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Creative numbers and division by zero

Published: april 10th 2011 (thursday), 14:27

I'm sure you've heard at least once in your life that division by zero is impossible. At schools they've probably teached you some nursery rhymes to remind you that you can't do that, and to convince you once for all that you shouldn't even try. Like if dividing by zero were to cause some enormous cosmic catastrophy; eg. create a big black singularity which will suck in the whole Universe in an instant. Like in these funny pictures.

Side note:
Well… It's true that sometimes division by zero can cause problems: when a computer program tells the CPU in your computer to divide by zero, the CPU of course doesn't know how to do it (because none of his creators knew either), so it'll just give up and break the execution with an error. If the programmer hadn't foreseen that, the program will usually crash. If this program controls some serious process, eg. a nuclear chain reaction or a space shuttle, it might truly cause problems :-P But these problems lie not in the sole division by zero, but in the architecture of contemporary computers, which are not designed to do it.

Every self-respecting mathematician (the more self-respecting the worse ;-P) will tell you that it's impossible and absurd to divide by zero. He will show you thousands of mathematical proofs which show that such an operation is nonsense, illogical and it even couldn't be sensibly defined.

But is it really that way? Could the number system really exist with such a serious breach related to only one single number? (the zero) Is it right that such a "nothing", as zero, could really lead to so much problems?

When someone tells me that something is impossible, I usually translate it to myself as: "No one so far has figured out how to make it possible". I have a similar approach to the problem of division by zero.

In this article I'll try to answer you to that question and solve that mystery, which mankind struggle with through millennia without any effect. I'll try to convince you that perhaps the solution exists, and additionally it's very simple :-) So simple that it's a shame that no one have figured it out so far.

Before I'll be able to show it to you, we'll look at all numbers we have discovered so far, and we'll try to find some common unifying factor which connects the circumstances of their discovery. It will help us to use the very same pattern to discover how to divide by zero.

Where the numbers come from?

Every number comes to life to solve some equation; to make its solution possible. And we create equations to describe mathematically some problem we faced in our life. The stranger the problem we come across in Nature, the stranger equations we create and the stranger numbers we invent to reflect this problem and solve it. Hmm... we create... or do we?

Amir D. Aczel in last words of his book The Mystery of the Aleph: Mathematics, the Kabbalah, and the Human Mind has wondered whether the numbers are merely a human invention or maybe there's some real and tangible feature of our Universe that stands behind them? For a long time I was a follower of the former version (that of the human invention), but these words of Aczel upset my confidence. I mean, if numbers are only a human invention, then why from time to time we are forced to "invent" some new kind of numbers? Why such new numbers often surprise us? Nay, sometimes even so much that we cannot stand up to the facts which Mathematics reveals before us! How's it possible that our own creation could surprise us and do something we didn't made it for? Maybe the truth is different?: We do not invent numbers at all, but discover the numbers pre-existing in deeper and deeper levels of the structure of Nature and reality?

So let's see now how did we discover different kinds of numbers and what problems they were supposed to solve.

Natural numbers

Here the case is pretty simple: if we want to count some uniform, discrete objekts, sheeps for example :-), we invent a new name for every subsequent cardinality of the herd of sheeps. One sheep, two sheeps, three sheeps… <yawn> Let go with these sheeps, because I'm starting to become sleepy ;-D

So we find a new symbol (a number) to write down each subsequent, increasing group of objects.

Of course that way it would be difficult to deal with bigger numbers, because it would be necessary to invent newer and newer symbols for each subsequent cardinality. And who could remember them all! Therefore people created several number systems, in which numbers can be encoded in an organized way with use of small set of symbols (digits). I'll write more on this subject in a separate article. For now we simply take it for granted that natural numbers exist and everyone is familiar with them ;-)

Negative numbers

{...This gap I will fill when I find good examples...}

Rational numbers (fractions)

I've mentioned earlier that every number is a solution to some equation (yes, I know, there are also transcendental numbers which allegedly don't come out from any equation; we'll get back to this later, and for now just breathe ;-J ). Then let us look at the following equation:

Equation (1)
2x = 1

If to this time your experience has been limited to sheeps :-D (or some other discrete objects) and you know only integers, such an equation can appear to you as impossible to solve. For no matter you try, you cannot get one full sheep from joining two equal groups of sheeps.

But you cannot find the solution not because it doesn't exist, but because your tool (integer numbers), and your current perspective (counting uniform, discrete objects), are too much limited for this job. However, if you don't worry with it and you try to solve the equation for x anyway (i.e. divide both sides by 2), then what will you get? Let's see:

(2x)/2 = 1/2

Twos on the left side will cancel and what's left is:

Equation (2)
x = 1/2

On the left we have now only x, so the equation can be considered solved properly for x. Nonetheless on the right side we have an expression which doesn't look pretty, but we cannot simplify it any further. Nothing simplifies, because it's impossible to divide 1 into 2 whole-number parts without any remainder. 2 even doesn't fit into 1! We don't have such a whole number, which we could replace this expression with. So it seems as a hopeless case!

But you can also look at this all from different point of view: We cannot simplify it further by replacing it with some single whole number, because the solution is none of the numbers known to us to this point! It is a totally new number, which is created from a combination of numbers known to us. So we can assume that this is our solution, and use this very expression as a symbol for our new, just discovered number.

This symbol will remind us that our new number is a ratio of two whole numbers, but it's not a whole number itself – it's a fraction of a whole. Thus we'll call this symbol a vulgar fraction, a new kind of number called rational number.

Notice that we don't reduce this expression any further, because it cannot be reduced in terms of the symbols we know. Therefore we use the whole expression as is, as a symbol of the new nubmer. And the whole expression represents a number which is our solution. Keep this observation in mind.

Having this symbol we can now try to use it in our original Equation 1 and see what will happen:

Equation (3)
2(1/2) = 1

We cancel these twos and we get a true statement:

Equation (4)
1 = 1   (true)

We can thus freely use such symbols (fractions) in our equations and convince ourselves that they behave nicely in algebraic manipulations and allow us to solve real problems. For example, the above Equation 3 tells us that two halves of an apple taken together gives us one whole apple. It's enough to find such a reference to the reality to convince ourselves that these new numbers really exist and make sense.

Irrational numbers

Ancient Pythagoreans believed that everything in our world can be described by numbers. If not by whole numbers, then at least as a ratio of two such whole numbers. So they used rational numbers we have just re-discovered. But it soon became obvious to them that not everything can be expressed as a ratio of whole numbers. It turned their theories upside down and broke their thrifty built order. It scared them to the point that they kept this discovery as one of their greatest secrets. For revealing this secret one could have paid with his life (Hippasos had paid).

So let's now find out what we shall do to discover the secret of Pythagoreans. Here we have a unit square:

Unit square, which diagonal we don't know.

What is the length of the diagonal of such a square? To figure it out, we can notice that it makes a hypotenuse of a right triangle, whose "legs" are two sides of unit length, and use these facts in the famous Pythagorean formula (though it's more ancient than Pythagoreans):

x^2 = 1^2 + 1^2

x^2 = 1 + 1

Equation (5)
x^2 = 2

And we encounter a problem: Any whole number, nor any rational number, when we multiply it by itself (that is, square it), doesn't give us 2. Again, among the numbers we know, there is no such number which would solve this equation!

But if you look carefuly at the picture of the square which led to this equation, then you surely say with astonishment: "Heck, there must be some number which designates the length of this diagonal down here! For the diagonal exists, it's real!" It is of course true. So if you, inspired by geometry, decide to try to solve this equation anyway, by taking a square root of both sides, then you'll get:

Equation (6)
x = +- square root of 2

And again we cannot simplify the expression on the right side any further. We can only assume that square root of 2 is a symbol for a new number, which we just have discovered, and that this number has support in reality as the length of the diagonal of the unit square.

If you substitute this symbol to the original Equation 5, you'll find that it properly fulfills the equation:

Square root of 2, squared = 2

Square root of 2 times the square root of 2 = 2

2 = 2   (true)

Thus the new symbol and the new number represented by it behaves nicely in equations and allows us to solve problems which we earlier considered impossible to solve :-) For example we now know how to designate the length of a diagonal of a unit square:

A unit square and a known length of its diagonal: the square root of 2.

Besides the square root of 2 we can find many other such numbers. We call such numbers irrational numbers. Not because they're illogical, or because we can't measure it, but because we cannot express them with any whole numbers or their ratios (fractions). (In my opinion, the direct translation from Greek, "incommensurable numbers", better describe their meaning: that they don't have any common measure – you can't find any measuring stick which would fit in both of them a whole number of times.) But, as you can see, we can use these numbers with success for solving real problems. Many polynomial equations can be solved with their help.

Transcendental numbers

It turns out that all the numbers we have found to this point don't exhaust the subject yet. Because there are numbers which we cannot "reach" by any polynomial of rational powers. If you put on one number line all the whole numbers, rational numbers and irrational numbers, you'll find that despite at first glance it seems to be densely packed with points, nearly continuously, between these points there are plenty of small gaps.

What's worse, when Cantor tried to count which of these points is more: these representing rational numbers, or these gaps, he discovered to his amazement that there are as many rational numbers as whole numbers, but irrational numbers are more in number there: an incountable number! (so called continuum) But still, when one shoots into random places on the number line, the chances to hit a rational number, or even irrational number, are nearly zero!

So there's still something missing here… This "something" is transcendental numbers. Such a number is for example the famous pi ("pi", the ratio of a circumference of a circle to its diameter), or another famous number "e" (base of the natural logarithm). Mathematicians say that those numbers are algebraic numbers (that is, they are solutions to some polynomial of rational powers). But in my opinion it doesn't mean at all that they can't be tamed into formulas! Because what else is the formula to obtain "pi" if not the very definition of it?:

C/D = pi.

where C is the circumference of a circle, and D is its diameter. After all, this equation is not so different from what we have seen so far: pi here is a symbol of a number which fulfills this equation and stands for its solution. Admittedly C and D can change, but they always change simlultaneously and proportionally, so their ratio is constant and the fraction will always boild down to pi. Similar equations we can find to define "e" and many more transcendental numbers. Joseph Liouville had found quite a lot of them.

All rational and irrational numbers, along with transcendental numbers, readily make a full continuum along the whole number line. The whole set of these numbers we call real numbers.

But is it the ultimate end? If we have just tightly covered the whole number line with numbers, can we finally breathe a sigh of relief and conclude that we have discovered all possible numbers?

The mysterious square roots of negative numbers

Well, it appears that we're still not at the end of that trip :-P Still some polynomial equations don't have solutions. Sooner or later you'll come across a trivial quadratic equation like this one:

Equation (7)
x^2 = -1

and soon you'll see that it's not that trivial at all: If you'll experiment a bit with squaring different numbers, positive and negative, soon you'll see that a square of any real number, positive or negative, will always give you a positive number. Never a negative one! That's the reason why mathematicians for centuries sweared themselves by all sacred things that such equations don't have solutions. It's hard not to agree with them; after all, if you rearrange this equation as below:

x^2 + 1 = 0

and next you turn it into a function of x, that is, y(x):

x^2 + 1 = y

and you plot it:

A plot of the function x^2 + 1 = 0

then you'll see that it doesn't cross the x axis at any place. From looking at this graph one can get an irrefutable impression that those mathematicians are right and this equation cannot have any solutions. But it turns out that such a graph is deceiving and it strongly limits our perception! It limits our perspective in such a way that we nowhere see the solution, even if such a solution actually exists.

It resembles a way of thinking of someone who regards his own backyard as a whole world and believes only in things he can find there. If you tell him about the elephant, he will respond: "Absurd! There's no such animal! I've never seen any such thing on my backyard! Are you trying to convince me that elephants exist? Then show me one on my backyard!" It's clear that he gives us a challenge impossible to fulfill: Although we know well that the elephant exists, we won't be able to prove it to him!

From these reasons mathematicians for a long time ignored such equations and the square roots of negative numbers. Even when such square roots had been appearing in the solutions of some cubic equations. Up to the time when someday Girolamo Cardano noticed that when he, no matter the difficulties with these square roots, bravely continues his calculations, then at the end of the day he comes to a "nice" result: a real number!

The conclusion was inescapable: These numbers, square roots of negatives, must have some meaning. They behave nicely in equations and lead to correct answers. The only problem was that no one yet knew what did these numbers mean. It was hard to imagine. Too hard! Therefore these numbers didn't have much appreciation from mathematicians, and they has been called quite offensively imaginary numbers. Their bad fame hasn't changed to the present days!

I'll expend a whole separate article to these new imaginary numbers (square roots of negatives), and to their geometrical meaning, because their story is fascinating and studded with intrigues ;-) But for now let's do what Cardano did: let's try to continue nevertheless our calculations and see where they'll bring us. If you take a square root from both sides of Equation 7, you'll get:

Equation (8)
x = +- square root of -1

We don't know what that square root of -1 is, nor what it means, but let's presume that this is the solution to the Equation 7: this symbol, square root of -1, designates a new mysterious number which we have just found; an imaginary number. Let's find out now what will happen if we substitute this number back into the Equation 7:

square root of -1, squared = -1

square root of -1 times the square root of -1 = -1

-1 = -1   (true)

Again we have found that everything is dandy: we've got a true equation. Our new number allows us to solve this one equation and many other similar equations, and it behaves nicely in algebraic manipulations. Its geometric meaning will become clear to you shortly (in a separate article) and you'll see for yourself that there's nothing unreal or imaginary in it – it has a very robust geometrical interpretation.

Is it finally the end of this research? Or maybe there are some other exotic numbers waiting for a discoverer?

All the numbers we have discovered so far allow us to solve every polynomial of rational, irrational or even imaginary powers and it turns out that all their solutions lie back in the set of the numbers we know, which we discovered to this point (so called algebraic numbers). No matter how we try, no equation throws us out of this set of numbers (mathematicians say that the set of algebraic numbers is a closed set for all arithmetic operations). Thus it looks like no additional numbers are needed. In the article about imaginary numbers I'll show you why does it work that way, by using geometry.

But, taking lessons from the history, I would still recommend you to be cautious in making statements that we know everything. It may turn out someday that there is some new kind of numbers which we discover only when we face new kinds of problems and write them down with some new kinds of equations. Actually, we can recall right now that there are such equations we still cannot solve! I speak of the equations containing division by zero! So it's time to get back to our main topic and finally solve this age-old mystery :->

Time to draw conclusions

Let's now look back at all these numbers we discovered so far, and let's try to find in the circumstances of their discoveries some common, unifying pattern which have led to these discoveries. Please read the following paragraph very carefully! The clues contained there will allow us to solve the enigma which mankind struggled with for millennia!

We can notice that every new number has come out of some equation. From time to time we run across an equation we cannot solve with the numbers discovered to that point. We've been able to recognize it from the fact that we have had that equation correctly solved for x, but on the right side the expression remained, which we couldn't reduce any further. And the reason was that it's been impossible to replace it with any number among the ones we have known to that point. It's been an entirely new number which has been created as a combination of known numbers. Then we have used that remainding, irreducible expression as a symbol for that new number. Since then we've been able to use this new symbol in our equations to solve similar problems. From the fact that this symbol behave nicely in equations and fulfill them (solve), resulting in true equations, we've been able to judge its correctness and usefulness.

So let us apply the knowledge just obtained to finally disentangle this division-by-zero conundrum.

At last: division by zero

Example equation which leads to the problem of division by zero is this:

Equation (9)
0 times x = 3

Such equations are usually pointed by mathematicians as a best proof that the division by zero doesn't make sense. Mathematicians say that the above equation is a contradiction, or false, because every number times 0 gives 0. It's impossible to get 3 this way. They multiply x times 0 and they get the logical contradiction (false):

Equation (10)
0 times x = 3   (false)

On these grounds they argue that it's impossible to find such an x which would fulfill this equation. But doesn't it remind you something? :->

After all, they've been saying exactly the same things about those square roots of negatives! Among all the numbers known to them (positive and negative real numbers) they couldn't find any such number which squared would give a negative number, so they claimed on these grounds that the whole equation is false, because there's no such number! But after some time it has turned out that they were wrong! And such numbers actually exist, but outside the set of the numbers they have known then: those are imaginary numbers. So, as you can probably see, they did exactly the same error they make here:

It's of course true that Equation 10 is a logical contradiction (or false): zero will never equal to three. However, Equation 9 is not contradictory any more! "Why?", you may ask? Because of that blank called x which you can fill with some number.

The error lies in the fact that we don't know all possible x's to be able to tell for sure that there's no such one x somewhere, which could fulfill this equation. We can only tell for sure that there's no such x among the numbers we know so far! Sounds familiar? :-> This is the key to this enigma.

Let's presume that maybe, just maybe, there is such a number which fulfills this equation, and let's try to find it by solving for x. To do this, we have to do what? (Attention now, mathematicians, hold on your pants, because a heresy comes in ;-D) Both sides of Equation 9 we have to divide by zero! :-D Look:

(0/0) x = 3/0

Let's not worry of what mathematicians say, that division by zero is infeasible, and let's just do it. On the left side the fraction will cancel and we're left with:

Equation (11)
x = 3/0

I don't know how about you, but for me it looks like an equation properly solved for x! :-) Similar to how it worked for the numbers discovered earlier, we've got on the right side an expression we cannot reduce any further, because we don't know how to make this division to replace it with some of the numbers we know. But maybe it's because this is none of the numbers we know? :->

And SasQ said: "Let there be Creative Numbers!" ;-)

So let's assume that this is a completely new number, and the expression 3/0 is its symbol.

What will happen if we substitute this symbol into Equation 9 which have led us to it? Let's see:

0(3/0) = 3

On the left side the zeroes will cancel and we'll get:

3 = 3   (true)

which is a true equation! This means that 3/0 is just such a number which can create something out of nothing: in this case it makes three from zero. (But remark that it's not always the case! Only if you multiply it with 0!). On the ground of this creative power of this number, let's call it creative number :-D

Additionally we can notice that this number always "creates" not some arbitrary number, but exactly the value from its numerator, in a totally unique way! You can't get any other value this way. For that, you'll need a completely different creative number, which fulfills a different equation. E.g.:

Equation (12)
0x = 5

0(x/0) = 5

x = 5/0

And we've got a totally different solution. When we substitute it into the original Equation 12, again we'll get a true equation:

0(5/0) = 5

5 = 5   (true)

And SasQ saw they were good ;-)

As you can see, creative numbers, which we have just discovered, allows us to solve equations which previously were "impossible" to solve :-) They also behave nicely in different algebraic manipulations. For example, you can multiply them together:

0 (2/0) (3/0) = 0 (2*3)/0 = 0 (6/0) = 6

and add them:

0 (2/0 + 3/0) = 0 (5/0) = 5

They are subject to the distributive law:

0 (2/0 + 3/0) = 0 (2/0) + 0 (3/0) = 2 + 3 = 5

So, as you can see, they behave nicely in algebraic manipulations. They make sense.

I don't know yet what these numbers could mean geometrically and what could they be useful for. Nevertheless I have a feeling that something what stood a problem throughout the millennia must have some very important meaning. Maybe it will allow somehow to explain another great mystery: How our enormous Universe could came to life out of nothing? Maybe it'll turn out useful to scientists studying the mysterious singularities in the gravitational field, called black holes? Maybe it will allow us to investigate the problem of infinity somehow? I think it might be somehow connected to the problem Cantor struggled with?: That the same infinite number of points fits in two line segments having different lengths, as in any planar figure or solid, or even in a single point.

Creative numbers may also come in handy in differential calculus which I'll describe soon, because in my opinion it relies on a clever way of avoiding division by zero by using the mathematical notion of limit. The bottom line is that every differential is a kind of a ratio, which denominator we tend to decrease more and more closer to zero, to get in the limit a number as close to this limiting value as possible. This limiting value means, for example, the speed of an object in some instant of time, that is, an interval of zero length. "Now". In the definition used these days people use limits to avoid the division by that zero-length interval and skip stright to that limiting value. Maybe the use of creative numbers could allow to work directly on such values, by using division by zero explicitly instead of jumping out of the blue to the limiting value.

As you see, everything figures that the problem of division by zero can have solution. Moreover, the solution is very simple! But now it's necessary to look carefuly at this solution and to make sure that it behaves nicely in every algebraic manipulations we know, and if not, then why, and how to improve the theory. I left this problem open to you. Let others also have opportunity to discover something :-D Maybe you wanna try? :->

I recommend you to play creatively with setting different equations with these new creative numbers and to check how well they behave in there, and whether they lead to some interesting solutions. You may also try with some problems which you couldn't solve in the past because of mathematicians telling you that it won't work. I'm curious to see your ideas. If you'll find any error in my line of thought, please contact me. I'll read about it readily. I'm also curious about the possibility of representing these new numbers by some geometry.

Stay tuned and watch my website, because maybe I'll soon get back to this subject and show some new evidence related to this :-) And for now, good bye and thank you for reading! :-)

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