5i X 7 is -35i. These two are going to cancel out. And then -5i X 5i is -25i^2 ("negative twenty five i squared"). -25i^2 is the same thing as -25 times -1, so that is +25. Now let's simplify it. These guys down here cancel out. Our denominator simplifies to 49 + 25 is 74. And our numerator: we can add the real parts, so we have a 42 and a -15. Let's see: 42 - 5 would be 37, minus another 10 would be 27. So that is 27. And then we're going to add our 30i, plus the 21i -- so 30 of something plus another 21 of that same something is going to be 51 of that something, in this case that something is the imaginary unit; it is i. (We'll do this in magenta; o, I guess that's orange.) So this is +51i. And I want to write it in the form of "a+bi," the traditional complex number form. So this right over here is the same thing as 27/74, 27/74 + 51/74 times i. (I'm going to write that i in that same orange color.) And we are done. We have a real part, and we have an imaginary part. If this last step confuses you a little bit, just remember, if it's helpful for you that this is the same thing. We're essentially multiplying both of these terms times 1/74. We're dividing both of these terms by 74. And we're distributing the 1/74 times both of these, I guess is one way to think about it. And that's how we got this thing over here, where we have a nice real part and a nice imaginary part.