Talk:Division by zero

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1 Early discussion
2 0/0=0
3 my 2 cents
4 0/0 and differentiation
5 Difference between 1/0 and 1/infinitesimal and x/0
6 three impossible things before breakfast
7 Let's use the correct definition of division!
8 Star Trek Reference
9 Rewrite to remove bias
10 Completed (more or less)
11 Fractions
12 Formal calculation
13 Signed Zeroes
14 Unicode Mathematical Symbols?
15 Just in from Lewis Carroll
16 Reversion of Psb777's rewrite of the intro paras
17 Added a section which may help
18 Note on the "limits" section
19 I did it through, and...
20 Wow, 2=1?
21 I don't like how it's worded.
22 12 / 0 = 0, says school division chart
- 22.1 Update
23 Abstract algebra
24 Trigonometry in the real projective line
25 0/2=0.0.5 ?
26 10/0 = infinity
27 0/0 = All Real Numbers, n/0 = No Real Numbers
28 Forget about all that rubbish!
29 Removed paragraph saying 0/0 is well-defined
30 0/0=1
31 Nullity
32 Interesting article, but needs references
33 The Zero-Infinity(zi) Theory
34 Problem with Section 3.1 "Proof that 1=2"
35 Fraction formatting (again)

[edit] Early discussion

In particular, it is incorrect to say that a ÷ 0 is infinity. The argument that any number a, divided by a very small one, becomes extremely large is unconvincing: a negative number a divided by a small positive number does not become large, and neither does a positive number a divided by a small negative number.

Fine. But how about, a ÷ 0 is infinity if a is positive, and negative infinity if a is negative?

I have no idea what 0 ÷ 0 would be, of course. Evercat 23:19 6 Jul 2003 (UTC)

(Disclaimer: IANAM; poorly remembered high school algebra follows!) Lemme try to show why that doesn't work (aside from the definitional problem already explained in the article)... If you plot y = a/x (where a is positive) you'll see the trouble:

                  |* <- approaching 0 from positive x
                  |*    we get a limit of positive infinity
                  |*
                  | *
                  | *
                  |  **
                  |    **************
------------------+------------------
**************    |
              **  |
                * |
                * |
                 *|
                 *|   approaching 0 from negative x
                 *| <- we get a limit of negative infinity

If a is negative, the curve is upside-down but has the same split. a/x approaches positive infinity when approaching x=0 from negative x, and negative infinity when approaching from positive x.

If we try to just come straight out with x=0, we can't fake it with the limit, since the curve is discontinuous. There's no reason to favor positive infinity over negative infinity, whatever the sign of a.

Now, depending on what you're doing, positive or negative infinity may be a useful way to treat a divide-by-zero case, but which is appropriate probably depends on which side of 0 you're approaching from in the denominator, as well as the sign of the numerator.

It may be useful to have a pretty graph in the article to illustrate this, as the quoted sentence tries to say it in words and I think is even more confusing than my attempt here. :) --Brion 23:45 6 Jul 2003 (UTC)

What you say may be insightful (and the italicized passage quoted at the beginning of this thread mentions this briefly.) However, I don't think a lengthier discussion than the passage above is warranted, and this why.

The reasoning that this explanation attempts to demolish only applies to a problem which is tangential to the question "what's 1/0?" What the above does say is: "the limit of 1/x as x tends to zero does not exist, even in the extended sense when limits are allowed to be +∞ or -∞."

The remainder of the article makes this clear. Division is the inverse of multiplication. 0*x is 0 for any x, therefore 0*x isn't invertible (because 0*x isn't injective.) This is the core of the matter.

When you ask a politician a question they don't like ("will you provide transmogrifiers to all the citizens?") their reply is sometimes tangential, sometimes even nonsensical ("I will work so that all citizens are treated justly. My opponent doesn't support the fluxification of space, but I will put a flux on the moon if it kills me. Remember the Alamo.") You then have two choices. You can take that long statement, and make a longer statement explaining how that's nonsense and doesn't answer the question, or you can try to give a more correct answer. To the question "what is 1/0?", the answer "if x is small and positive, 1/x is large, thus 1/0 is +∞" is at best tangential, and at worst nonsensical, and I don't think it's worth a lengthy rebuke.

Since it is in fact a widespread fallatious reasoning, it is worth mentioning as it is now, but wasting electrons on a long exposition about what is essentially bollocks (as I'm doing now) isn't ideal.

Ok, I'll shut up now. -- Loisel 05:16 7 Jul 2003 (UTC)

Wow. all that over a simple idea like 1/0. Seriously, guys, I always made the assumption that x/0 was either 0 or null since if you have any quantity "x" and you place equal fractions of x in a number of containers equal to 0 then how many pieces of x are there in each containter? Zero. But, you actually don't even have any containers, so really "null". And I never believed in the concept of infinity so the other statements about 1/0 = ∞ were insane at the get-go.

Regardless, the question I always wanted to know was: what's the practical application (if any) of the concept of x/0 anyway? No mathematician has ever been able to answer that, and I doubt they ever would.

Um, integration. Morwen 22:51, Dec 18, 2003 (UTC)

While true, one must wonder if this is just a result of our mathematical system. If x*0 is 0 (defined), why then isn't 0*0=x ?(as it should be if you reverse the process) Yet, x/0=(undefined) because the undefined * 0 also doesn't = x. Almost a little broken at points. Maybe one day (certainly not in my lifetime) it will be fixed.

I'm not sure if I understand what you mean; modifying x*0 = 0 would lead to 0/0 = x, which makes x undefined, which is why one can't divide by zero. Er.. I dunno. Evil saltine 23:01, 18 Dec 2003 (UTC)

I look at this question as is being similar to how for example

$x = 2 \Rightarrow x^2 = 4$

$x = -2 \Rightarrow x^2 = 4$

yet

$x^2 = 4 \Rightarrow \mathbf W$

the bold W meaning whatever or simply "?" indicating unknowledge. Using sets, however, it can be argued that

$x = \{-2, 2\}\!$

Should this analogy be applied to division by zero

${0 \over 0} = \mbox{number}$

such that

$\forall x,\ x \sub {0 \over 0}$

In other words zero divided by zero is the most ambigious number; it could be any number of any type: scalar, vector, or tensor; set or matrix; real or imaginary; differential (Rⁿ); albiet combinations thereof. Zero is an amazing number. As for the division of any nonzero number by zero,

${x \over 0}\ \mbox{DNE} \Leftarrow x \ne 0$

this being due in part, if not mainly, to the fact that there is more than one infinity: positive, negative, imaginary, complex; others perhaps.

I recommend avoiding the usage of the words "possible" or "impossible" in formulating conjectures about division by zero. To be a step ahead of human knowledge, one should acknowledge the possiblity of that which we accept as true being false.

--Lindberg G Williams Jr 19:30, 22 May 2004 (UTC)

[edit] 0/0=0

Try reading http://members.lycos.co.uk/zerobyzero and see how certain logical approach can result in solution to real life problems otherwise un solvable due to your error theory.

0 / 0 = 0

Try this simple formula for average speed.

Total Distance Traveled / Hours Traveled = Average Speed

10 Kilometers / 2 Hours = 5 Kilometers Per Hour

What happens when you're not traveling

0 Kilometers / 0 Hours = 0 km/h

0/0=0

There is no other solution. We have created a world of limited mathematics, eliminated the limit of zero, and told the world it was unlimited, providing you did not divide by zero which limited it. Seems you all forget there are other applications and formulas that ABSOLUTELY NEED IT. Anyhow that is just one of thousands of them.

ZBZ

Well, so much for the theory that 0/0 belongs here rather than at indeterminate form. Charles Matthews 18:25, 24 Sep 2004 (UTC)

world of limited mathematics? CSTAR 19:44, 24 Sep 2004 (UTC)

This is actually a good example of why 0/0 is not defined!

If you go at 0 kilometers for any amount of time (say 1 hour), the 0/1=0, as you would expect. Now imagine you are going at X km/hr, for any X. How far will you go in 0 hours? you will go 0 kilometres. Therefore 0 kilometers / 0 hours = X km/hr for any X!

Think of it this way, - say you ran 20 kilometres. To define the speed, if you ran over it in 2 hours, it would be 10 km/h. If I took a "snapshot" of a zero-second timeframe (nil), and you ran 0 km during this period, it does not mean your speed is 0 km/h. Because the timeframe taken is insufficient/invalid and isn't an accurate reflection of actual speed, and therefore the value given is undefined because it could be anything. -- Natalinasmpf 09:41, 25 Apr 2005 (UTC)

You are partially correct, 0/0 does in fact equal 0 but that isn't the only answer. Let's say a/b=x, that would mean bx=a. So when this is applied to the equation 0/0=x you could truthfully say that 0x=0 and this would mean that x is every number (except for any other a/0 equation of course). - RyanAH

I'm interested in this conversation, but I think that in reality, 0/0 should equal 1 (as long as the 0's are the exact same 0's). What I mean is that, in reality, any 0 is only apporximately 0, and is just a very small number - mostly no useful quantity can be 0. Thus x/x is 1 even when x is infinitely small, ie. 0.

Also, I wanted to point out a flaw in the shpeil about average speed. "0 Kilometers / 0 Hours = 0 km/h" isn't quite right. Average speed requires a change of time. Speed is a change in distance over time, dL/dt. A derivative is by definition a limit, and a limit cannot occur if you don't have a spot from which to approach the limit. Thus 0 Km / ( 0 hrs) is not a valid speed. In reality such a quantity would be impossible to find. But if 0 was instead thought of as an infintesimal amount, then 0 Km / 0 hrs = 1 Km/Hr, as long as (0 Km)/Km = (0 hrs) / hrs. I hope i made some sort of sense. Fresheneesz 00:52, 4 February 2006 (UTC)

let me end this dumb attractor

Two two branches converge at ∞. ∞ is nonpositive and nonnegative, just like 0. There is no sense in equivocating its sign with its side; there are only ∞⁺ and ∞^-, not +∞ and -∞. So 1/0 = (1)R∞, where R is any real number. Zero distance over zero time does not produce zero speed; it produces any speed. If any mathematicians disagree with me, and they do because they don't know me, they're wrong. lysdexia 19:44, 13 Oct 2004 (UTC)

[edit] my 2 cents

I look at this question as essentially one of general topology, in which case the answer depends on which topology you choose. In this case, "1/0" is just shorthand for "what is the limit of 1/x as x approaches 0?" It depends on the space and topology. If you're in the real numbers R, then 1/0 doesn't "equal" anything, since 1/x has no limit as x approaches zero. If you throw in plus and minus infinity as separate points (the so-called "extended real number system") with the obvious neighbourhoods, then it still doesn't have a limit, so 1/0 still is meaningless. If, however, you identify plus and minus infinity to a single point, then it's perfectly legitimate to say that 1/0 = infinity. The same thing goes for the case of the Riemann sphere. The limit of 1/z as z approaches infinity (z complex) is infinity. Here, infinity is a well-defined "point" that exists just as surely as 0, 1, or pi. My whole point is, it doesn't really make much sense to ask, "what does 1/0 mean??". It's a meaningless question, a red herring. Stop whoever is asking it, and demand that they ask a more well-defined question which has an answer. Revolver 09:11, 15 Oct 2004 (UTC)

(1) I guess I see it as an algebraic problem -- that is, it has mostly to do with sets & operations. It's not necessary to bring limits into play; the motivation to do that is to attempt to guess a value for 1/0, but whether the assigned value was found by a limit or some other means is ultimately irrelevant, since what you want to investigate is whether 1/0 = x implies 1 = 0 x and that sort of algebraic thing. A beneficial side effect is that the article becomes accessible to anyone with some knowledge of algebra, if we can avoid making limits central to the discussion. (2) I'm not sure that assigning 1/0 = infinity on a Riemann sphere solves the problem entirely. Since (presumably) a/0 also = infinity for any nonzero a, one can't conclude a = 0 infinity. But perhaps there is more that can be said here. (3) I'm not convinced that 1/0 = ? is any more "meaningless" than any other question in mathematics. In any event, "stop whoever is asking it" seems a counterproductive tactic for an encyclopedia. Wile E. Heresiarch 14:23, 15 Oct 2004 (UTC)

Looking at the current article, it would be suitable to add a short section on the algebraic geometry case - where rational functions to a complete variety extend to honest functions on a blowing-up. This effectively says the maximum possible, about the case of polynomials which vanish. It also includes the Riemann sphere = complex projective line case, without privileging it unduly.

Charles Matthews 14:49, 15 Oct 2004 (UTC)

Sounds good to me. Have at it! Wile E. Heresiarch 16:17, 15 Oct 2004 (UTC)

Wile, by "stop whoever is asking it", I didn't mean "silence the person", I meant "demand whoever is speaking to ask a better question". I do think that "1/0 = ?" is a meaningless question, simply because, as our responses indicate, it has multiple interpretations. I interpreted it as a topological question, you as algebraic. I don't think there's anyway for either of us to "prove" the other's interpretation is "wrong". This is precisely what I mean by "1/0 = ?" not being a "well-defined" question. You say that "what you want to investigate is whether 1/0 = x implies 1 = 0x, etc." but that is presupposing an algebraic interpretation. Your interpretation comes from viewing the problem algebraically. Mine comes from viewing it from a calculus point of view (i.e. removing discontinuities, etc.) Charles' interpretation is even more broad and fundamental, and further supports my contention that the question must be recast before addressing it. As for the "1 = a for any nonzero a implies contradiction" argument, again that's presuming an algebraic perspective. There is no contradiction from the calculus perspective, in fact in complex analysis classes, the function 1/z is often defined from the Riemann sphere to itself, in the usual way for finite complex numbers, with the stipulation that 1/0 = infinity and 1/infinity = 0. This is not a statement about binary algebraic operations, but a statement about extending the function 1/z on C to be continuous on the Riemann sphere. Revolver 21:22, 15 Oct 2004 (UTC)

Well, I could quibble about various things. Instead I'll just ask how you would like to improve the article. I think that will help us stay focused. Regards, Wile E. Heresiarch 22:36, 16 Oct 2004 (UTC)

OK, time for some mathematics. For X and Y compact spaces, and F: X → Y a function but not everywhere defined, one can take the graph G of F and its closure G* in XxY. The main idea is to project G* back onto X and say it must be a closed set. Which is true if spaces are Hausdorff. For something like 1/x this suggests Y should be some compactification. What we are looking for is that G* should actually be the graph of a function. In this purely topological case one can't really argue that compactifying the real line to the extended real line is better or worse that to a circle (i.e. we could add two points or one to compactify R).

What the algebraic geometers do is a bit more definite, in that projective space acts as a compactification (non-Hausdorff, but everything is covered by the complete variety/proper morphism properties of projective space, plus the fact that the Zariski topology on XxY is not defined as the product topology). The added feature here is that there is more control of G* when it has a 'vertical component' projecting down to X.

Charles Matthews 08:22, 16 Oct 2004 (UTC)

OK, thanks for the information. How would you like to use this to improve the article? Wile E. Heresiarch 22:36, 16 Oct 2004 (UTC)

I would improve it by first noting that there are multiple ways of addressing the problem. Then, I would address it by interpretation, starting with the algebraic (binary operation) way. This includes much of the article (e.g. "incorrect arguments in dividing by zero" is essentially a misuse of algebra) and is easiest to understand. Then, maybe a mention of the other interpretations, but clearly set off in other sections. Again, I don't object to what's there, or object to the usual algebraic interpretation, just to its priveledged status. Revolver 22:50, 16 Oct 2004 (UTC)

Well, it doesn't seem like what you want is all that different from what's already in the article. Why not just go ahead and make the changes you want -- at least it will give us something different to talk about. Wile E. Heresiarch 16:36, 17 Oct 2004 (UTC)

I worry a bit about going back and forth between "concrete" and "abstract" discussions, that it might be something that only a relatively few who read it will be interested in. (Or worse, that it might confuse... "2*4 = 2"?) The only way to do this seems to be to break up the organisation by interpretation. Revolver 06:12, 19 Oct 2004 (UTC)

[edit] 0/0 and differentiation

I can see a great deal of serious thought went into this article and it is extremely well and formally reasoned. However, it perpetuates the (formally accepted) statement that 0/0 is "just as" undefined as, say, 127/0. I'm quite sure this is formally correct, and I'll thank you not to spank me for saying it is not correct at a lower level, somewhere down among primitive life-forms such as engineers. Here's the text:

Limits of the form

$\lim_{x \to 0} {f(x) \over g(x)}$

in which both f(x) and g(x) approach 0 as x approaches 0, may converge to any value or may not converge at all.

Okay, this is all true, but still, what you are doing, in a sneaky sort of sliding-up fashion, is 0/0. Please don't throw a blizzard of complicated math at my statement; I know what you're after. My point is that 0/0 is what differentiation feels like when I imagine the limit approaching. It is useful -- to me, if to nobody else -- to believe in 0/0 as the limit: the point at which anything becomes possible.

Look at this from the amphibian point of view of an engineer. Think of the tangent to the curve as a long, massive stick; at an initial approximation, it is fixed to the curve at two points, (x,y) and (x1,y1). These points are widely spaced and the tangent does not wiggle. But, as the limit is approached, the distance between the two points decreases and the stick is liable to wiggle more and more; at the limit itself, the tangent is only attached to the curve by a single point and is free to rotate throughout a complete circle. However, we know better than to apply any torque to the tangent-stick as we approach the limit; we very carefully edge (x1,y1) up to (x,y), so when we finally reach the limit, the tangent is left pointing in the correct direction.

All I'm saying here is that it is a useful tool to the student to suspend disbelief and imagine 0/0 = the tangent. It makes differentiation real.

By the same token, when integration is demonstrated by area under a curve, first numerical integration approximates the area as a large number of thin slices. As the limit is approached, the student can think of an infinite number of zero-width slices, which add up to the total area. This is ∞ × 0 = the area. This may not be formally correct, but it is a good and useful way to think about it.

I remember I once listened to my 8th grade math teacher lecture my class on this subject. He said it was "impossible" to divide any number by zero, since no number could be multiplied by zero to yield anything except zero. I raised my hand and said, What about 0/0? He said that was "undefined" and looked pissed off. I accepted that 0/0 was mathematical nonsense. Later, when I got into the calculus, I had a tough time of it, despite being very good at algebra, until I freed my mind and allowed myself to simply imagine the process as a clever way of defining the undefined.

I would like to see if there is anyone bold enough to step away a moment from the purely theoretical and formally correct. Who will join me in wondering where these thoughts belong? I truly believe they are of value to the beginning student of the calculus, who is ill-served by adamant denial of 0/0. --Xiong 06:27, 2005 Mar 11 (UTC)

[edit] Difference between 1/0 and 1/infinitesimal and x/0

Can someone explain in simple terms, if 1/0, 1/infinitesimal, or 1000000/0 or 1000000/infinitesimal or any different?

In standard analysis, an infinitesimal dx is just an independent variable, while any fraction symbol with zero in the denominator is undefined. (Exception: 0/0 is defined as a L'Hopital Form, but is not a number -- really it's is just a way of remembering the hypothesis of L'Hopital's Rule.) Thus anything/0 is a meaningless symbol, while anything/infinitesimal is a function of the independent variable infinitesimal (usually denoted dx). But for another way of looking at things, see the article on Non-standard analysis. Rick Norwood 17:42, 23 December 2005 (UTC)

More simply, an infintesimal is some value that is approximately 0. 1/0 is undefined because 1/x has two limits as x->0 . However, 1/infintesimal is simply a positive really really big number (as long as the infintesimal is positive) - such a big number it can be thought of as infinity. One thing to remember though, is that if x = infintesimal, then x/x = 1. but if the infintesimals are different values, than their ratio might be somthing other than 1. This is the basics of derivatives. Imagine that .002 is an infintesimal and .001 is another. .002/.001 obviously = 2. That sort of thing would happen in an equation like Zdy = dx where Z= 10^9999. .002 would most definately be approximately infintesimal, so we could set dx = to .002, in which case dy would = .002/10^9999 which = approximately 0. Maybe that got convoluted.. but its a different explanation Fresheneesz 01:06, 4 February 2006 (UTC)

Simply to clarify, it is merely because an infinitessimal represents the smallest possible number greater than zero. The existence of two limits at zero for 1/x is because the reciprocal of a number og signum x will be of signum x. Since an infinitessimal is greater than zero, yet theoretically equal to it at once, x/dx will equal infinity, and only positive infinity. He Who Is 20:24, 7 June 2006 (UTC)

An infinitesimal is NOT "the smallest possible number greater than zero." One can divide a positive infinitesimal hyperreal number by 2 and get a smaller positive infinitesimal. Michael Hardy 18:40, 9 November 2006 (UTC)

There exists no infinitesimal element other than 0 in real numbers. --Kprateek88(Talk | Contribs) 14:11, 9 November 2006 (UTC)

[edit] three impossible things before breakfast

I was going to let my Star Trek reference go, though I tend to favor interesting articles over dull ones, but then came a mathematically incorrect edit, and since I was reverting anyway... : )

Division by 0 is in no way analogous to squaring the circle. Division by zero is a question of definition. Any attempt to define division by 0 forces us to give up more than we are willing to give up -- if we define division by 0, no matter how we define it, then we have to give up the cancellation property, because 0 * 2 = 0 * 3 but 2 does not equal 3. On the other hand, squaring the circle is easy, just not with straightedge and compass. You need other tools. Rick Norwood 17:37, 23 December 2005 (UTC)

Yes. I don't see how anyone would see those as analogous... That's to saying "2 has no square root because its irrational... He Who Is 20:30, 7 June 2006 (UTC)

[edit] Let's use the correct definition of division!

Intermediate Algebra by Barnett and Kearns has the actual definition of division and the explanation as to reason for division by zero is not defined:

"We say that a divided by b equals c if and only if there exists a unique value c such that b times c equals a."

Division by zero (b=0) is not defined because if a is nonzero, c doesn't exist, and if a is zero, c is not unique.

B.Wind 21:41, 24 December 2005 (UTC)

There is more than one "correct" definition of division. There is an article in the current issue of Focus that shows just what a great variety of definitions have been proposed. Rick Norwood 01:47, 25 December 2005 (UTC)

So I'm curious - is there a "correct" definition of division that allows for division by zero in the real numbers? The division algorithm is pretty clear, and does not allow division by zero. This is key, isn't it? It IS after all a definition issue. 12/29/2005 DB

Any definition of division that allows division by zero necessarily results in giving up something, usually giving up lots of things, for example, left and right cancellation, closure, well defined, the relationship between multiplication and division, and/or the commutative, associative, and distributive laws. Except for a few very simple systems (example: the only number is 0 and for every binary operation * we define 0*0 = 0, so 0/0=0) defining division by zero looses you more than you gain. However, see non-standard analysis.

All "standard" definitions of division give the same answer for all divisions, and all leave division by zero undefined. Rick Norwood 00:46, 30 December 2005 (UTC)

You can't divide by zero in non-standard analysis either. Michael Hardy 02:11, 30 December 2005 (UTC)

But the simple extension of reals, $\mathbb{R}\cup\{\infty\}$ , where $\infty$ is an unsigned infinity, so $\frac{1}{0} = \infty$ and $\frac{1}{\infty} = 0$ allows division by zero (for non-zero numerator). True, that does give up a few things, but in my opinion is very elegant. --Meni Rosenfeld 13:53, 17 January 2006 (UTC)

The big problem with that is that then you lose $\frac{a}{b} = c$ implies

a = b c

, which is in my opinion fairly serious...Mrjeff 14:18, 17 January 2006 (UTC)

Well, you win some, you lose some... As I said, I think this is a very elegant extension, although arguably not very useful (though this is a common extension to the complex plane, that neatly unifies some analytic properties of functions). --Meni Rosenfeld 18:39, 17 January 2006 (UTC)

[edit] Star Trek Reference

I previously removed a star trek reference from this page. It got re-added, so I thought I'd have a quick discussion about it here.

I don't feel "In some episode of star trek, someone blew up a computer by asking it to divide by zero" is interesting or relevant. Star trek did many stupid things, and also jokes about dividing by zero have appeared in many different areas. Also, I really disagree about this being in the "computer architecture" section. How about a new section called something like "Division by zero in popular culture"? (there is probably a better and/or more standard name for this section).

I certainly think the Star Trek reference should state which episode this happened in -- unfortunately I don't remember. I do remember, in "Return of the Archons", a computer breaking down on being asked to compute pi to the last digit. Rick Norwood 14:31, 4 January 2006 (UTC)

That kind of backs up my point. Do you want to add that to the pi page? I wonder why so many computers on star-trek break down trying to do things that the laptop I'm working on don't crash on at all...Mrjeff 15:37, 4 January 2006 (UTC)

Hows that?Mrjeff 15:31, 10 January 2006 (UTC)

It is for the same reason that the alien computer in Independence Day is the only computer in the universe that will interface with an Apple. Rick Norwood 15:47, 10 January 2006 (UTC)

Another way to think of it is the Macintosh is the computer that can interface with alien systems. :P --127 12:16, 23 January 2006 (UTC)

From my memory, and from searching on the web, it seems that the reference to dividing by zero in Star Trek is incorrect. The incident was actually computing pi, in the episode "Wolf in the Fold" [1], [2]. So I am going to be bold and make the change. 1diot 17:53, 10 January 2006 (UTC)

The evidence that the statement was incorrect notwithstanding, I would have argued to keep it. In the article on Pi for example, there is a section called "Fictional References," containing many similar references.

[edit] Rewrite to remove bias

This article currently seems to have a bias against division by zero, something in the sense of "There are those who foolishly define $\frac{a}{0} = \infty$ , but the ultimate truth is that division by 0 must never be defined." I have a mind to rewrite several sections (some of which have factual errors) to give a more balanced view, that defining division by 0 doesn't make much sense in algebra, but can be useful in analysis if defined properly. That is, of course, as long as no one objects. --Meni Rosenfeld 06:39, 18 January 2006 (UTC)

I'm sorry to say it, but the way you have phrased this does not give me confidence that you can handle this successfully. Why not try it out here, first? Rick Norwood 13:32, 18 January 2006 (UTC)

You're not giving me much credit are you? Well, if you insist, I'll try to compose something and put it up here. But I sure would like to know what made you uncomfortable. --Meni Rosenfeld 13:58, 18 January 2006 (UTC)

What made me uncomfortable? Accusations of bias. Putting words such as "foolish" in other people's mouths. Assertions of factual errors without examples. Rick Norwood 19:07, 18 January 2006 (UTC)

I did not put the word "foolish" in anyone's mouth. I thought it was clear that the sentence inside the quotation marks is an exaggeration, and I apologize if that wasn't clear. All I said is that this is more or less what several phrases in the article sound to me. As for bias, I did not mean it in as strong a sense as you have possibly interpreted it. Since I'm not interested in politics and the like, to me it doesn't carry such a strong negative connotation, as it possibly does to you. As for factual errors, I could of course have given examples such as

For any nonzero a, it is known that

Which is actually true only for positive a (a small error, but an error nonetheless). But I thought it would be much better to fix such errors than talking about them.

All in all, it seems that you have been offended by my comment, as I have been from yours. I meant no harm, and I'm sure you didn't either, but I wish you will be more careful when putting potentially offensive comments (as I will when putting potentially controversial comments).

Like I said, I will try to compose an improved version, but this could take a while. --Meni Rosenfeld 10:39, 19 January 2006 (UTC)

Let's agree that both of us want to improve the article. The error you point out certainly needs to be corrected. Rick Norwood 15:29, 19 January 2006 (UTC)

[edit] Completed (more or less)

There, I've made the changes I wanted (though not on as large a scale as I originally intended). Any format corrections are welcome, as are any comments regarding the content. -- Meni Rosenfeld (talk) 19:04, 21 January 2006 (UTC)

Most of your changes seem good, though I may want to adjust the wording slightly. Also, in reading over the article again, I see a need to say a few words in the introduction for the non-mathematical reader. Rick Norwood 21:07, 21 January 2006 (UTC)

I've done a little rewriting, caught one capitalization error. I also corrected a few other minor errors that had somehow gone unnoticed for a long time -- for example, the assertion that division by a non-zero number was always defined in the ring of integers.

I'm going to pause here, because I do not like to change too many things at one time. Rick Norwood 21:24, 21 January 2006 (UTC)

I've mad some more changes, mostly slight fixes, but also another reword of the Bhaskara thing, adding a few points that I believe your version missed. -- Meni Rosenfeld (talk) 06:32, 22 January 2006 (UTC)

[edit] Fractions

Zack, what you did with the fractions doesn't look very good to me (the shrinking thing). I think they are much less readable this way, and also, using a matrix construct to make them smaller seems not very systematic. Please reconsider this change. Putting the fractions inline and consistently using \frac was great though. -- Meni Rosenfeld (talk) 19:24, 22 January 2006 (UTC)

It is substantially more readable for me this way. I find the enormous gaps between lines produced by the old markup to be a worse readability problem than the tiny print.

\begin{matrix} ... \end{matrix} is the documented method of doing this (see Help:Formula#Fractions, matrices, multilines). It would be better if the distinction between $ ... $ and $$ ... $$ in TeX were directly available in wiki markup, but this is what we have now.

Zack 20:17, 22 January 2006 (UTC)

I agree with Zack. Rick Norwood 20:25, 22 January 2006 (UTC)

[edit] Formal calculation

Rick, I don't agree with you that a formal calculation is a bad idea. Sure, since it's not rigorous it can lead you astray. But an experienced mathematician will usually know what is "right" or "wrong" to do (for example with 1/x^2 you know that "the zero is positive" and you get a positive infinity, whereas in 1/x you know "the zero is unsigned" and you can't do anything). And the point in a formal calculation is not to prove something, but rather to get an intuitive result, which you can later prove if you want to be convinced about it. Also, any criticism regarding this concept is of course welcome, but in its proper place, the article formal calculation which I have started. My point in this section was simply to present a useful substitution for a/0, not to discuss all the rights or wrongs of a formal calculation (which belongs, again, in the appropriate article). Please consider rewording the section in a less negative way. -- Meni Rosenfeld (talk) 19:44, 22 January 2006 (UTC)

And I thought I asked for comments regarding my changes, not mocking of them. I never said that the real projective line and the Riemann Sphere are fields, or that you can define $\infty - \infty$ in them. You can't, for the exact reasons you described. Nor did I say that these structures preserve all the properties of their respective original fields. All I said is that those structures are interesting, consistent, potentially useful, and you can define division by 0 in them. I do wish you will reconsider your last edits. -- Meni Rosenfeld (talk) 19:52, 22 January 2006 (UTC)

BTW, the article's name is "Division by zero", not "Division by zero in the real numbers field". In either case, effectively deleting my edits without discussing them first, is just uncivil. -- Meni Rosenfeld (talk) 19:54, 22 January 2006 (UTC)

You underestimated the problems involved in division by zero. I did not revert your changes, but I did remove statements that were misleading. I'm sorry if you were offended, but you should be sure of your facts. Rick Norwood 20:00, 22 January 2006 (UTC)

First, I will have to request that whatever the case, you will be civil in your edits. When you and Zack did changes I didn't agree with, I didn't go about reverting them (and what you did is effectively reverting); I started a discussion about them. I expect you to do the same.

Second, I'm not sure I am the one who needs to do my homework. If you take a look at Riemann sphere and extended real number line, you'll know that the fields of Reals and complexes can be extended, at some cost. In the extended real number line, you can't divide by 0, because positive and negative infinities are separated, so you can't give a sensible definition. However, you will see that the article comprehensively defines arithmetic operations in it, leaving those that cannot be sensibly defined, undefined. For example $a + +\infty = +\infty$ where $a \neq -\infty$ , but $+\infty - +\infty$ is undefined. The advantage of the extended real number line over the real projective line is that you have the order relation; The advantage of the latter is that you have division by zero. Again, you can't expect everything to be defined; $\infty + \infty$ , $\infty * 0$ and $0 / 0$ are all undefined. This may seem like a great sacrifice, but the advantages of having division by zero are many.

It all comes down to what you want to have and what you are willing to sacrifice. If you want an ordered field (which is a big plus, explaining the popularity of reals), you will use the reals numbers. If you want to add in elements representing infinity, while keeping the order, you'll use the extended real number line. If you are willing to sacrifice order, and some of the properties of the field, while adding infinity and the option of division by zero, you'll use the real projective line. And if you dealt with complex numbers in the first place, and want to add infinity and division by 0, you'll use the Riemann sphere (which is very popular).

These stuctures aren't "wrong" because they lack field properties, just as the reals aren't "wrong" because they don't have division by zero. You can't win them all; Each structure has its own features, and you choose which one to use depending on the features you desire. If you want to write an article "The proof that it is impossible to define division by zero in the field of real numbers", go ahead; But the purpose of an article named "Division by zero" is to discuss structures where it is and is not possible to define it.

To conclude, I hope you will have the common sense to restore the content that you have erased. But I'm more than willing to do it myself if necessary. -- Meni Rosenfeld (talk) 07:09, 23 January 2006 (UTC)

And one more thing: Please do check out [3] for additional insight. -- Meni Rosenfeld (talk) 08:29, 23 January 2006 (UTC)

I did not delete any of the sections you added. I'm certainly familiar with the Riemann sphere. In fact, I'm teaching a course in Complex Analysis this semester. I just added warnings that your sections lacked, and also removed contractions and what seemed to me to be unnecessarily informal adjectives. Of the various sections you have written, the one I like least is "formal calculation", which I have seen too often used, especially by students but sometimes by people who should know better, to take limits "the easy way", often resulting in wrong answers. Note that the link you directed me to on Mathworld is very careful to say that if you define division by zero equals (unsigned) infinity, then you must make infintiy minus infinity undefined. Your section lacked that cautionary note.

In any case, I'm not trying to fight with you on this, just to keep the article mathematically accurate. Rick Norwood 14:12, 23 January 2006 (UTC)

Okay, maybe I'm beginning to understand you. A few points though; What I've written is not instructions for a nuclear plant or anything, it doesn't require words of caution at every step in the way. For example the $\infty - \infty$ thing, is just one example that the Riemann sphere is not a field, and that not all aritmetic operations are defined in it. Such notes are great for the article on the Riemann sphere (and indeed, that article in Mathworld is about the real projective line), but this article is about division by zero. So perhaps a few words that these structures aren't fields (which isn't a bad thing) are in order, but specific examples are too much to be put here. And one definitely should not imply that it is wrong to use these structures because they are not fields. The "many problems arise...so $0 = \infty$ " comment certainly has no place here, it's inaccurate and non-encyclopedic, and in fact added considerably to my irritation. And writing that " $+\infty = -\infty$ is nonsense" is just as aggrevating and inaccurate - The equality is just another way of saying that having division by zero must imply that infinity is equal to its negation, that is, it is an unsigned infinity.

About the formal calculation, it is a tool, and like any tool, what you get from it depends on how you use it. And it is well known that misusing this tool can lead to errors. So perhaps a reminder that care should be taken can be placed, but not your comment that is basically an attempt to discredit it, which can be considered POV (and I'm sorry if that word offends you). And again, any neutral critcism of formal calculations is welcome at that article.

All in all, I really don't like the way the article looks now, and will improve it, taking into consideration your concerns. I can't do this currently, perhaps tomorrow. -- Meni Rosenfeld (talk) 20:31, 23 January 2006 (UTC)

I hope that you do improve the article, but feel strongly that the cautions I provided, or something like them, are necessary. Rick Norwood 21:22, 23 January 2006 (UTC)

I made some changes, now it looks more reasonable to me. I hope you will approve. Too bad about the small fractions though, I still don't like them but will not argue with you. -- Meni Rosenfeld (talk) 08:45, 24 January 2006 (UTC)

I've made a few small changes, of which the most important is an example in which formal calculation fails. Rick Norwood 15:24, 24 January 2006 (UTC)

Looks reasonable. But unsurprisngly, I'm not comfortable with your version of the formal interpretation section. A counter-example is, maybe, appropriate (though I'm inclined to think it does not belong here), but I very much doubt that a definition of the concept is. I think you are making too big a deal out of a small section. I will re-scrutinize this section in a few days, after I've made sure your ideas have sufficient representation in the formal calculation article. And I wish you will re-read my comment on the matter at the very beginning of this thread, to remind you what I think of the issue. -- Meni Rosenfeld (talk) 16:20, 24 January 2006 (UTC)

I reread your comments -- and discovered that I had remembered them correctly. Of course, mathematicians use "formal calculations" all the time. But beginners need an example of how they can go wrong. Rick Norwood 17:00, 24 January 2006 (UTC)

Can I just ask a question, what "formal calculation" is going on here? From my nearby maths book..

lim(x -> a) f(x) = b if and only if: (exists d > 0) such that (forall 0 < e < d) there (exists f > 0) such that |f(e) - b| < f.

Thats the only formal definition of lim I could find. That clearly doesn't apply in this case. What is being used? Mrjeff 12:11, 25 January 2006 (UTC)

See formal calculation. "Formal" as in "formal definition" is not the same as in "formal calculation". And also, you should check the definitions for infinite limits, the one you give is for finite limits. -- Meni Rosenfeld (talk) 12:44, 25 January 2006 (UTC)

Formal calculation, as I understand it, means you go through the forms of doing a calculation without worrying about the meanings of those forms. It is almost the exact opposite of a formal definition, which is a definition where care is taken with both the form of the definition and the meaning of the symbols. Rick Norwood 01:31, 4 February 2006 (UTC)

[edit] Signed Zeroes

The article mentions how one can use signed or unsigned infinities to try to make sense of division by 0. It seems that you can also use signed zeroes. As long as $\pm \infty \mp \infty$ , $\frac{\pm \infty}{\pm \infty}$ , $\frac{\pm 0}{\pm 0}$ , and $\pm 0 \times \pm \infty$ are left undefined, then it seems consistent. For example, if $a$ is a positive number, then:

$\frac{a}{+\infty}=+0$

$\frac{a}{-\infty}=-0$

$\frac{-a}{+\infty}=-0$

$\frac{-a}{-\infty}=+0$

$a \times +\infty = +\infty$

$a \times -\infty = -\infty$

$-a \times +\infty = -\infty$

$-a \times -\infty = +\infty$

and so on. Is there a name for this type of system? --BrainInAVat 02:55, 13 February 2006 (UTC)

One can, of course, invent any system one likes. This would be a system in which only multiplication was defined, not addition (else you would have +0 = +0 + -0 = -0). Also, you loose the betweenness property -- there is no number in between +0 and -0. It will then be isomorphic to the closed interval [-1,1] with one additional point -0. I don't know a name for such a system. Rick Norwood 13:44, 4 March 2006 (UTC)

I also don't know a name, but these definitions are written at −0. I guess you could call them "IEEE 754". Melchoir 17:26, 4 March 2006 (UTC)

[edit] Unicode Mathematical Symbols?

I was wondering what everyone's reaction would be to the idea of using Unicode characters in place of images of mathematical symbols in this article (where possible). My connection has been stubborn this evening and it took forever for the equation images to load. 67.142.130.32 05:50, 4 March 2006 (UTC)

The vertical fractions can't be changed, but there is indeed too much math-mode in the article. I'll try to cut it down. Melchoir 06:43, 4 March 2006 (UTC)

[edit] Just in from Lewis Carroll

The late reverend C. L. Dodgson just channelled me a message that he loved this phrase: "Nothing can be divided into zero parts." He added: "And doing it requires no effort at all!". Lambiam ^Talk 23:39, 6 April 2006 (UTC)

[edit] Reversion of Psb777's rewrite of the intro paras

OK. What's wrong with my edit that it needs reverting? Paul Beardsell 23:35, 24 April 2006 (UTC)

The very first sentence. "division by zero is not defined" is directly contradicted by the section "Other number systems". Melchoir 23:41, 24 April 2006 (UTC)

You are right but at what cost? If you define division by zero then it is no longer undefined. But, as is shown again and again in the article, defining it reduces the usefulness of the resulting mathematics. The remainder of the article is simply a set of examples demonstrating why we choose to leave division by zero undefined. To avoid confusion I think we should be plain about this at the beginning of the article. Maybe I didn't do a good job of it. Paul Beardsell 23:55, 24 April 2006 (UTC)

But see, we don't always choose to leave it undefined. That's kind of the point. See Riemann sphere. --Trovatore 23:56, 24 April 2006 (UTC)

I think we taint maths by such construction. What I do get annoyed by are ongoing discussions here and elsewhere (outside of Wikipedia) as to what the value of 1/0 is. And what 0/0 is. It would be very useful to simply have an authoritative reference I can point to that says "division by zero is undefined". I would like to revert to my version and then edit it to say "Division by zero is undefined except in certain rarified, esoteric branches of mathematics. Division by zero is always undefined with real numbers." Would that be acceptable? Paul Beardsell 00:08, 25 April 2006 (UTC)

That's POV; besides, IEEE 754 is hardly rarified mathematics, and limits are high-school material. Now, the article already says "In ordinary (real number) arithmetic, the expression has no meaning." Perhaps you'd like to emphasize that statement somehow? Melchoir 00:18, 25 April 2006 (UTC)

NPOV does not require or consider desirable that all possible interpretations need be given the same weight. IEEE 754 might not be difficult but it is both esoteric and rarified BY DEFINITION. (Strange how definition keeps popping up!) See http://dictionary.reference.com/search?q=esoteric and http://dictionary.reference.com/search?q=rarified Paul Beardsell 00:34, 25 April 2006 (UTC)

The Riemann sphere is to maths what rubber balloons is to architecture. Yes you can construct with balloons but we don't give it much weight in an encyclopedia article on architecture. Paul Beardsell 00:34, 25 April 2006 (UTC)

No one is suggesting that the Riemann sphere be given equal weight with the real numbers. And it is perfectly correct to examine the properties of rubber balloons in an encyclopedia article on rubber balloons. Melchoir 00:49, 25 April 2006 (UTC)

Yes! But this is an article on division by zero not on Riemann spheres. I am not saying do not mention balloons in the architecture article! And I am not saying do not mention Riemann spheres here. Paul Beardsell 00:58, 25 April 2006 (UTC)

Eh, I meant something else; such are the perils of analogies. Melchoir 01:23, 25 April 2006 (UTC)

I agree that we need to say clearly and unambiguously that in the real number system, division by zero is undefined. I know personally one grade school teacher who teachs her students that any number divided by zero is zero, and she smugly tells me where I can take my opinion and shove it when I try to convince her otherwise. It would be nice if her students, who must number in the hundreds by now, could come to wikipedia and get good, clear advice. Starting the article with the statement that division by zero is undefined in the real number system does not prevent other number systems in which division by zero is allowed being discussed further down in the article. Rick Norwood 00:24, 25 April 2006 (UTC)

It does say that unambiguously, in the very second sentence. Let's get our priorities straight here. This is an encyclopedia, not an educational aid for high school students. The first priority is being mathematically correct. --Trovatore 00:37, 25 April 2006 (UTC)

So: Is what I propose above acceptable to you? Paul Beardsell 00:44, 25 April 2006 (UTC)

No, not remotely. The existing article is fine. --Trovatore 00:46, 25 April 2006 (UTC)

Until this very moment I thought we agreed. No Wikipedia article is ever finished. Paul Beardsell 00:48, 25 April 2006 (UTC)

Let me clarify. Of course I am not saying the article should never change. I am saying it should not change in the direction of saying more unequivocally that division by zero is not defined. It is defined, in important contexts. Saying that it is never defined is the sort of laziness that makes it easier to teach a high-school algebra class, and I have some sympathy for algebra teachers (having been one) but it does not need to be any part of our purpose to make life easier for them. --Trovatore 01:18, 25 April 2006 (UTC)

Surely the article has room for improvement... that doesn't involve belittling its own topic or simply being wrong. Any ideas? Melchoir 00:53, 25 April 2006 (UTC)

Is the idea I proposed acceptable to you? I think so: You have suggested a re-emphasis in line with my suggestion. Paul Beardsell 00:58, 25 April 2006 (UTC)

If you're still referring to "certain rarified, esoteric branches of mathematics" then no. Melchoir 01:05, 25 April 2006 (UTC)

Let's get away from the high school. You are appointed mathematical advisor to the prime minister. You are called in urgently. "Tell me about division by zero." What do you do? Talk about Riemann spheres first or last? I think first you say that the operation is not defined on real numbers. "Real numbers? What do you mean?" barks the impatient prime minister. No one is suggesting any material be culled, just re-ordered and re-emphasised. Paul Beardsell 00:58, 25 April 2006 (UTC)

Are you saying we should shorten "In ordinary (real number) arithmetic, the expression has no meaning" to "In ordinary arithmetic, the expression has no meaning" for the benefit of readers who can't deal with parentheses? Melchoir 01:08, 25 April 2006 (UTC)

If the prime minster asked me that, I would say "It is undefined in normal arithmetic". Then I might talk about strange and unusual arithmetics in which it is defined. I would definatly talk about riemann spheres last. First I would talk about arithmetics where A/B = C means A = B*C, which is I feel an important rule of arithmetic, and one reimann spheres throw away. —The preceding unsigned comment was added by Mrjeff (talk • contribs) 08:30, 25 April 2006 (UTC)

I remember a baseball announcer talking about a certain pitcher who had allowed an earned run in the majors, but had not gotten anyone out. He said something along the following lines:

The next time this pitcher gets someone out, his ERA will go down [from what it was when the announcer was speaking]. Because it is infinity.

The announcer wasn't wrong. I don't think he knew anything about the map x |-> 1/x being an anti-order-preserving autohomeomorphism of [0,∞], or at least not in those words. But he was correct that this pitcher's proportion of earned runs to outs was worse than any finite value, and would be better than that as soon as he got someone out. The announcer had intuitively chosen a mathematical structure that was right for the job, rather than one that satisfied some fixed list of algebraic properties. --Trovatore 14:58, 25 April 2006 (UTC)

There are perfectly reasonable algebraic systems on the closed infinite interval

$[ 0, \infty]$

for which divison by zero is defined and useful.--CSTAR 15:10, 25 April 2006 (UTC)

I think that's what I just said. But more entertainingly :-). --Trovatore 15:11, 25 April 2006 (UTC)

Hmm, entertaining, assuming you know what an ERA is. That's a bit too culturally specific to th US. Also, in reference to Paul Beardsell's question, there are no prime ministers (yet) in the US where baseball is played. There many be a King, but no prime minister.--CSTAR 15:20, 25 April 2006 (UTC)

I wouldn't be shocked to see the Prime Minister at a Blue Jays game, though I know his sport is really hockey. --Trovatore 15:49, 25 April 2006 (UTC)

[edit] Added a section which may help

I have added a section on elementary arithmetic. There are many people who read encyclopedias whose needs are met by simpler explanations; inclusiveness would indicate that "naive" is not a good term for those needs. It is also the case that those who teach elementary concepts require support with concrete examples. 1diot 18:59, 25 April 2006 (UTC)

I agree; the "naive" bit is mostly a language barrier issue though, since mathematicians don't intend the word to be pejorative. (I can certainly see how it can be interpreted that way.) The introduction could use a bit more meat, so I'll try restoring the explanation with a different adjective. Melchoir 19:28, 25 April 2006 (UTC)

Err... on second thought, I'm not sure what to write. The introduction still ought to provide a brief overview of the difficulty, though. Melchoir 19:32, 25 April 2006 (UTC)

I'm not too sure if this section is necessary, but for the time being I've made some work on it. -- Meni Rosenfeld (talk) 14:32, 26 April 2006 (UTC)

[edit] Note on the "limits" section

This section makes a lot of hay out of the sign ambiguity. That's irrelevant in the case of the real projective line, and it's also irrelevant if you look at the interval [0,∞], leaving out the negative numbers.

The latter point is important because it has real-world interpretations; there are lots of situations where a quantity that you use as a denominator can meaningfully be zero, but cannot meaningfully be negative. The example of earned run average I've already given (you can retire no batters, but not a negative number). For an example that might resonate more with Europeans—how do they measure automobile fuel efficiency? In Canada it's done in litres per hundred kilometres. Again, a negative denominator is not meaningful (because a car driving in reverse uses positive fuel), but a zero denominator is, and an idling car has infinite fuel consumption per kilometre. --Trovatore 19:43, 25 April 2006 (UTC)

Even better, sometimes it's useful to switch between two different systems. In everyone's favorite example, negative temperature, the inverse temperature resides on the extended number line with signed infinities and an unsigned zero, while the temperature scale has signed zeroes and an unsigned infinity. I'm not sure if it would be original research to mention this stuff, but in a perfect world, it wouldn't be! Melchoir 20:13, 25 April 2006 (UTC)

Temperature has signed zeroes and an unsigned infinity? This seems strange to me. Absolute zero is the lowest possible temperature, and so temperature can only approach positive infinity, not negative infinity. And a temperature of -0 is the same as a temperature of +0. Rick Norwood 21:04, 25 April 2006 (UTC)

I think negative temperature will clear all that up. Melchoir 21:12, 25 April 2006 (UTC)

(Actually, the quote in that article doesn't make the case for unsigned infinity, but it's made elsewhere in the text.) Melchoir 21:13, 25 April 2006 (UTC)

Very interesting. I learned something new today. Can the temperature "wrap around, and pass -0, if it gets "hot" enough? Rick Norwood 22:32, 25 April 2006 (UTC)

Hmm... I have to doubt it. To try to connect back to our topic of dividing by zero, at a temperature of -0, any state of the system with an energy E less than the maximum energy E-star will be suppressed by a Boltzmann factor of

$e^{-(E-E^*)/-0}=e^{(E^*-E)/-0}=e^{-\infty}=0.$

So the system is confined to its highest-energy state(s). Intuitively, this means that it's already as hot as possible. Melchoir 22:55, 25 April 2006 (UTC)

This section, as does most of the article, assumes that the starting point for the discussion is the reals. It shows a possible line of thought for defining division by 0 in it, identifies a few difficulties with this approach, and gives a hint as to what can be modified in order to make something out of it. It demonstrates that negatives are a problem, so we should either identify +∞ with -∞ - leading to the real projective line, or dismiss negatives altogether - leading to [0, ∞], which was not trivial at the starting point (we had to throw away half the objects of our original discussion). The latter should probably also be added to "other number systems".

In short, the section makes a lot of hay out of the sign ambiguity exactly to show why we should restrict ourselves to structures where that's irrelevant. -- Meni Rosenfeld (talk) 14:45, 26 April 2006 (UTC)

[edit] I did it through, and...

I have come up with the following: A divided by 0 is infinity. That's right, infinity. There. Does that count as defined? (no, I'm not trying to sound smart) Random the Scrambled 19:12, 19 May 2006 (UTC)

That does count as defined, but the question is how consistent this definition is with the properties we wish division to have. Defining it as infinity is discussed in the article (you should read it again, and perhaps also Real projective line), but there are some subtleties involved (you'll have to let +∞ = -∞, and 0/0 is still a problem). -- Meni Rosenfeld (talk) 09:04, 20 May 2006 (UTC)

[edit] Wow, 2=1?

It's amazing how 2 can equal 1, very postmodern. Before you know it, pigs will start flying and I'll be 7 feet tall! 152.163.100.201 19:50, 24 May 2006 (UTC)

The point is of course that if you can divide by 0 the way you divide by other numbers, then 2 = 1; but since 2 ≠ 1, you cannot so divide by 0. (Most people see this particular argument when they're in about 9th grade, don't they?) Michael Hardy 21:22, 24 May 2006 (UTC)

I believe the logic is usually the other way round. You cannot divide by 0, therefore the proof is fallacious. Elle _{vécut heureuse} ^{à jamais} (Be eudaimonic!) 21:50, 24 May 2006 (UTC)

Doesn't this 2 = 1 proof prove that 0/0 is 1/2 :P

[edit] I don't like how it's worded.

It says:

However, this definition fails for two reasons.

First, positive and negative infinity are not real numbers.

This is as absurd as saying the square root of -1 isn't i because i isn't a number. I think this should be reworded. (I don't know how it should be reworded, though, so I'll leave that to someone else) --Zarel 01:16, 28 June 2006 (UTC)

To say it's not a real number means it's not an element of the set R of real numbers in the same way that i is not a real number. I'm probably missing you're point.--CSTAR 01:40, 28 June 2006 (UTC)

I realize that it's not a real number. However, the article implies that the aforementioned is a reason why the definition fails, which is, in my opinion, absurd. --Zarel 21:21, 2 July 2006 (UTC)

How about this? Melchoir 02:21, 3 July 2006 (UTC)

Much better; thank you. --Zarel 04:30, 5 July 2006 (UTC)

[edit] 12 / 0 = 0, says school division chart

Your local K-4 school can easily buy a division chart for classroom use (either wall-sized or notebook-sized), which "proves" the opposite:

  Any number divided by zero equals zero

https://www.highsmith.com/webapp/wcs/stores/servlet/Production/Search.jsp?category=88747&A=15015&NSave=0&Au=CategoryId&catalogId=10060&NaoSave=0&An=0&langId=-1&NttSave=division&storeId=10001

Wow, that's both funny and awful. I guess if you want to be generous, you could trace their error back to Brahmagupta... Do you suppose we should mention this in the article? Melchoir 18:41, 2 August 2006 (UTC)

If evidence is found that there actually are schools which have sunk so low as to use this chart, it just might be worth a mention. -- Meni Rosenfeld (talk) 18:46, 2 August 2006 (UTC)

It's actually highly educational. It's a good early lesson that grownups don't always know what they're talking about, and just because you see something on a laminated chart doesn't make it true. --Trovatore 18:50, 2 August 2006 (UTC)

I looked at that web page, and I can't see anything on the chart that says 12/0 = 0. It's too small to read. How do you know it's there? Michael Hardy 20:29, 4 August 2006 (UTC)

Save the image to disk, and blow it up in ImageMagick or Eye-of-gnome or whatever you use. --Trovatore 20:50, 4 August 2006 (UTC)

That is really scary raptor 14:07, 19 August 2006 (UTC)

OK, here it is. BARELY legible:

Image:H731660F.jpg

Blatant educational malpractice. If I had a child attending a school that used this, I'd sue. Michael Hardy 21:55, 19 September 2006 (UTC)

(I've now deleted the image for copyright reasons.)

At this web site where the chart saying 12/0 = 0 is sold, the publisher is now soliciting opinions of the product. Everyone: please go there and tell them what you think. Michael Hardy 20:42, 9 November 2006 (UTC)

[edit] Update

They have now taken the product off the web and report that they will put up a corrected version. Michael Hardy 03:30, 19 November 2006 (UTC)

[edit] Abstract algebra

From Abstract algebra

Similar statements are true in more general algebraic structures, such as rings and fields. In a field, every nonzero element is invertible under multiplication, so as above, division poses problems only when attempting to divide by zero. However, in other rings, division by nonzero elements may also pose problems. Consider, for example, the ring Z/6Z of integers mod 6. What meaning should we give to the expression 2 / 2? This should be the solution x of the equation 2x = 2. But in the ring Z/6Z, 2 is not invertible under multiplication. This equation has two distinct solutions, x = 1 and x = 4, so the expression 2 / 2 is undefined.

This makes no sense to me, I understood everything in this article till this section. This should be explained better, or atleast cited. . HighInBC 23:34, 12 September 2006 (UTC)

As the name of the section suggests, knowledge of abstract algebra is required to understand it. Links to ring and field are provided for readers without such knowledge. -- Meni Rosenfeld (talk) 05:10, 13 September 2006 (UTC)

[edit] Trigonometry in the real projective line

(copied from User Talk:Meni Rosenfeld) Why does the comment you removed not belong there? The range of the tangent and cotangent functions should be viewed as the real projective line with only one point at infinity. Michael Hardy 17:45, 19 September 2006 (UTC)

It just seemed to me that, while this may be perfect for the real projective line article, it is too much detail for this article, where I think that no more than a mention of what it is and its relation to division by zero is required. If, taking this into account, you still think it should be mentioned here, then I have no objections. -- Meni Rosenfeld (talk) 18:00, 19 September 2006 (UTC)

OK, I'm going to try to rephrase it to alter the emphasis. Apparently you though my comment was about the real projective line rather than about how to interpret division by 0 in a certain particular case. Michael Hardy 21:22, 19 September 2006 (UTC)

[edit] 0/2=0.0.5 ?

I think if you look at it from a quantum perspective (if that's how to word it), zero divided by two, would be zero point zero point five. 0/2=0.0.5 What do you think? - Infurnus 02:40, 23 September 2006 (UTC)

Would you mind clarifying this? For starters, what does a.b.c mean? -- Meni Rosenfeld (talk) 05:59, 23 September 2006 (UTC)

I'm not sure, but let me put it like this: There is no such thing as nothing in our universe, it is all made of SOMETHING. Zero is a placeholder of sorts, so if you're trying to divide "zero" into two, then you would get "half of zero", whatever zero may be. I hope this helps. EDIT: Also: There's no such thing as nothing because everything is made of an infinite amount of things that get infinitely smaller. - Infurnus 11:03, 24 September 2006 (UTC)

Nothing is not a thing, it is the lack of things. How many humans live on the sun? That's right. 0. There are humans, and there is a sun, but there are no humans living on the sun.

The closest mathematically meaningful statement to your idea I can find, is that if ε is a positive infinitesimal (which, of course, cannot be a real number), then ε / 2 is less then ε. -- Meni Rosenfeld (talk) 11:39, 24 September 2006 (UTC)

By the way, I'm not sure there is scientific evidence supporting your claim that "everything is made of an infinite amount of things that get infinitely smaller" (for all we know, the universe could be discrete), although it's a nice idea. -- Meni Rosenfeld (talk) 11:51, 24 September 2006 (UTC)

[edit] 10/0 = infinity

Another approach to teaching divison to elementary students is to discuss equal parts. To divide 10 by 2, we explain that we can take ten objects and put them in groups of 2. The quotient will tell us how many groups we have. Therefore, to understand, ten divided by zero we can say with ten objects how many sets of zero can you make? You can make an infinite number of sets.

Yes, but this subject is too subtle to be dealt with using purely intuitive notions. For a more serious discussion, take a look at real projective line. -- Meni Rosenfeld (talk) 12:38, 26 September 2006 (UTC)

That example makes me want to answer, "0 with a remainder of 10." Hmmm... --BekiB 17:19, 16 October 2006 (UTC)

Not quite, the remainder must be less than the divisor. That would be the correct answer to "10 divided by infinity", though. -- Meni Rosenfeld (talk) 21:21, 16 October 2006 (UTC)

[edit] 0/0 = All Real Numbers, n/0 = No Real Numbers

Let us consider two simultanious linear equations, with a, b, c, and d representing constants and x and y representing variables: $\begin{cases} y = a x + b \\ y = c x + d \end{cases}$ Then, using substitution,

a x + b = c x + d

a x - c x = d - b

$x = \frac{d - b}{a - c}$

In the case of these two equations to be lines on the cartesian plane, then a and c represent their respective slopes, b and d represent their respective y-intercepts, and x and y represent the coordinates of the intersection of the two lines. Therefore, when the two lines are parallel, meaning that they have the same slope but different y-intercepts, they will never intersect. And what will the above equation give you as the coordinates of their intersection? Some non-zero number divided by zero. And if the two lines are the exact same, they will have the same slope and the same y-intercepts, and they are constantly intersecting. And what does the equation give you? Zero divided by zero. Therefore, wouldn't zero divided by zero mean "for all x" and a non-zero number divided by zero mean "For no x"? For further proof, consider a logarithm base-1. By the law of changing base, we can dictate that

$\log_{1} x = \frac{\ln x}{\ln 1}$

And, as the natural log of 1 is equal to zero,

$\log_{1} x = \frac{\ln x}{0}$

Therefore, the log base 1 of 1 would be equal to 0 divided by 0, and the log base 1 of any other number would be 1/0. This means the same thing as the aforementioned statement of the terms for division by zero. Just wondering if my logic is flawed or anything. --User:ThatOneGuy

Your logic is correct, but uses an argument much too complicated than necessary for the conclusion - 1/0 should be a number x such that 0x = 1, and of course no such real number exists; And 0/0 should be a number x such that 0x = 0, and every real number satisfies it. Since the result of every arithmetic operation is expected to be a unique number, neither is defined in the context of real numbers. -- Meni Rosenfeld (talk) 19:52, 21 October 2006 (UTC)

[edit] Forget about all that rubbish!

Can't we just say that division by 0 is undefined, full stop, and there's nothing we can do about it?--67.10.200.101 02:52, 15 November 2006 (UTC)

No. --Trovatore 03:04, 15 November 2006 (UTC)

Yes!--67.10.200.101 03:23, 17 November 2006 (UTC)

Well, literally speaking, of course, we can say it. But it would be incorrect, as the article explains at length. --Trovatore 03:29, 17 November 2006 (UTC)

Why do you say that? I always thought that division by 0 was always undefined, no matter what.--67.10.200.101 03:27, 18 November 2006 (UTC)

Ya thought wrong, friend. As I say, read the fine article. --Trovatore 03:32, 18 November 2006 (UTC)

You're saying that 0 divided by 0 is any number? That's not true!--67.10.200.101

What I said was that you should read the article. If you want more specific direction, concentrate on the bit about the real projective line and the Riemann sphere. --Trovatore 01:40, 23 November 2006 (UTC)

That is, of course, as long as you're satisfied with dividing a / 0 where a ≠ 0. If you want 0/0 to exist, you'll have to resort to wheel theory or something. -- Meni Rosenfeld (talk) 20:22, 25 November 2006 (UTC)

No, it's not a number. That by itself doesn't mean it's undefined in all contexts. Also, even when it's undefined, there's more to say: if y and z both approach 0 as x approaches c, then y/z can approach a specific number, whose value depends on how y and z depend on x. Michael Hardy 02:46, 26 November 2006 (UTC)

[edit] Removed paragraph saying 0/0 is well-defined

I removed this :

However, 0 divided by 0 is well-defined. Suppose you had an equation $\frac{a}{b}=c$ . This statement is always true: $\frac{a}{b}=c$ if and only if $b c = a$ . So if you set both a and b as zero, the quotient, c, would be all numbers, because 0, which is the value for b in this case, mutliplied by anything, would always equal 0, which is also the value for a in this case.

There seems to be a misconception about the notion of "well-defined". One need a unique real number c for a/b to be well-defined. --Kprateek88(Talk | Contribs) 06:21, 23 November 2006 (UTC)

[edit] 0/0=1

0/0 = 0¹/0¹= 0⁰

Since any number to the power of 0 = 1 it can be shown that 0/0 = 1—The preceding unsigned comment was added by 86.131.253.224 (talk • contribs) 13:55, November 26, 2006

0/0 = (2*0¹)/0¹ = 2*0¹/0¹ = 2*0⁰

= 2

So 0/0=2. That's why 0/0 is indeterminate - it can be pretty much anything. --Zarel 02:48, 27 November 2006 (UTC)

I was experimenting with that. a^0 = 1 when a =/= 0. If you've ever that qualifier in algebra books ("when x =/= 0"), its usually there in any fraction problem where they must make sure it is defined by not making the bottom zero. The very premise of dividing by zero is undefined. Jaimeastorga2000 05:14, 7 December 2006 (UTC)

[edit] Nullity

Take a look at this: http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/divide_zero_feature.shtml It seems someone has solved the problem (at least sort-of). Medevilenemy 19:30, 6 December 2006 (UTC)

Meh. I find that article sensationalistic and naive to the point of plain ignorance. It doesn't explain how "nullity" is anything but a synonym for "indeterminate form" with a fancy symbol, or how a software implementation would be any different from using NaN. I'm willing to keep an open mind and believe it's worthwhile, but only if someone shows me. Or, to put it in Wikipedia terms, if we come up with a decent source. Has anything related to this "solution" been published in a mathematical or educational context? Melchoir 20:34, 6 December 2006 (UTC)

Check out [4], I suppose. I still don't put much into it. He actually mentions one of the Wikipedia examples of division by zero in the 2nd paper, and says it is fallacious because of the assumption that 0/0 = 1. I don't like how he uses infinity as a number - he has defined infinity as 1/0, and negative infinity as -1/0. Not only is infinity not a number, but isn't it trivially demonstrated, since 0 = -0, that this statement is meaningless (+infinity != -infinity)? --RandomPrecision 09:50, 7 December 2006 (UTC)

Is it different from NaN? --MarSch 14:46, 7 December 2006 (UTC)

Aha, I knew I've seen something like this before: Wikipedia:Articles for deletion/Transreal number line. Melchoir 15:47, 7 December 2006 (UTC)

take a look at the howls of derision in the comments box beneath the story. it seems he's passing something trivial and of not much use (rather than something non-mathematical) as something spectacularly original --Mongreilf 15:10, 7 December 2006 (UTC)

I've seen them. The link above to the article[5] is more enlightening though. I don't see anything which is much dfferent from NaN. Not that I checked thoroughly. It is the authors job to do such things and present them clearly. For now I'm going to assume it's just NaN by another name. --MarSch 15:19, 7 December 2006 (UTC)

His biggest fallacy is when he says 1/0 is positive infinity, yet division of a positive number by a negative sign yields a negative answer, and since 0 has no sign, you cannot divide by it and get EITHER positive OR negative infinity. The concept is flawed, and should be promptly deleted from wikipedia.220.253.57.87 13:27, 10 December 2006 (UTC)

He defines 1/0 to be positive infinity and definitions cannot be wrong in the sense of false. --MarSch 14:02, 11 December 2006 (UTC)

[edit] Interesting article, but needs references

This is a pretty interesting article. However, I noticed that it currently appears to have no references at all. It really needs citations for its statements and claims to meet Wiki standards for verification and to show that the article isn't mostly original research. Dugwiki 21:30, 7 December 2006 (UTC)

[edit] The Zero-Infinity(zi) Theory

"The Zero-Infinity(zi) Theory" by David Tulga is a complete algebra for dealing with division by zero. It's way more interesting and much more important than the "easy" identity that professor Anderson "discovered", although his identity and solution is also interesting. [6]

ZI and Nullity are much like the complex number system opening new horizons, well, dimensions really.

Please add the external link to David's site to this article, about division by zero, as it is most amazing and people need to find out about it.

- Peter

Sorry, but Anderson's material isn't in this article either. Also, Wikipedia's external links aren't there for promotional purposes, and they have standards: see Wikipedia:External links. Melchoir 00:33, 8 December 2006 (UTC)

I'm not affiliated with David Tulga. I don't know him. I'm not "promoting him". He has made a significant breakthrough in mathematics and I'm trying to have something about this breakthrough included in Wikikpedia on the appropriate page - which is this page. All the material he has published on this amazing theory is on the linked page as far as I know. In what way can I add material and get by the snobish wikipedia attitude that I'm promoting him or am some sort of spamer - these are not the case. This is frustrating, especially so since I've aready been to the web page that you linked to and that page directed me here. Arrrgg.. Would someone please add something to the this page about this breakthrough. Thank you. —The preceding unsigned comment was added by 69.251.244.134 (talk) 00:39, 8 December 2006 (UTC).

This is not a breakthrough, at all. The idea has been brought before several times, and it just doesn't work. We even have a reference desk question about this subject. Also, unless it can be cited through peer-reviewed papers, we can't mention it on Wikipedia. That link in this page would violate the no original research rule. ☢ Ҡi∊ff⌇↯ 01:43, 8 December 2006 (UTC)

First, let me assure you that none of us is against division by zero, and as you can see in the article, there are well-defined mathematical structures in which such a thing exists.

What we are against is inconsistent mathematical inventions, and while I'm sure that David Tulga is a very nice person, unfortunately he doesn't really know what he's talking about. For starters, he doesn't define his structure in a way that makes sense mathematically; and even if you try to put some sense into it, you will end up with multiplication not being associative (0*(2*Zi) = 2 ≠ 1 = 0*Zi = (0*2)*Zi), which is not what we want. -- Meni Rosenfeld (talk) 10:47, 8 December 2006 (UTC)

[edit] Problem with Section 3.1 "Proof that 1=2"

Has anybody noticed that although this 'proof' shows how dividing by zero leads to absurdities it's very first step is fatally flawed? It claims that you can factor $x 2 - x 2$ (which is itself zero) in two different ways the first of which being;

(x 2 - x 2)(x 2 + x 2)

Am I the only person who has noticed that if you factor this out it becomes

2 x 2 - 2 x 2

so really even the second line claims 2=1!

Perhaps this derivation (based on the same principle) would be better

a = b

a 2 = a b

a 2 + a 2 = a 2 + a b

2 a 2 - 2 a b = a 2 - a b

2(a 2 - a b) = 1(a 2 - a b)

2 = 1

Again because a=b you are dividing by zero in the last step. —The preceding unsigned comment was added by 83.216.146.141 (talk) 12:24, 12 December 2006 (UTC).

Well, yes and no. The general formula

(a + b)(a - b) = a 2 - b 2

is used to deduce that

(x + x)(x - x) = x 2 - x 2

. True, this is also equal to

2 x 2 - 2 x 2

, but this is obvious since all those expressions are 0. If you ask me, any variation of this argument is silly, it's just an obfuscated way to state that 1*0=2*0, so if you allow division by zero then (under certain assumptions) 1=2. -- Meni Rosenfeld (talk) 18:16, 12 December 2006 (UTC)

[edit] Fraction formatting (again)

There are three ways to format fractions that occur inline within text:

Fraction ^a/_b within text → <sup>a</sup>/<sub>b</sub>.
Fraction $\textstyle\frac{a}{b}$ within text → <math>\textstyle\frac{a}{b}</math>.
Fraction $\begin{matrix}\frac{a}{b}\end{matrix}$ within → <math>\begin{matrix}\frac{a}{b}\end{matrix}</math>.

The first form is the standard HTML way, but does not always mix well with <math> markups. And while it's been pointed out that the third form is the documented way to do inline fractions in T_EX, the second form appears to work just as well, and has the benefit of being easier to edit. So that's what I've used. Purists may want to revert my changes, but please choose a consistent format if you do. — Loadmaster 16:42, 12 December 2006 (UTC)

There are other formats, but they are not really fractional forms and look terrible:

Fraction a/b within text → a/b (no markup at all).
Fraction $a / b$ within text → <math>a/b</math>.

These kinds of formatting that should be fixed. — Loadmaster 16:47, 12 December 2006 (UTC)

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