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Prologue to "COMPLEX NUMBERS" (Written by Thomas Anton Gobold in 1999 and 2001)

Editorial notes: "COMPLEX NUMBERS" was my topic when I made the Matura in Vienna, it was written in January 1999.

For the characters Anton, Sandy (Sandy-rat) and Jutta (Jutta-mouse) are all rights reserved! Any copying of their behavior or manners etc needs my authorization! If you'd like to read more adventures (not only containing mathematics) feel free to use the links to the Barney-hexology within my home page.

PROLOGUE

Anton (a little bit grumpy, handles his slide rule): Well, well, well! It seems that I have to teach the idea of the complex numbers to the two of you!

Sandy-rat (manicures on her nails): Yes we'd like to hear that from you!

Jutta-mouse (stretches her legs): We have heard rumors that you are good at it!

Sandy-rat (views to her now manicured nails): To be precisely: You are.... you should be an expert, according to this rumors .

Jutta-mouse: Now we'd like to get our own picture of these complex numbers.

Sandy-rat: ...and we could probably check if there is some truth in these rumors!!

Anton (takes a deep breath): About the complex numbers: We will discover very soon that the possibilities of the real numbers are not unlimited!

Sandy-rat (takes out two chocolate drops from her tool bag): We know that so far!

Jutta-mouse (takes and eats one of Sandy's chocolate drops): But we're all ears when you're going to tell us, for instance, how often zero in 65, your age, is!

Anton (silently, just for himself): Definitely not the amount of your brain cells, (to both of them) the subject of the complex numbers is somewhat different. You may get this very soon when you try, for example, the equation 0 = x� + 16 !

Sandy-rat (swallows the other chocolate drop): That's very easy, it's just 4, a very normal real number, even a natural number!

Jutta-mouse (licks the rests of her chocolate from her fingers): Could you positively, my dear Sire, after this level-one-equation, continue with something more suitable!

Sandy-rat (looks onto her bare hands because the chocolate drop isn't there anymore): We are no second class students anymore!

Anton: Something what suits you more? Certainly! As soon as we have got the correct solution for the latter equation. I'm afraid to tell you (he's obviously not afraid) that "4" is not the right solution!

Sandy (while she takes a cookie from her tool bag): Of course, it is minus four!

Jutta (she tastes a potato chip): Of course, naturally, it is -4!

Anton (touches with his left hand Sandy's right shoulder and with his right hand Sandy's left shoulder): Just check and see if the function x�+16 with -4 as x is indeed zero (smiles).

Sandy and Jutta (both stop playing their gimmicks or eating their snacks, get busy and begin calculating and to think. It shouldn't last too long, even for them).

Anton (proudly with a laugh in his face): Of course, if this does not overstretch the limited capacity of your brains!

Sandy and Jutta (both move on and get rid of Anton's two hands from her shoulders)

Sandy: The result is 32, not zero... Not zero indeed.

Jutta (to Sandy): I told you, he's a wizard. I told you! (to Anton, while she's touching Sandy's tool bag) In both cases, which one does fit?

Anton: Yes, we have to fit x� = - 16 with a suitable x.

Sandy (rummages in her tool bag in search for a cookie): Couldn't we say it's minus four multiplied with plus four?

Anton: That's not four squared. But the squareroot of -16 has something in common with other squareroots like -25 or -49. All of them you could transform into 16*(-1) or 25*(-1) and so on...

Jutta (licks her salty fingers): Yes, then we can find easily the roots of 16 or 25v (intonates) and so on...

Sandy (recognizes that she has already, since a while, a cookie in her other hand and eats half of the cookie) Yeah, but there is still to find the squareroot of minus one (cheeky) and so on...

Jutta (grabs the rest of the cookie from Sandy's hand, to Anton): Would you now, my dear Sire of the eternal math-raiders, be good enough to tell us the result of the squareroot of minus one?

Anton: First things first: We have to think of such numbers which are carrying such a squareroot of minus one. Such a number which carries this makes no sense ...

Sandy (interrupts Anton): Then, why are we here?

Jutta: Well, when the lesson is over, play time starts, give us back our Lego bricks!

Anton (continues loudly): ...makes no sense in reality, now we start the task!

Sandy (plays with her toes): This squareroot of minus one is a strange number, isn't it?

Jutta (tickles one of Sandy's toes): Hey Anton, the complex numbers are misnamed, we should name them strange numbers?!

Anton: Now back to business: These numbers have no right for its real existence. There is no real number which stands for the squareroot of minus one.

Sandy (puts Jutta's hand away): Do not make a monologue!

Jutta (makes stupid notes on her paper): Show us your didactic skills! We have waited long enough. Now tell us: How much is this squareroot of minus one?

Sandy (points with a finger to Anton): Quickly, give us the answer!

Anton (catches Sandy's hand and moves the hand roughly away): In fact, there is no answer!

Sandy: Sire !!!

Anton: Well ?

Jutta: The fact, that you do not know the answer doesn't mean, that there is no answer!

Sandy: Yes, anytime he knows no answer, Mr. Anton always considers that there is no answer. (to Jutta) One time a tried to solve an equation of the fifth grade, and he just told me, that there is no formula to solve them. He does that frequently!

Anton: Listen well to my words: It is not the question to know what means this squareroot. It's the question how to calculate with these squareroots of negative real numbers. It happens sometimes that these numbers appear, just for a while. Then they disappear in a later point of the same calculation!

Jutta (impatiently): Yes, yes, yes, now this squareroot of minus one has appeared and suppose, just suppose, we have caught it. What exactly is it now?

Anton: You haven't listened very well. Enlarge or broad your spirit and inhale these additional numbers, just for our imagination. These are numbers which are not real. We only can imagine them...

Sandy (interrupts): As a matter of fact, even the natural numbers do exist in our imagination only!

Jutta (adds): The whole realm of mathematics does exist in our imagination only!

Anton: Cut that out! We wide the real numbers to something new and add these imaginary numbers. Numbers which are the sum of a real and an imaginary part are a composition, let's call them complex numbers!

Sandy (repeats): Imagine....imaginary... enlarge the real numbers... (to Jutta) Do you consider Anton as fine, honey, is all okay with him ?

Jutta (to Sandy): Darling, maybe he has such a big brain only because he has enlarged his own brain. Anton may have a small real brain and a big one which just exists in his imagination only.

Sandy (to Jutta): That must be something like that... roots of negative numbers??

Jutta (interrupts her): Even roots!

Sandy (continues): Maybe our whole Anton is just a product of his or our fantasy?

Jutta (to Sandy): With all his ideas of mathematics.

Anton: Enough small talk! For example: We try to solve 0=x�-10x+40!

Jutta (looks around): Will not, Cardano, a scientist of the future deal with this problem?

Sandy: Yes, the equation comes from having two numbers whose sum is 10 and whose product is 40.

Jutta: We can even prove to you, that the maximal effect by multiplying two numbers whose sum is 10 just is 25!

Sandy (collects crumbs and throws them away): Come on, tell us something new, something what is really interesting!

Jutta (finds a big crumb and eats it): You should do so. And what is more: I found a contradiction in one of your performances!

Anton (rapidly): A contradiction? Definitely not, my statements are totally clear. There is no room for a doubt. I can prove everything what I've said.

Jutta (takes another cookie from Sandy's tool bag): You told us just a moment ago, that this squareroot of minus one has lost the right for its real existence, if you do remember ?! (Anton nods)

Sandy : Then, my dear Sire, (climbs on the table, so she gets a better view) How could it be possible, that the result of squareroot of minus one power itself is a real number, approximately 0,20788 (looks deeply into Anton's eyes and smiles).

Jutta: And do not tell us now, for instance, that the squareroot of minus one stands for i or that i� makes -1. That's just a substitution...

Sandy (puts her feet onto the table too): ...but no answer!



i^i = e^(-Pi/2)