Okay, so this time I want to tell you a little bit more about complex conjugation and some of its arithmetic properties, so how it behaves with respect to some of the operations we've been looking at, like multiplication and addition, subtraction and division. First of all, all, tell you how to imagine parts using complex conjugation. So what should we start with? So we had that z was typically going to be a complex number and we can write that as x plus or x plus yi depending.
They're both the same. And then the complex conjugate z bar is equal to x minus iy. So what happens if we add a number, complex number z to its complex conjugate?
Well, let's see. So z... plus z bar is equal to, well, just add the real part and we add the imaginary part. So x plus x is 2x, and then we get plus iy minus iy.
So it just gives us 2x, which is just the real part. So that's twice the real part. So x... Z is equal to x, which is equal to a half of z plus z bar.
So just adding a number to its complex conjugate, divided by 2, you just get the real part. And similarly, how do we get the imaginary part? Well, if we subtract one from the other, then we're going to cancel out the real part.
So z minus z bar. It's x minus x, which is nothing, and then we get iy minus iy, two minuses make a plus, so that gives us two times iy. So what can we do to get y, which is, remember the imaginary part is just y, so we can get the imaginary part by dividing by 2i.
So that imaginary part of z, is the real number y which is just oops 1 over 2i times z minus z bar okay so the real imaginary part you can just get both you know z and its conjugate okay So, what we want to do now is look at some other properties of the complex conjugate. So let's start with addition. So what happens if we add two numbers together? So suppose we've got, this time we're going to have two numbers.
So let's say z1 is going to be x1 plus iy1, and z2 is going to be... x2 plus i y2. So what is the complex conjugate of z1 plus z2? Well, this is the complex conjugate of x1 plus x2 plus i times y1 plus y2. So that's just the...
I'm just using the rule for the addition of two complex numbers. I'm just adding the real part and adding the imaginary part. So I can just use now the rule for defining the complex conjugate.
I just change the plus i into minus i. So that's just x1 plus x2 minus i into y1 plus y2. And then this becomes, I can rearrange this to become x1 minus iy1 plus x2 minus iy2. So all I've done is just rearranged that there.
And now we can spot that this is just a complex conjugate of z1. So I've just, z1 with just a minus sign in place of the plus, so that's just a complex conjugate. So that's equal to... equal to z1 complex conjugate plus, and this is similarly just z2 complex conjugate. So the conclusion is that the complex conjugate of the sum of two numbers is just the sum of the complex conjugates.
So we then have similar rules for multiplication. So what happens now if we try to multiply these two numbers together? Well, I'll spoil the surprise by telling you there's no surprise here. Okay, so z1 times z2, and we take the complex conjugate. So this is the complex conjugate of z1.
Sorry, the complex conjugate of... x1 plus i y1, x2 plus i y2. And I want to take the complex conjugate of the whole thing.
So how do I work out what the complex conjugate is? Well, I have to express this as something plus i times something. So just multiply out. Well, I'll just whiz through it this time.
But if you multiply out the brackets, we get... x1, y2 minus y1, y2 plus i into x1, y1, x1, y2 plus x2, y1. And we want to take the complex conjugate of the whole thing there, so that's the same as...
x1, x2 minus y1, y2 minus i times x1, y2 plus x2, y1. Okay, so I've just taken the complex conjugate there. I've just changed the plus i into a minus i. Now, it takes a little bit of imagination to see what this actually is.
This is actually the same as... x1 minus iy1 times x2 minus iy2. Okay, so you can check what's happened here.
So if you multiply this out... you're going to get exactly that. So all I've done is I've just changed the i's to minuses here, and that's going to just contribute to that minus i.
So you'll also get a minus minus when you multiply out in terms of the y1, y2 term, but that'll just give you a... plus. So you can just check, you should verify that that's exactly what you get when you factor that.
But of course this is the complex conjugate of Z1 and this is just the complex conjugate of Z2. So this is equal to Z1 complex conjugate times Z2 complex conjugate. So the complex conjugate of the product is the product of the complex conjugates. So as I say, no real surprise there.
Okay, so we saw the addition of complex conjugate is the same as the complex conjugate of the addition, and similarly with the product. And just two I will leave you to work out for yourself. So we also have that the complex conjugate of the difference is equal to the difference of the complex conjugates. And the more complicated one to check, but the one that's similarly true, is that if we take z1 divided by z2, you have to remember how to calculate that using realising the denominator, and we take the complex conjugate of the whole thing, I'll just try to emphasize that by putting that in brackets. So I do the quotient, and then I say the complex conjugate.
That's equal to the same as if I do the complex conjugate of Z1 first. I'll draw a big line for the quotient line. and then divide by the quotient of Z2. So these two are also true, and I'll leave those as exercises for you to have a go at.
Okay, so that finishes the little bit of simple arithmetic of complex conjugates.