Lost numbers

[mellowtigger]

warning:  geek question impending

With the equivalent of a minimum wage job, it's going to take a while to get myself out of debt and save up money for classes.  So what to do in the meantime?  I've been pondering two things.  A) Programming a game.  (a decades-old interest of mine)  B) Defining division by zero.

Division by zero is not "impossible", as I understand it, because it's merely "undefined" at the moment.  That makes it an attractive topic of interest.  A place to leave an intellectual mark, so to speak, by solving the problem.  Here follows my attempts to find appropriate metaphors for what I imagine is division by zero.

I think the solution has something to do with rotation.  When I imagine division, I end up "turning" my perspective.  Division by zero is a really intensely fast rotation... so fast that my orientation is lost altogether.  Is it possible to measure anything (scalar) without even a single dimension for orientation?  Some information is lost but not all of it.  Division by zero has a "place" (in my mind) but it lacks a "value" that I can measure in the usual way.  It may take a bit of mental trickery to create a "placeholding marker" (like i for imaginary numbers) where division by zero occurs in a formula, with neat tricks for working it out of the formula again.  So division by zero wouldn't necessarily make a formula unsolvable any more.

Zero, in my mind, has both a place (at the origin on a scale) and a value (distance measured from the origin along a dimension).  Division by zero, however, has only a place (related to the origin) but no value.  It has lost all dimensionality so it can't have a "value" in the usual sense.  That information is lost.

question finally: So does this metaphor sound familiar to anyone?  Could you point me in the direction of a book or an author (mathematician?) that describes stuff like this?  Has somebody already explored the concept of Lost numbers?  I'd like to learn more.

Comments

[furrbear.livejournal.com]

Division by zero has a value. It's just that that value is undefined, because it's infinite

It's a limit problem. lim 1/x as x→0 = ∞.

[mellowtigger]

edit: expressing myself backwards, ugh!

Yes, but that's exactly the bit (where the denominator < 1) that the rotational acceleration (in my mental metaphor) increases to the point that I get "lost". (1/(1/2), 1/(1/10), 1/(1/100), etc) The denominator gets smaller and smaller and smaller until suddenly it stops being measureable in the usual way in its particular context (denominator). At that point, infinite or zero for the whole expression neither makes much sense, because scalar definition itself is "wrong" there.

I still think it's possible to give a "meaning" to that particular scenario. It wouldn't necessarily be any more "real" than the square root of negative one, but at least it would be a productive kind of imaginary value.
http://en.wikipedia.org/wiki/Defined_and_undefined

How big is one part of nothing? Infinitely large? That can't be right. We need a new definition of it. I don't think the opposite of zero is infinity. Instinct tell me that there's at least a third possibility. I want to learn the math to pursue the question.

Hrmm. Do I want to spend $30 for the opportunity to read this article? *ponder*

[mellowtigger]

p.s. There's more than one kind of infinity. I think what I'm proposing is more than one kind of zero.

[mellowtigger]

p.p.s. I see now why the limit definition seems "wrong" to me. It's circular... self-referential.

It explores the limit of 1/x, as x approaches zero. But how does x approach zero? 1/2, 1/3, 1/4, 1/5... using the same fractional representation that it's trying to explore.

I want a metaphor to use instead of fractions on the usual scalar grid. I want something about rotation and its velocity change. Maybe I'll convince myself to spend that $30 and just pretend that I'm signing up for a month to a porn website. Geek porn. Yay!

[cipherpunk.livejournal.com]

Speaking as someone who sometimes (okay, rarely, but sometimes) earns his bread and butter with limit analysis, it's not self-referential. :)

Limits are formally defined in terms of δ and ε and fundamental theorems of calculus which are best not talked about here. :) Speaking informally, John is correct: if for any arbitrary error (however small) you can find a value for x which will give you a value within your error bar, and those values of x converge from both sides, then your limit is well-defined.

There's nothing self-referential to it. Have a defined delta and epsilon? Can they be made arbitrarily tight? Do they converge from both sides? Bang, Bob's your uncle.

Even if it was self-referential, it still wouldn't be a problem. One of the major forms of mathematical proofs, inductive reasoning, uses self-reference. Prove it works for the first value, assume it works for the Nth value, prove it works for the N+1th value, and bang, Bob's your uncle.

Inductive proofs in math are the direct analogue of recursive algorithms in computer science. Math geeks love proofs by induction, since they tend to be elegant, insightful and beautiful.

[furrbear.livejournal.com]

The denominator gets smaller and smaller and smaller until suddenly it stops being measurable in the usual way

It never just stops being measurable. That why the language is "as x approaches zero", it never becomes zero. No matter how small the distance between x and zero, that distance is always able to be split smaller,eg halved.

How big is one part of nothing? Infinitely large?

Try that as "How many times can I take nothing from something before I've taken all of it away?" (Division defined in terms of subtraction)

The math works. If the metaphor doesn't explain the math, look for a new metaphor, not new math.

I suspect the $30 may be better spent for something at a more fundamental level. That article looks to be deep PhD-level number theory.

Also the square root of a negative number may be referred to as "imaginary", but they do indeed exist and are exceedingly useful; just ask an Electrical Engineer. Without i, there would be no radio or television because the math wouldn't be possible without "imaginary" numbers. And if you dislike "Imaginary" numbers too strongly, you may wish to reevaluate the rotational metaphor, i is central in converting rotational polar coordinates, (r,θ), to Cartesian (x,y):

r e^i&theta = r cosθ + i r sinθ (http://www.faqs.org/faqs/sci-math-faq/specialnumbers/eulerFormula/); x = r cos&theta, y = r sinθ.

[mellowtigger]

Yes, I remember a proof for square roots (a funny name given the method) that used polar notation for solutions. It showed graphically why square roots actually have 4 solutions, but they reduced mathematically into the traditional two. I think there were two "real" solutions and two "imaginary" solutions.

Yes, that's what I want to know more of.

[cipherpunk.livejournal.com]

Strictly speaking, this is not quite correct. (It's close, though.) lim 1/x as x approaches zero from the right is infinite; lim 1/x as x approaches zero from the left is negative infinite. The infinite discontinuity there (on one side the slope is negative infinity, on another the slope is positive infinity, and there's an infinite gap between them) makes this sort of division by zero generally inaccessible to calculus methods.

There are some tricks you can do involving the reducing of fractions, L'Hopital's rule, etc., which can sometimes yield limits when dividing by zeroes. E.g., the limit as x approaches 3 of (x**2 - 9)/(x - 3) can be found, despite the fact you're dividing by zero. The fraction can be reduced to (x + 3)(x - 3)/(x - 3). The (x - 3) terms cancel, leaving just (x + 3) with a discontinuity at 3. Evaluating it at 3 from both directions yields 6 as a final result, meaning the limit can be found despite the fact there's a discontinuity from the zero division.