 Let's talk about complex division. We've learned how to do additions, attraction, multiplication of complex numbers. How does one do division? Well, before we can introduce division, we're gonna introduce something we call the complex conjugate of a complex number. So let's say that we have some complex numbers, z equals a plus bi, then we define its conjugate to be the complex number where you switch the sign of the imaginary part. So plus goes to minus and minus goes to plus. And you'll have to denote this by drawing a bar over the complex number. So as you can see here, if I take two plus three i, its complex conjugate is just two minus three i. You switch the imaginary part. And then the complex conjugate of negative six minus two i would be negative six plus two i. You just switch the sign of the complex conjugate, of the imaginary number. Positive goes to negative, negative goes to positive. When you calculate the conjugate, you ignore the sign of the real part. If the real part's positive, it stays positive. If the real part's negative, it stays negative. You're just gonna switch the imaginary sign. And why do we care about this? Well, this is gonna be helpful as we do complex division here because of the following property. If you take a complex number and you multiply by its conjugate, this is actually equal to a sum of squares of its real imaginary parts. And you can see this very quickly. If you take z times z bar, let me switch to a different color here, you're gonna take a plus bi and you multiply by its conjugate a minus bi. If we foiled this out by the rules of complex multiplication, you're gonna get a squared minus a bi. Then you're going to get a bi positive this time and then a minus b squared i squared. And you'll notice that there's a negative a bi and a plus a bi, they're gonna cancel out. And so this then gives you a squared minus bi squared. But as we recall, i is the squared of negative one, which means i squared is equal to negative one. And so then this expression becomes a squared minus a negative b, a negative b squared there, which then becomes the a squared plus b squared like we wanted to. So when you multiply a complex number by its conjugate, you always get a sum of real numbers, a sum of squares. And so let me show you how you can use this to compute complex quotients. If you have a fraction, a complex fraction, like one over three plus four i, the strategy that we're gonna employ is you're gonna take the denominator, which in this case is three plus four i, and you're gonna multiply the top and bottom of the fraction by its complex conjugate. So you get, you're gonna multiply top and bottom by three minus four i. Notice how I switched the sign right here. Then you're gonna multiply the top and bottom using this complex conjugate. One times anything will just be that number. So we get three minus four i. And then in the denominator, since you're multiplying a complex number by its conjugate like we saw before, you're gonna get the real part squared plus the imaginary part squared, three squared plus four squared. So you get three minus four i. Then this will be over nine and 16, which adds together to be 25. So we get three minus four i, and this will sit above 25 just right there, which we couldn't break this up into a complex number. This looks like three over 25 minus four over 25 i. And so this then would be the reciprocal of the complex numbers. And I'll leave it up to you to check if you take three plus four i, and you multiply it by three over 25 minus four over 25 i. This product is equal to one, thus showing that we did find the reciprocal of the complex number. Let's do a few more examples of this. If you had one plus four i over five minus 12 i, the idea would be multiply the top by the conjugate, which in this case should be five plus 12 i. You don't really have to worry about the denominator because the denominator, when you pull it out, you'll always get the real part squared plus the imaginary part squared. That's always what it's gonna be. In the numerator, it will take a little bit more effort as we foil this. You're gonna get one times five, you'll get one times 12 i, you'll get four i times five, and then you'll get four i times 12 i, which is gonna be a negative 48. i squared is negative one. Combining like terms now in the top, you're going to get five minus 48, which is a negative 43. 12 i plus 20 i is a 32 i. In the denominator, we get 25 plus 144. And so always writing these in its real and imaginary part, you're gonna get negative 43 over 169, and then you're gonna get 32 over 169 i. When you're working with complex numbers, it's always our goal to write this as a real part and an imaginary part. Now let's do another example. Take two minus three i divided by four minus three i. So we're gonna multiply the top and bottom by the conjugate of the denominator, four plus three i. The numerator, you'll foil it out. So two times four is eight. Two times three i is a six i. Negative three i times four is a negative 12 i. And then you're gonna get negative three i times three i. Three times three is nine. i squared is a negative one, and since there's already negative, it becomes a plus right here. In the denominator, you're gonna get a sum of squares, four squared, which is 16, plus three squared, which is nine, like so. Combining like terms, eight and nine are gonna give us a 17. 16 plus nine is a 25 for the denominator. And then six minus 12 is gonna give us a negative six over 25 i. So it's not so bad. And then maybe we do one more example just to really send it home for us, right? If you wanna divide by five plus two i, you're gonna times top and bottom by the conjugate, which would be five minus two i. Foil out the numerator. So you get two times five, which is 10. Two times negative two i, which is a negative four i. Negative three i times five, which is a negative 15 i. And then this next one, you're gonna get negative three times negative two, which is a six, and then i squared is negative one still. And then this will sit above five squared, which is 25, plus two squared, which is four. And so combining like terms, 10 take away six is a four. This will sit above 29. And then negative four minus 15 is a negative 19 sitting above 29, and then you times that by i, which then gives us the quotient right here. And so it's always our goal to write this as a real part plus an imaginary part. Every complex number can be written as a plus bi. And so we're gonna break up the fraction when we do division, spread the denominator across the real part and the imaginary part, like we've been doing in these examples here.