All right, so let us now come to the solutions to the exercises so he can see once again that three
tasks.
So if you have not done so, please stop the video and please take a sheet of paper and calculate these
tasks one, two and three.
So now let's come to the solutions.
So our first task was to calculate to some difference and a product of two complex numbers.
So that was pretty easy, I hope.
So let's start with task number eight.
So here we have to calculate the sum.
So we have to calculate three plus two I plus minus one, plus four I and that is of course that you
have to add the real parts first two and then the imaginary parts.
This is six I then four B we have here the same thing and then we have to subtract minus one and four.
I so we get for the real part four and for the imaginary part we get minus two I.
Now for the product, you have to calculate three different terms, so you have to.
Multiplied at two real paths to imaginary parts and to two mixed terms is what we get here is minus
three, then we get three times for EI, which is 12 I and we have minus two EI for the other mixed
term and for the imaginary product we have two times fours eight and items ie is minus one, so we get
minus eight.
So in total we get minus 11.
Plus 10 I.
OK, so while the first task was really easy, the second task is a bit more difficult.
So we have again two numbers.
First of all, we have here one number that is already separated into the real and the imaginary part.
And then we have here a number that is represented in this oilor representation.
So each of the power of AI and then something.
So let's start and calculate, first of all, the complex conjugate.
So this is C one star and this is just minus one.
Minus two I.
And for the other number, in the earlier representation, it is said to Star is equal to two times
E to the minus five, PI over eight.
This is because all of the numbers here are real numbers.
And just this eye is a complex number and we have to calculate the complex conjugate.
Also, another trick that I can show you here is you can write down that this is equal to two times
cosine PI over eight plus I sign PI over eight.
And so now you can see we have already separated this into the real and into the imaginary part, and
this means we could write down that C two square, C two star is equal to two times cosine PI over eight
minus.
I sign PI over eight.
And now, since cosine is an even function and sine is an odd function, we can also write that this
is equal to cosine minus pi over eight plus I sign minus Pi over eight.
And now this you can see is really what we have written down here with the minus sign.
All right.
So this little trick here I have only shown you to to prove that this is really correct, in case you
didn't believe me.
But also it helps us for the next task where we have to calculate the real part of the equation and
see two and also the imaginary part.
So here, obviously, it's very easy to have to write down these two numbers and please be careful.
The imaginary part is a real number.
So it's minus two and not sorry, it's of course, plus two because we want to calculate it of this
number.
That's plus two and not plus two eye.
The eye is just the imaginary number, but the imaginary part is just a real number.
All right.
So here we have a real part of C two is equal to two times cosine of PI over eight.
And the imaginary part of C two is two times sine PI over eight.
All right.
Now we want to calculate the absolute value.
So this is the square root of the real part squared plus the imaginary part square.
So this is one squared plus two square.
Square root is a square root of five.
Very easy.
And here we have absolute value of C two.
And here again, I want to show you two methods.
First of all, the simple method which you should use.
This is just absolute value of two times absolute value of E to the power of I pi over eight.
And so the absolute value of each of the eyes, something is always one.
So the absolute value is two.
I mean, that's the whole trick and the whole purpose of writing it down in such an hourly the representation
that you write, the absolute value in front of it.
But if you didn't believe me, you could also write down that this is equal to the square root of.
And now we have to use our real and imaginary parts.
So we have two square times cosine PI over eight squared plus two square.
Right down with these brackets sine square.
Pi over eight.
And, of course.
If you have a cosine square plus assigned square, no matter which argument you have, it's always one.
So you have that this is equal to two square square root is equal to two.
So it's the same thing.
OK, and then the only thing that's left is the reciprocal.
So we have to calculate one of as you want.
So here we have or we we can do multiple things.
But what I like to do is multiply by one.
So this is equal to one.
And now what is written down here is the one star divided by absolute value of C one square.
And we know, what, seven squares and we know what city one stars, so we can just write it down from
above minus one, minus two, I divided by square root of five square, which is five.
So this is equal to minus one five as the real part and minus two over five I or minus two or five as
the imaginary part.
And here we have one over Z.
Two is equal to.
Yeah, here we can do it really simple, we can just ride one of A to E to the I Pi over eight and so
this is one over two times one over here to the I pi over eight.
And if you know the rules for exponential functions, you know that this is one of a two E to the minus.
I pi over it.
So as you can see these, this is, are the representation has some advantages.
For example, you can quite easily calculate the absolute value and the inverse, but it also has disadvantages.
It was a bit more difficult to calculate the real and the imaginary parts compared to the left hand
side.
OK, and let's come to the last part where you should practice drawing these complex numbers as vectors
in the complex plane.
So first of all, let us draw the two numbers, C1 and Z two.
And so let's start with the one in red.
We have a real part of minus one and we have an imaginary part of two.
So our vector points like this, sorry, it's a bit ugly, but hard to hard to draw on this tablet here.
So then we can look at C to and draw in blue.
So we know that the length of this vector is two and we know that the polar angle of this vector is
PI over eight.
So we know that here it is, imaginary axis is PI over two, which would be ninety degree, so PI over
eight would be one quarter.
So it would be approximately let's say.
Would look like something like this.
All right, so this is our too, and now we can draw all of these combinations, you had the some the
difference, the product and the quotient.
So first of all, let us calculate the sum of these two numbers so we have to draw a parallelogram so
we could, for example, draw like this.
So I think should it should look something like this.
See one plus see two, then for C, one minor C to this is where we have to draw this arrow here into
the opposite direction.
So we have to draw the parallelogram along.
Like this.
So our Victor, I think, would point like this, this is the one liners minus to.
All right, now it's a bit more difficult now we need to calculate the product and we have learned that
the product of two complex numbers can be calculated by multiplying their lengths and by adding the
polar angles.
OK, so the length of these two vectors, so we know that the length of C2 is two and we know that the
length of this vector here is a bit larger than two, but smaller than three.
So that's estimated something like 2.5.
So we get as a product the length of five.
OK, so our vector must be approximately the length of five and now we just have to add up these two
polar angles.
So this angle here is a bit small.
It's it's 90 degree over force or twenty two point five.
So we have to add twenty two point five to this angle here.
So I would say it's pointing along this direction and it has a length of five.
So, yeah, it's hard to say maybe.
Maybe here.
OK, I hope that this is correct.
Let's see later see time two and then we calculate the quotient.
So here we have to divide this length here, which is we said approximately 2.5.
We have to divide by this length, which is two.
So we get our length.
That's a bit longer than one.
And for the polar angles, we have to take this polar angle and we have to subtract this polar angle.
So I think this fact is pointing something like this.
So this is sea one divided by Sea two.
And I told you, you should compare these vectors with the actual numbers.
And I have calculated the actual numbers.
So let's see.
All right, so let's check we have the one plus two is on zero point eight plus two point seven nine,
OK, your report was maybe a bit smaller, but two point seven is OK.
So I think it's quite good.
Then the one minus the two, we have minus two point eight plus one point two.
I OK here.
Yeah.
The the rear part is a bit too bit of should have been actually a bit longer this whole thing.
Yeah.
OK.
But I think it's OK and it's just a practice for two to get the feeling it's much more important that,
you know, it points along this direction roughly.
And then for the product, we have minus three point four and two point nine K minus three point four
is actually really good.
But two point nine here would be you would be rather here, but yeah, it's fine I guess.
And Fawzy one divided by two, we have almost no real part.
Yeah that's good.
And we have one point one as the imaginary part.
That's also great.
OK, so as you have seen this, this practice was not about getting the perfect estimation, but to
get a feeling when someone gives you two numbers, especially if these numbers are in different representation,
where it's quite difficult to multiply them to or to calculate the quotient, then you can just draw
them and come up with the idea where this product or this quotient points along.
OK, so this was the exercise number three and we have completed the whole three exercises and I hope
now you know how to calculate with complex numbers.