Let us now go ahead and talk about more difficult operations, so here we will discuss the multiplication
and division of complex numbers and how this works using the polar representation of a complex number.
So the multiplication of a complex number is not so difficult if you just do it straightforwardly by
writing down the two numbers.
So you have the real part, imaginary part, real part, imaginary part of the two numbers.
And so the result is that you have to calculate your four terms.
You have to calculate X times, you plus X times V.I. plus Y times you, plus Y times the eye.
So the only thing that may be a bit tricky here is the term Y times by eight times VI.
So what we get here is why V Times Square and I square.
It is of course minus one.
So this means this product here, y Times VII becomes actually a real part because I square is minus
one, which is a real number.
So we get as the real part x2.
Minus five.
And then there's the imaginary part.
We get the other two terms, we get X, times V plus you times Y.
Now, the next thing we want to do is to calculate the quotient of two complex numbers.
So what we have to do is we have to take the first numbers one and have to multiply by a reciprocal
number.
So here we first have to find out what is the reciprocal of a complex number.
And it's not that straightforward.
But instead, we have to use a trick.
We have to multiply by one and instead of one, we write seastar, seastar divided by seastar.
So this is here, the complex conjugate, where you change the sign of the imaginary part of the number.
And so we have learned in the previous lecture that Z Times the star is equal to the absolute value
of C squared.
And the absolute value of C squared is, of course a real number because the absolute value is a real
number.
So this, in fact, here is the real part squared plus the imaginary part squared.
And then on top here we have a complex number.
So this is a perfectly fine, complex number.
We can write down all of this mathematics here.
And so we get X minus Y times I, which is seastar divided by X squared plus Y square, which is this
one here, as I mentioned previously.
And then if you really want to split it up into the real into the imaginary part, you write it down
like this.
So the real part is the real part of the first number.
Yeah, part of the number divided by the absolute value squared minus the imaginary part times.
I divide it by the absolute value squared of the number.
OK, so now that we have the reciprocal of the number we can calculate Z two divided by zero one.
So here I have used again U plus V for a real imaginary part of C two and X and Y for the real and imaginary
parts obviously you want.
And now we can just use this relation here for one over each one.
So we have this one and we have to multiply with you plus V times I.
So we will get in all terms the denominator X squared plus Y square, which is the absolute value square
of C1 and you we will get four terms.
So these are of course these terms here and the only difficult term is probably V times I times minus
Y times I.
And so this gives us V Times Y and I square and then another minus one minus sine.
And so since I square is minus one and we have this minus sine here we get here the positive sign and
we get this entry as a real part.
So this is what the quotient looks like.
So now we can go ahead and look at what this looks like in the complex plane.
So what we will we write down is, first of all, we write down to two complex numbers.
Let's go ahead.
We have Z.
One is equal to one plus three times I and we have the two is equal to minus two plus PI.
And then we take our equation for the multiplication so we can write down the one times C two is equal
to.
And no, I will not really rely on this equation.
I will just show you how I would do it.
First, let's write down these two numbers.
Just copy them from here and then minus two plus one times.
I know you calculate the four terms first to calculate one times, minus two, which is minus two,
then you calculate one times I, which is I, then we have three times I times minus two, which is
minus six I and then we have the last term, which is three times I, times I and I.
Square is minus one.
So we get minus three and now we can take the real parts which are minus two and minus three.
So we get minus five and here we get one times I minus six times I which is minus five times I.
And you would also get this if you would just try to use this formula.
But I rather prefer do it like this.
So now we can look at what this vector looks like.
So here we have now our new vector Z one times Z two.
So that is in fact really difficult to.
Yeah.
To think about what how we can get to this vector by by a simple vector operations.
And in fact, it's not really possible because this is not a sum, not a subtraction, it's a product.
And calculating the product of two vectors is not really that simple.
We have to dot product, which is a scalar number, so not a vector.
And we have the vector product which works just in three dimensional space.
And what in this case point out of the plane.
So how do we get this vector?
Let's come back to this later when we have to pull our representation.
But first, let us go ahead and calculate the quotient Ziyuan divided by Z2, so please don't be confused
here.
I used different nomenclature for Ziemann and Z2.
So here's you want as you plus V.I., which was previously easy to.
So what we need first is this, this absolute value square X squared plus Y squared.
So this is the absolute value of C two square.
And just for your information, we can also write down the real value of C two squared plus imaginary
part of C two square.
And so this is minus two square, which is four plus one square one.
So we get five here and now we can write C one divided by C two is equal to one over five.
Then we need these two terms, which are the products of the real parts and the imaginary parts.
So we get one times minus two plus three times one, then we get plus I.
And first we must calculate X times of V, which is the real part of C, two minus two times three.
So this is minus six and then minus the other term, which is then just one times one.
So we get one of a five
um minus two, plus three is one and each here for the imaginary part we get minus six, minus one is
minus seven.
So we have C one divided by two is one over five as three apart and minus seven over five as the imaginary
part.
And this means in the complex plane, our vector is now here.
And once again, it's really difficult to see how this vector is related to the two initial vectors.
And we will find out now how they are related.