All right, so let's start our section on complex numbers so maybe you already know what complex numbers
are, but maybe there's also the chance that you have never, ever heard of them.
So let us start with the question.
What are complex numbers?
Why do we need them and what can we do with them?
And here I want to start with a very simple example that you can probably solve yourself so a to function.
So here you have to function.
Y is equal to X squared minus two.
And in general we were minus C where C is a positive, constant can be any real number.
So now we want to solve this function and determine the zeros.
So in our case, rewrite zero is equal to X, not square minus two.
Now of course we can bring this minus C or D minus two to the other side so we get X, not square is
equal to two and then we just calculate to square it.
So as you can see in our case, it's plus minus the square root of two, which is approximately one
point for a bit larger.
And in general we have X not as equal to plus minus the square root of C, where once again C is a positive
real number.
So we have no problem here.
So a solution for the for the zeroes is always a pair of real numbers.
Now let's discuss a different quadratic function, so in this case, we have the function.
X squared plus two, so we have now here plus see why once again, C is a positive, real number.
So now we do the same game as here.
We we set the left inside side to zero because we want that F, F of X or Y is equal to zero.
So we have zero is equal to X, not squared plus C or in this case plus two.
This brings us to X, not square is equal to minus C or minus two.
And then formally we can write down that X not is equal to plus minus the square root of minus C and
in this case the square root of minus two.
And in high school or elementary school you would now say, OK, the square root of a negative number
is not defined.
We we're finished.
It means we do not have any zeros of dysfunction.
And if you look at the function, it's true.
We do not have any zeros because the minimum the point of the lowest Y value is located here at X equal
to zero, and it has a value of positive two.
So it means there is no value that slower and especially there is no value of Y equals zero.
So there is no Xeros of this function.
However, it's a bit unsatisfying that sometimes you can determine zeros of a quadratic function and
sometimes not.
So what we can do here formally is we can define something new.
We can first of all use this as law for the square roots that we write that X not is equal to plus minus
square root of C minus eight times the square root of minus one and the other case.
We do a similar thing, but we write here the square root of two.
Now, we have not really changed the problem, we still have a square root of a negative number, but
we can just define formally that the square root of the negative one is something we call.
So we write X not as equal to plus minus the square root of two times AI.
And we define that AI is equal to the square root of minus one.
It's just something like a like abbreviation of of this square root of minus one.
And it reminds us that this is something like an imaginary solution because here in this real plane
we do not have any zeros, but formally we can write that X not as equal to plus minus two times I or
I is defined as the square root of minus one.
So you could imagine it's something like a unit in physics, for example, if you have a length, for
example, one meter and you want to add another quantity, like a time, for example, three seconds,
you ride one meter plus three seconds and the one meter and three seconds, they are defined by different
units, which makes them unique.
And you cannot really add them up.
So you cannot ride.
That's equal to four times some other unit.
You really have one meter and three seconds.
And here in this case, we have the unit of AI.
So we have here a quantity that's a square root of two times this imaginary unit.
And the only different thing about this unit compared to a physical unit is that we know how we can
transform this unit into a real number.
Of course, we have to square it.
So the square of AI is equal to minus one.
Now, this trick also works for different problems or for more difficult problems, so let's use here
this quadratic function where we have another term, the linear term, the X, and in this example where
I've plotted it, it's X squared plus two, X plus two.
Now, of course, again, we said Y equal to zero to determine the X not.
And then we can use this famous famous formula minus B of A two plus minus the square root of B square
over for minus C.
And here in this example, we have minus one plus minus the square root of one minus two.
So once again, you can see here we get minus one, we get and negative number under the square root.
So, again, we could say there is no solution, but we could also use the trick and say, OK, let's
introduce our imaginary unit.
And this brings us to the square root of minus one times square root of two minus one, because here
we have changed to sign.
And so this gives us minus one plus minus I.
So it means we can really add up real numbers and imaginary numbers.
And in this case, we call this number a complex number.
And of course all of the imaginary numbers, for example, square root of two I and also all of the
real numbers, for example, square root of two.
They are also included in the in the complex numbers.
They are just special cases.
So it means we have a complex number that can be defined as the real part of the number, plus the imaginary
part of the number of times I and we always have to think about and I is equal to the square root of
minus one.
And what this also means is we can, um, think of a complex number in terms of a vector for two dimensional
vector, because now this number is a pair of two real numbers.
Just one of them has a different unit.
It has the imaginary unit.
So what we can do is we can consider the so-called complex plane that rides the real part on the x axis
and the imaginary part on the Y axis.
And you indicate this number Z as a vector.
So in this case, our number is two plus three times I.
OK, so our number is sometimes to make it a bit shorter, also written down as X, why times I wear
access to report Y is the imaginary part, or as I already mentioned, we write it down as a vector
where x.
But what very real part is the first component, Numerati imaginary part is the second component.
The only problem with this vector notation is that you kind of forget that you have here this imaginary
unit AI.
So you really have to take care and remember that you have this unit.
So far, this is all just a mathematical trick, but it can become really useful, for example, in
electronics or in electrodynamics where you write down alternating current or waves in terms of exponential
functions with complex number and the arguments, and especially in quantum mechanics, the imaginary
part of a quantity really becomes a physical property.
And it becomes very important.
For example, if you have the wave function where the real part and the imaginary parts are essential,
for example, for concepts like quantum computing.