All right, so now that we know what complex numbers are, let us continue and work on some simple operations
like addition and subtraction, and I will also show you how you can do this in the complex plane.
So previously we have introduced the complex number as some of a real part and the imaginary parts times
the imaginary unit.
I would use the square root of minus one.
And we have also learned that we can represent a complex number as a vector.
So adding to complex numbers is actually really simple.
We just write them down and do it like with real numbers.
So we have the first complex number that I have written down as X plus Y times.
I read this is the real part and this is the imaginary part.
And then we have another complex number which is determined by you and V as the real and imaginary parts.
And now you can do it like with physical quantities that have a unit.
You can only add up to the quantities that have the same unit.
So you can add up X and you and you can add up Y and B because Y and we these parts here have the same
unit, which is I so our sum is X plus you as the real part and Y plus B as the imaginary part.
That's really nothing new, nothing special, just like for real numbers and for physical quantities.
Now of course the subtraction is exactly the same because any subtraction is basically just an addition
with a negative number here.
So we get the same thing, but here have different signs.
So we get X minus Uster apart and Y minus V as the imaginary part.
So let's see how that works.
So we take here two complex numbers and we want to add them up.
So our complex numbers are C one is equal to two plus three times I.
So that's that one here, C one.
And then we have a Z two, which is just this one and the two is equal to minus three.
Plus, one times I and of course, we do not really have to write down the one times, we can just write
three plus I and now our our new number, zie one plus the two is, of course, the sum of the two real
parts.
So we have two minus three and then we have the sum of the imaginary parts which is three plus one times
I so I was the one plus two is equal to minus one plus four times I.
So what we have to do and what we can do is we can add these two numbers in terms of a vector addition.
So you, you take this vector here and plotted like this and you take this vector plotted like this
and where the two vectors meet, this is where our sum ends up being.
So as you can see, we have the minus one as the real part and we have the four as the imaginary part,
just like we calculated.
All right.
So let's do the same thing for the subtraction and take again our two numbers, C one and the number
C two.
Now, if we want to calculate the difference between these two numbers, see one minus C two, we must
calculate the difference in the real part and in the imaginary part.
So we have the difference here and the difference here.
So we have in the rear part two minus minus three, and here we have three minus one.
So the difference C one minus C two is equal to five plus two times by now.
Again, we can do the same thing in terms of a vector addition or subtraction.
So we have to do is we have to take to C to Vector here and plotted along the opposite direction.
So basically mirror it.
And then again, we have to draw our parallelogram and what the two vectors meet.
This is where our difference C one minus C2 is.
So we get five as the real part and we get to see the imaginary part, just like we have here, five
plus two I.
Now, we can also do other things with complex numbers, so calculating products is possible, also
calculating quotients is possible, but that's a bit difficult.
We will do this in the next lecture and this lecture.
We will also discuss operations that are only possible for complex numbers and not for real numbers.
However, they are also only based on addition and subtraction.
So first of all, what we can do is we can calculate to so-called complex conjugate of a complex number
and that is indicated by the star up here.
So the complex conjugate, it means that you take your complex numbers, Z, and you just reverse the
sign of the imaginary part.
So instead of writing down X plus items, I, we get now X minus Y plus Y times I.
So just switch around this sign here and the real part of this number is invariant.
That remains the same.
Now we can use this complex conjugate of a number to to do special operations and two of them are to
calculate the real and the imaginary part of a number quite easily, because not all of the time you
will find a complex number that has such a simple shape.
So you ideally want to have to shape real part plus imaginary part.
But when someone gives you a complex number, it could also be that you have some products and some
exponents and you first have to really simplify and an easy way to find out what the real and the imaginary
parts of a number are, is to just calculate the sum of the number.
Plus it's complex conjugate and then divide by two.
Because as you can see, what happens then is you add up the real parts, but the imaginary parts they
cancel out because here you have this negative sign.
So you will get here two times X divided by two is equal to X and X is our real part.
For the imaginary part, you take the difference.
This is because here you will then get this minus sign here, and this means the real parts will cancel
and this time the imaginary parts will add up.
So because you have here y time as I minus minus Y times AI, which gives us two times Y times.
So when we take this and divide by two AI, we get Y, which is the imaginary part.
Now, another thing we can do is we can calculate the absolute value, and this is exactly as in, you
know, in vector algebra.
So you take the absolute value of this number X Y times eye by squaring the individual components,
the real part and the imaginary part and adding them up.
So please note this.
Here is not the absolute value yet.
This is the absolute value square and this is equal to X squared plus Y square.
And then if you know about the complex conjugate, you can also just take the number itself and multiply
it with its complex conjugate.
And you can also do this in the opposite order because then you would get by using the nominal formulae
formula, you get X squared minus Y squared, Times Square and I square is minus one, so you get X
squared, plus Y square.
And now when you have this, it's very easy to get the absolute value you just take to square root.
So you have you to square root of X squared plus by square or you have the square root of this product
of the complex conjugate and the complex numbers.
So let us now go ahead and see how this works in practice.
So let's take our number Z, which is equal to two plus three times I.
And now we want to calculate the complex conjugate.
So we write seastar is equal to two minus three times.
So we have here there's minus sine where we had here the positive side.
And now what this means in terms of writing down complex numbers as vectors is you take to the vector
after complex numbers and you move right at the real axis.
This is because the sign of the imaginary part has changed.
So now our complex numbers seastar is here.
So now that we have the numbers and it's complex conjugate, we can calculate the real.
So let's do this according to this formula here.
So our real part of C is equal to one half times and then we add up the real parts.
So this is then.
Yeah, two times or let's let's write it down like this, two plus two and the imaginary part, it's
of course three plus minus three I.
And this is equal to one 1/2 times four plus zero, and this is equal to two.
And I mean, of course, in our case, we already knew before that this was true.
So we have now that the real part is in fact, too.
But here it's a bit useless, to be honest.
But in general, you can have very difficult looking expressions here where it's not really separated
in real and imaginary part.
And you first have to calculate this.
So now we can go ahead and look at what this means for this vector, for these vectors here.
So what we have to do is we have to take the numbers and it's complex conjugate, we have to add them
up, which brings us here and we have to multiply it by one half.
So this means the real part is exactly here.
And of course, that's really the real part.
It's to.
Now, let's do the same thing for the imaginary part here, we calculate the difference.
So this means the imaginary part of Z is equal to one over two AI.
And then we have the difference of the, you know, the real parts of the axis.
So we have two minus two plus three minus minus three times AI.
And this is of course one or two AI zero.
Plus six times I and this is equal to three I divided by I is one, so we get three as our imaginary
part.
So let's that one here.
And again, we can practice by subtracting these two vectors, C and seastar.
We mirror this vector here and draw the parallelogram.
So we end up being here at the top and now we just have to multiply by one half or we could also multiply
by one over Twohy, which would rotate this vector here in the real plane.
But we will talk about how to multiply in the complex plane later.
All right.
So now we know how we can determine the real part and the imaginary part.
In the next lecture, we will talk about multiplication, division and DiPaola representation.