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All right, so let's start our section on complex numbers so maybe you already know what complex numbers



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are, but maybe there's also the chance that you have never, ever heard of them.



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So let us start with the question.



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What are complex numbers?



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Why do we need them and what can we do with them?



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And here I want to start with a very simple example that you can probably solve yourself so a to function.



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So here you have to function.



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Y is equal to X squared minus two.



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And in general we were minus C where C is a positive, constant can be any real number.



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So now we want to solve this function and determine the zeros.



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So in our case, rewrite zero is equal to X, not square minus two.



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Now of course we can bring this minus C or D minus two to the other side so we get X, not square is



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equal to two and then we just calculate to square it.



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So as you can see in our case, it's plus minus the square root of two, which is approximately one



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point for a bit larger.



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And in general we have X not as equal to plus minus the square root of C, where once again C is a positive



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real number.



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So we have no problem here.



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So a solution for the for the zeroes is always a pair of real numbers.



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Now let's discuss a different quadratic function, so in this case, we have the function.



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X squared plus two, so we have now here plus see why once again, C is a positive, real number.



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So now we do the same game as here.



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We we set the left inside side to zero because we want that F, F of X or Y is equal to zero.



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So we have zero is equal to X, not squared plus C or in this case plus two.



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This brings us to X, not square is equal to minus C or minus two.



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And then formally we can write down that X not is equal to plus minus the square root of minus C and



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in this case the square root of minus two.



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And in high school or elementary school you would now say, OK, the square root of a negative number



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is not defined.



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We we're finished.



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It means we do not have any zeros of dysfunction.



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And if you look at the function, it's true.



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We do not have any zeros because the minimum the point of the lowest Y value is located here at X equal



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to zero, and it has a value of positive two.



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So it means there is no value that slower and especially there is no value of Y equals zero.



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So there is no Xeros of this function.



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However, it's a bit unsatisfying that sometimes you can determine zeros of a quadratic function and



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sometimes not.



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So what we can do here formally is we can define something new.



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We can first of all use this as law for the square roots that we write that X not is equal to plus minus



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square root of C minus eight times the square root of minus one and the other case.



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We do a similar thing, but we write here the square root of two.



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Now, we have not really changed the problem, we still have a square root of a negative number, but



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we can just define formally that the square root of the negative one is something we call.



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So we write X not as equal to plus minus the square root of two times AI.



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And we define that AI is equal to the square root of minus one.



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It's just something like a like abbreviation of of this square root of minus one.



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And it reminds us that this is something like an imaginary solution because here in this real plane



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we do not have any zeros, but formally we can write that X not as equal to plus minus two times I or



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I is defined as the square root of minus one.



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So you could imagine it's something like a unit in physics, for example, if you have a length, for



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example, one meter and you want to add another quantity, like a time, for example, three seconds,



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you ride one meter plus three seconds and the one meter and three seconds, they are defined by different



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units, which makes them unique.



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And you cannot really add them up.



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So you cannot ride.



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That's equal to four times some other unit.



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You really have one meter and three seconds.



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And here in this case, we have the unit of AI.



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So we have here a quantity that's a square root of two times this imaginary unit.



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And the only different thing about this unit compared to a physical unit is that we know how we can



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transform this unit into a real number.



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Of course, we have to square it.



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So the square of AI is equal to minus one.



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Now, this trick also works for different problems or for more difficult problems, so let's use here



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this quadratic function where we have another term, the linear term, the X, and in this example where



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I've plotted it, it's X squared plus two, X plus two.



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Now, of course, again, we said Y equal to zero to determine the X not.



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And then we can use this famous famous formula minus B of A two plus minus the square root of B square



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over for minus C.



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And here in this example, we have minus one plus minus the square root of one minus two.



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So once again, you can see here we get minus one, we get and negative number under the square root.



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So, again, we could say there is no solution, but we could also use the trick and say, OK, let's



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introduce our imaginary unit.



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And this brings us to the square root of minus one times square root of two minus one, because here



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we have changed to sign.



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And so this gives us minus one plus minus I.



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So it means we can really add up real numbers and imaginary numbers.



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And in this case, we call this number a complex number.



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And of course all of the imaginary numbers, for example, square root of two I and also all of the



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real numbers, for example, square root of two.



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They are also included in the in the complex numbers.



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They are just special cases.



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So it means we have a complex number that can be defined as the real part of the number, plus the imaginary



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part of the number of times I and we always have to think about and I is equal to the square root of



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minus one.



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And what this also means is we can, um, think of a complex number in terms of a vector for two dimensional



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vector, because now this number is a pair of two real numbers.



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Just one of them has a different unit.



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It has the imaginary unit.



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So what we can do is we can consider the so-called complex plane that rides the real part on the x axis



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and the imaginary part on the Y axis.



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And you indicate this number Z as a vector.



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So in this case, our number is two plus three times I.



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OK, so our number is sometimes to make it a bit shorter, also written down as X, why times I wear



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access to report Y is the imaginary part, or as I already mentioned, we write it down as a vector



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where x.



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But what very real part is the first component, Numerati imaginary part is the second component.



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The only problem with this vector notation is that you kind of forget that you have here this imaginary



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unit AI.



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So you really have to take care and remember that you have this unit.



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So far, this is all just a mathematical trick, but it can become really useful, for example, in



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electronics or in electrodynamics where you write down alternating current or waves in terms of exponential



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functions with complex number and the arguments, and especially in quantum mechanics, the imaginary



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part of a quantity really becomes a physical property.



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And it becomes very important.



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For example, if you have the wave function where the real part and the imaginary parts are essential,



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for example, for concepts like quantum computing.