Okay, so this time I want to say something about complex numbers. So, what are complex numbers? Well, there's something missing in the ordinary numbers that we have in a way that I'll try and explain. So, if we have a...
an ordinary number, a real number, 7, 27, 3 and a half, whatever, called x, and we square it, well, we know that we're always going to get a number bigger than or equal to 0. So that's for all real numbers. x. So you multiply a positive by a positive, you get a positive.
Multiply 0 by 0, you get 0. Multiply a negative by a negative, you get a positive. So what that means is, so this, so the equation that z squared equals minus one has no solution there could be no real number no number on the number line that when you multiply by itself gives you a negative number so there is there is no such number so this has no solution in the real numbers So the real number is just that numbers are minimalised. However, it turns out that it's actually often useful to pretend that this does have a solution. So these complex numbers which we're going to invent, if you like, are going to have many applications in things like electrical circuits, in fluid mechanics. They're a convenient mathematical tool, so it's often useful to have a solution to this equation.
So we're just going to pretend that such a solution exists, and we're going to call the answer i. So we're going to define number i to be a solution of this. So obviously it's not a normal number, it's not a real number.
So we have i squared equals minus 1, or another way of saying that is that i is the square root of minus 1. So some engineers... call this thing J. It's just a convention that some people use. In this course we're going to be calling it I.
I or J. Mathematicians always call it I, but sometimes engineers will call it J. So this is sort of a bizarre thing, and we just call it the imaginary unit.
So this is sort of, we've invented it from our imagination, so this is the imaginary unit. or the square root of minus 1. So then it becomes useful when we mix that in with ordinary real numbers. So a complex number is a mixture of imaginary numbers and real numbers. So a complex number...
It's something of the form x plus i y. It's something where x and y are both real numbers. x and y are both just ordinary numbers, 27, minus pi, 3 and 4, whatever.
Well, sometimes we write it like that, or sometimes we might write it x plus y i. but it has a part which is real, so that's just an ordinary number, and this bit is sort of imaginary. This involves this magic number i.
So we call this, the real part, and this the imaginary. So often we'll use the letter z. It's a pretty common letter to use for complex numbers, rather.
And so we can say that this has a real part and an imaginary part. So we have that x is the real part. the real part of z and y is the imaginary part of z. So there's a little subtlety here. So the imaginary part of a complex number is a real number.
So there's no i there. So let's just give you an example. So if we take z to be 3 minus 4i, then what's the real part? Well, the real part is just 3. So the real part of z is 3. And what's the imaginary part?
Well, the imaginary part is the y thing here. So the y is going to be minus 4. Ah, like that. is minus 4. So it's not minus 4i, it's just minus 4. So that's just a little slip that people often make when they first do this. They think the imaginary part must be an imaginary number, but it's a real number.
Okay, so we want to do arithmetic with our complex numbers, and it turns out you can do arithmetic pretty much with complex numbers as you can do with real numbers. So you have the usual sort of laws and rules for... adding and subtracting but before going on to that I guess there's one one thing I should mention is that how do we know that two complex numbers are the same I mean so it sort of seems like an obvious question with an obvious answer but it's worth spelling it out because this has to be very useful so when the two complex numbers So if we have... Complex number Z1, which is going to be X1 plus IY1, and Z2 is going to be X2 plus IY2.
So when are they equal? Well, they're equal when they have the same real part and the same imaginary part. So Z1 equals Z2 if and only if...
We have that x1 is equal to x2 and y1 is equal to xy2. So we call this the comparison of the real imaginary parts. So we're comparing the real parts of z1 and z2, and we're comparing the imaginary parts of z1 and z2. So this is called comparing real and imaginary.
So, let's move on to a little bit of arithmetic, as I was promising. So suppose we've got two complex numbers again, so I should have left them up. So we've got z1 is x1 plus iy1, and z2 is x2 plus iy2.
And we want to add these and subtract these. So what we're going to do with this is... So let's just give you some numbers.
So we define Z1 plus Z2. We're just going to add the two real parts together to get the new real part, and then we're going to add the two imaginary parts together. So this has real part X1 plus X2, and it has the imaginary part Y1 plus Y2. And similarly...
for the difference of two complex numbers. So what do we do? We just subtract the real part of Z2 from the real part of Z1 and the imaginary part from the imaginary part. So that's just going to be X1 minus X2 is the real part, and Y1 minus Y2 is the imaginary part.
Okay, so those two are pretty straightforward, and what we want to do next is look at multiplication. So division is really going to be the slightly weirder one, or the most... the most new, the most novel part of the arithmetic.
So how do we multiply two of these together? Well, if we just write down what z1 times z2 is, we can write that out in terms of... the x is x1 plus Iy1 times x2 plus Iy2 and we can just multiply out the brackets. So just multiplying out the brackets. x1, x2, then we get, well we can do it various ways, x1 times i, y2, x1, i, y2, and i, y1, x2, and then finally i, y1 times i, y2.
Okay, now we can just move these things around. I mean, x1, i, y2 is the same as i times x1 times y2, etc. And the important one to notice here is this. So this is equal to i squared times y1 times y2.
But what is I squared? Well, that was the big thing, that was the whole point of defining I, was I squared is equal to minus what? So I squared times y1, y2 is just equal to minus y1, y2.
So this last term is just minus y1, y2. So let's just write this and collect all the terms together. So we've got two real terms and two imaginary terms. So that's x1, x2 plus... sorry, minus y1, y2, plus, we've got i times x1, y2, plus i times y1, x2, so we can just take out the i there, so that's x1, y2, plus x2, y1.
Okay. And so we've got the real part and the imaginary part. Okay.