Talk:Division by zero

Creative Numbers...

Creative numbers have purported to solve the problem of division by zero. I'm no mathematician but perhaps someone who understands the subject matter more critically could check out http://science.mistu.info/Math/Numbers/Creative_numbers_and_division_by_zero.html and see if it warrants expanding the article. 71.207.183.254 (talk) 18:22, 4 January 2013 (UTC)

Its original research, with no rigorous proof or peer review. It doesn't merit inclusion here. Mindmatrix 18:40, 4 January 2013 (UTC)

I guess you would know better than I would. But, it makes sense to me as a layman. 71.207.183.254 (talk) 19:31, 4 January 2013 (UTC)

I know how to properly divide by 0 and do 0 math in general, so I reviewed the reference for you

 No.
Victor Kosko (talk) 19:03, 3 June 2018 (UTC)

Grade seven revisited

Long ago, way back when a four-function calculator was consider high tech, I got into a sparring match with my grade seven math teacher, which did not end well so far as I was concerned.

He made completely valid points, but he still seemed to be missing something essential about how the problem appears to the novice as yet unencumbered with mathematical convention. After our interaction, I remained as inarticulate about what my math teacher had skipped over as I was before, so I found his arguments correct, but nevertheless unsatisfying.

And so I suddenly get the urge to see how this is old chestnut plays out on Wikipedia, and "whoa! time machine". Is that you Mr W? Are you editing Wikipedia from your retirement cabin on the lake? Because this is the same damn thing.

A compelling reason for not allowing division by zero is that, if it were allowed, many absurd results (i.e., fallacies) would arise.

...

The fallacy here is the assumption that dividing 0 by 0 is a legitimate operation with the same properties as dividing by any other number.

What that passage actually demonstrates is that defining 0 as its own multiplicative inverse (0/0 = 1) instantly leads to the absurd. I quickly refreshed myself on the axioms of field theory, and the axioms pertaining to a/b exclude the cases where b = 0. Defining 0/0 to some quantity does not appear to violate the existing axions of field theory, although we quickly ruled out 0/0=1 as leading to absurdity. By the same argument we can quickly rule out 0/0=a for any non-zero a.

But what about defining 0/0=0?

This isn't inconsistent with the axioms of field theory based on my quick review. Maybe it's inconsistent with a theorem of field theory, such as x=x^-1 having a single solution (er, a single non-negative solution) as proven from the existing axiom which excludes b=0 from any quantity a/b.

But that's kind of narrow-minded about axioms, and we ought to know better than to proceed on the basis of this kind of axiomatic narrow-mindedness because of non-Euclidean geometry.

So why don't we define 0/0=0 (and 0/x=0 and x/0=0 all around) and amend the theorem to x=x^-1 having a singular solution (the multiplicative identity) for x strictly positive.

While we're at it, we can also thoroughly subordinate division under multiplication, so that we can't actually divide both sides of an equation by any quantity; whereas we can multiply both sides of an equation by any chosen reciprocal. In this formalism, you can't divide an equation by zero, you can only multiply by 1/0=0, as defined in the previous paragraph. And you really can't get into much fallacy trouble multiplying both sides of an equation by 0 (be careful with strict inequalities such as a < b, however).

So why don't we do this? Here are my two primary guesses:

1. it's uniformly infelicitous to multiply both sides of an equation by 0 (wherever you were going, this doesn't help you get there, even if you salvage your proof later—leaving behind some foolish but non-fallacious deadwood).
2. when progressing to infinitesimals, this peculiar pointwise discontinuity clutters your statements and proofs to no useful effect

It appears to me on superficial review that defining 0/a=0 and a/0=0 for all a is not precluded by fallacy, nor by necessary rather than whimsical field theory axiomatization, but by pure infelicity: it simply doesn't buy you anything you can finally wield to your advantage; refusing to extend the definition of division in this way votes a certain kind of deadwood off the island at first point of contact.

I have far more background in computer science, and in this realm it is surely not a win in many contexts to lose your shit over deadwood that isn't causing any real trouble; better to let the computation proceed uniformly and deal with any issues that arise at the other end. What's a NaN here or there between friends that probably disappears again at a later step anyway? Did we really need to raise a bat signal to the overlord of the galaxy over one piddly NaN (the Toyota model where any task station can stop the entire assembly line in a heartbeat with one tug of a nearly rope, for 10,000,000 parallel arithmetic units spread over a 1000 distinct systems)? Perhaps exception handling at scale is already hard enough without the hair trigger?

Back to math, what I don't want in response is people crawling all over me because I'm lacking some vital pinprick of sophistication here. I'm deliberately standing in for a clever 14-year-old who will not be even slightly impressed with an answer that amounts to this: after you have worked within field theory for another decade, it will make you sick to your stomach to even think about defining a/0=0, because so much older and so much wiser.

Worst. Possible. Answer. Abominably unacceptable to clever, independent-minded 14-year-old.

Can we make this more usefully explicit, or am I smoking a crack pipe here?

What Mr W. ought to have told me: the seemingly obvious 0/0=1 leads to an instant quagmire, the weirder 0/0=0 (augmented with a/0=0) leads to no particular quagmire, but it's completely infelicitous and doesn't buy you a a darn thing as a working mathematician (proceed directly to tautology, do not collect $200).

My other perspective comes from quantum mechanics where destroying information is regarded as pissing away useful energy. Multiplying by zero is special in that way: it's the only multiplication that destroys information.

This is why fields don't promise for all a: a*b*b^-1=a when b=0 (b=0 is at least implicitly ruled out in conventional axiomatization because 0^-1 is undefined).

In the infelicitous extension of field theory where 0/a=0, a*b*b^-1=a except when b is an information-destroying black hole (b=0).

[*] See The Black Hole War (2008) concerning the controversial non-existence of information-destroying black holes in the real universe; only science book I've ever read where I got halfway through and said to myself "hmmm, better luck next time" as I glazed over into a perfect non-absorbent mirror. Coincidentally, next time is circa next week, because it's presently en route from my library for a second kick at the cat. — MaxEnt 00:49, 3 May 2021 (UTC)

This is clearly a subject that interests you. There are many books that will answer your questions. Essentially, we do not define 0/0 = 0 and abandon, for example, the other axioms of Field Theory which would make this "definition" fail to be well defined, because a great deal of interesting mathematics has already been written with the standard definitions. We could, of course, define something new, call it a quasi-Field, and try to come up with consistent axioms that would allow your definition. In fact, many people, ranging from the advanced mathematician to people almost totally ignorant of mathematics, have written extensively on this subject, and I know at least one grade school teacher who teaches your definition to her class, has for years, and isn't about to stop. Rick Norwood (talk) 11:43, 3 May 2021 (UTC)

Feel free to correct me, if I am wrong.

Wouldn't it be simpler to use examples such as Foucault's pendulum, as a real world example of dividing by zero to get an observable result? Sin 0 being at the equator. At the Equator, 0° latitude, a Foucault pendulum does not rotate. In the Southern Hemisphere, rotation is counterclockwise.

Or is this page simply concerned with the arithmetical concept? 49.185.200.59 (talk) 04:35, 18 May 2022 (UTC)

This is solely about the arithmetic concept, but it does address division by zero as a limit of a function. Your example is the limit where the period of precession approaches infinity because the precession rate approaches zero. –LaundryPizza03 (dc̄) 04:45, 18 May 2022 (UTC)

Wiki Education assignment: Computer Science Principles

This article was the subject of a Wiki Education Foundation-supported course assignment, between 19 September 2022 and 9 December 2022. Further details are available on the course page. Student editor(s): Annie.nguyen0811 (article contribs).

— Assignment last updated by Annie.nguyen0811 (talk) 21:33, 27 October 2022 (UTC)

Euclidean division by zero?

File:Gcd exercise.gif

Gcd exercise

In case I'm not asking this question in the right place, I'll be asking this same question at Talk:Euclidean division.

I'm curious why Euclidean division by zero is never discussed?

It seems like such a simple thing to assume that any dividend divided by zero yields a zero quotient plus a remainder equal to the dividend so that the multiplicative inverse is also true, that: the zero quotient times the zero divisor yields zero plus the remainder yields the original dividend. This may seem useless, yet it is not. For, it is useful to postulate division by zero as modulo zero when GCD factoring using a potentially limitless expansion of the GCD in two-dimensional format. -- Vinyasi (talk) 12:21, 5 January 2023 (UTC)