Talk:Parity of zero/Archive 1

Archive 1

To do

To do list:

• Education. I have to fetch a couple of old journal articles before doing this justice.
• Greek mathematics
• Other uses in mathematics
• Roulette and car license plates
• Real illustrations
• Lead section
• Wikipedia:Peer review
• Wikipedia:Featured article candidates
• Pare down History
• Harvard citations
• Separate explanations from educational strategy
• Wikipedia:Peer review 2
• Wikipedia:Featured article candidates 2

Early comments

This is an interesting page, but I am concerned that it's really an essay or explanation of some sort - not an encyclopedia article. The only section I really see that might be encyclopedic is the history section. Besides the general explanatory purpose of this article/essay, I might the "See also" section isn't really a "See also" - and it contains the phrase "some mathematicians," which is bothersome. I'm not trying to shoot this article down, but I'm not sure how it doesn't just get truncated and merged into Even and odd numbers. Maybe 0 needs its own section there, but this article may not be necessary. It's a great explanation, but it's not necessarily the point of Wikipedia to explain things in such thorough detail (maybe another Wiki would be better). --Cheeser1 07:47, 14 September 2007 (UTC)

At first, I also had the concern that this wouldn't turn out to be an encyclopedic topic, but the research I've done convinced me that it is. The education section will probably be the most encyclopedic section of the article in the end, because there is published research on how students and teachers approach the specific issue of the evenness of zero. Of course, I haven't written that part yet. But I've got probably half a dozen references for it, and I'm going to track down an article from the journal The Arithmetic Teacher that's actually called Zero is an even number. If you liked the history section, the education section should be much more substantial and satisfying.

The ideal relationship between this content and the general article Even and odd numbers is also tricky. Again, you'll kind of have to take my word on this, but this article has enough worthy potential material to dwarf the current general article. I wouldn't like to see it truncated, because that loses information. One could try making 0 into a running example in the general article, but I think it would be awkward. Also, it isn't immediately obvious to me why the article would be merged into Even and odd numbers but not 0 (number).

Does that make sense? Melchoir 08:13, 14 September 2007 (UTC)

Oh, and I'm not sure what to do about the "See also" section either. If it's possible to firm up the language and integrate those items into the body, I will be glad to see the section go! Melchoir 08:18, 14 September 2007 (UTC)

I'd have it merged into the even/odd article as its own short section. It seems like re-explaining why zero is even repeatedly sounds more like teaching than encyclopedic exposition. This is clearly a work in progress, so by all means, continue to work on it. We can revisit this when the article takes more shape. I just wanted to raise these concerns now so I didn't spring them on you later. --Cheeser1 09:29, 14 September 2007 (UTC)

Fair enough. I think when the article does take shape, you'll find that the re-explaining is actually confined to the Proof section. Melchoir 16:37, 14 September 2007 (UTC)

I've started up Education, but it still needs fleshing out. Melchoir 00:36, 17 September 2007 (UTC)

This entire article does not meet the standard for an encyclopedia. It is closer to an essay like "The earth is round" or "The universe is big" —Preceding unsigned comment added by Erikev (talk • contribs) 00:22, 19 September 2007 (UTC)

Please, do explain which encyclopedic standards are better met by the articles Spherical Earth and Observable universe than by this article. Melchoir 00:29, 19 September 2007 (UTC)

• The contents of the article should be merged with Even and odd numbers. Dlabtot 04:07, 19 September 2007 (UTC)

Why do you think that? Melchoir 05:48, 19 September 2007 (UTC)

This is an essay, imho, not an encyclopedic article. BTW, in response to some of your comments above, please read: WP:OTHERSTUFFEXISTS. Dlabtot 22:08, 23 September 2007 (UTC)

And this is a talk page, not an AfD vote. Of course the mere existence of Spherical Earth and Observable universe does not justify the existence of this article. However, they are certainly relevant. My expectation is that reading those articles, and comparing them to this one, will help visitors to this talk page better understand and communicate what is meant by calling this article an essay. For example, Cheeser1 is concerned that the article is primarily an explanation, and explicitly voicing that concern gives me the opportunity to argue otherwise. But there is no intelligent response to the charge "This is an essay." No it isn't?

In turn, a substantive discussion helps editors of the article — so far this means me — know what to concentrate on improving. If the article needs information that does more than explain the bare fact, it will be added. Potatoswatter was disappointed that the article did not explore divisibility by other numbers, which led me to find material in that direction and add it. Melchoir 00:02, 24 September 2007 (UTC)

Dlabtot, you should wait until the article is developed. It is clearly under development, and may offer content distinct from that at even and odd numbers. If the article is developed further and you still believe it should be merged, you can propose it (providing a rationale). --Cheeser1 06:06, 19 September 2007 (UTC)

Well, I feel perfectly fine with offering my opinion about the article at this time, but your suggestion that I should have held my tongue is duly noted. Dlabtot 22:08, 23 September 2007 (UTC)

The picture

It's arty. But, um, isn't it demonstrating the equivalence of two objects (the pans on the scale) rather than the equivalence of two lots of zero? Vashti 03:57, 19 September 2007 (UTC)

It is a comparison of the weight of half of zero, and the other half of zero. Zero can be evenly (balanced scales) divided in half. Thus it is even. --Cheeser1 04:03, 19 September 2007 (UTC)

Yup. The idea that the zero objects are evenly distributed is the necessary role of the balance. To explain my own line of thought, how else can you visually assert the equal size of two collections, especially when the collections themselves are invisible? Two dotted-line bags with nothing inside them and an equals sign? The only way to avoid such symbols is to let a third, visible agent do the comparison, and it has to indicate where the measured things fit even if they have no extent. A vacant double-pan balance is really perfect, and fortunately, we happened to have just such an image lying around.

Technically speaking, just because the balance is visible doesn't necessarily mean that it's not made of a weightless material. But the weight of the pans isn't even relevant. You could also complain that the crossbar has weight, as does the column of air above each pan. I think it's a well-understood convention that a balance is intended and designed to compare what is put into the pans, not anything else. Melchoir 06:07, 19 September 2007 (UTC)

Section on The empty set

Along these lines, I think that this image demonstrates well the ability to "split" the empty set. I feel like the section "The empty set" doesn't really catch the general reader: I mean, I'm a combinatorist, so I'm not a huge fan of involutions on n-spheres or what have you. I feel like it's much easier to explain this "split the empty set" idea combinatorially. It also provides an immediate explanation for why zero is also divisible by, say, three, since {} = {} U {} U {}. (Too lazy for TeX, but you get the point.) Maybe involutions on closed manifolds also does that, but it's not immediately apparent to me (or I think to anyone who isn't already familiar with that stuff). Should that section be expanded to include a combinatorial method that might be a bit more accessible? --Cheeser1 19:59, 6 November 2007 (UTC)

Well, you could also put a free Z/n action for any n on any odd-dimensional sphere, so there's a picture where you've got the empty set as a subset of the circle, and everything is just getting rotated around the origin, but by 120* instead of 180*. I was thinking about this stuff too much earlier today, hence my post on the reference desk. I added the manifold stuff mostly as candy more than explanation; the line of thought was "what is the set-theoretic structure of the evenness of zero" -> "okay, who cares" -> "oh look, topologists care" -> "hey, by adding in the Euler characteristic I can bring this topic back around to the number zero being even in the last sentence" -> "that's too cool to not do".

Anyway, I'm not sure I know what you're suggesting. By a combinatorial method, do you mean an interpretation where every {} in the sentence {} = {} U {} is counting something? Or a more explicit comparison to 0 + 0 = 0? Or maybe more detail on what empty sets and disjoint unions actually mean? Melchoir 02:55, 7 November 2007 (UTC)

Yeah, {} = {} U {} is a 2-part partition of {} into parts of equal size. Thus |{}|=0 is even. The same works for any integer (since 0 is divisible by any integer). It's just a trivial case of showing |A|=2n by finding a partition A=BUC where |B|=|C|=n (in more complicated cases by sometimes finding a bijection f : B → C - such an f is trivial here, but so too is computing |{}|=0). --Cheeser1 06:21, 7 November 2007 (UTC)

Right... and isn't that what the section already says in the first two sentences? I'm not sure what should be added. Maybe you could just show me. Melchoir 06:34, 7 November 2007 (UTC)

Well yes, but it seems to gloss right over that as if it is nothing but a lead-in to some topology that is, honestly, beyond the comprehension of almost any reader. I would suggest:

One way of interpreting the evenness of zero is to say that a set with 0 elements can be partitioned into two subsets of equal size. The cardinality concept of size requires that there exists a bijection between these two subsets. In general, a set A has even cardinality if a partition of A into disjoint sets B and C exists where |B|=|C|, and thus |A|=2|B|=2|C|. The empty set can be partitioned trivially as Ø=Ø∪Ø, which immediately shows that 0=|Ø| is even. This can also show that 0 is divisible by any integer n, since Ø = Ø ∪ Ø ∪ ... ∪ Ø (n copies).

That gives a more complete explanation of the combinatorial understanding of this, which to me is the most intuitive way to understand it. And if that becomes its own paragraph, it looks alot less like some sort of trivial lead-in to what is basically incomprehensible to the general reader. A picture demonstrating the evenness of 0, 2, 4, and maybe 6 might help, and I'm sure I could do one up in MS paint (although I imagine somebody good with svg might be better suited to make a quality image or something). --Cheeser1 06:41, 7 November 2007 (UTC)

Okay, I see. I'll throw that in! I was a little worried that too much exposition on the one point would lead too far away from the case of zero, but your paragraph doesn't. (At some point we're really going to have to seriously expand Parity (mathematics), just as a proof that there's plenty of material to go around.)

As for a picture, I'm pretty good with SVGs, if not very efficient. So if you want to make any rough sketches, I can fix them up. Melchoir 07:29, 7 November 2007 (UTC)

I've thrown up a pair of images for the section. Neither of them directly makes the analogy to 2, 4, 6, ... but the picture of the pencil holder could theoretically be said to present 0, 2, and 10 at the same time.

Unfortunately, there isn't room for every conceivable illustration in what is a fairly short section. If we come up with a compelling graphic that fits 0 in with 2, 4, and 6, it might be appropriate to include it somewhere in the education section instead. That section is going to be expanded, and I'm not sure exactly with what. Some of the material from Lichtenberg is purposed towards visual demonstrations of simple patterns for students, so there's certainly an opportunity there. Melchoir 08:14, 13 November 2007 (UTC)

bloody obvious

This article says that if you just ask a group of first graders, they will unanimously tell you that zero is even. Many words are spent proving the intuitive obviousness of the fact. Then it speculates as to the first person to discover the evenness of zero. Maybe the first person to bother dealing with zero as a quantity in the first place?

Really, I expected to see something to do with group theory or something of mathematical depth here, but it's just answering the question of "How obviously is zero even? Does anyone have trouble understanding this?"

Zero is "more even" than any other number because it is uniquely a multiple of anything. So it's even if you're dividing by threes or fives as well as by twos. This article's encyclopedic content entirely belongs to even and odd and group theory. Potatoswatter 06:21, 19 September 2007 (UTC)

• This article does not discuss any group of first graders.
  • Of the two groups it does discuss, they were not unanimous; some of them needed convincing or argued the other way.
    • One first grader was queried and produced an intuitive answer. A group of second graders produced the correct proof, and ended in unanimity.
• I count 28 words in the two-sentence proof. Is that so taxing?
  • No, just obvious. You can't understand "zero" and "even" and still need that proof. "Many words are spent" talking about the children is what I was referring to.
• The history of zero is complex and subtle, and zero was used as a kind of quantity long before mathematicians started doing algebra with it. There have been periods in history when 2 was not considered even, and when 1 was supposed to be both even and odd. It is far from obvious who first considered zero to be even, and the source cited is the best source I could find.
  • So maybe more depth could be added to zero?
• You wanted groups? How hard did you look? The section "Algebraic structure" uses the word "group" 16 times. Melchoir 06:37, 19 September 2007 (UTC)
  • That section doesn't demonstrate your understanding. It looks like a jumble of trivia and jargon. Being a subheading of "algebraic structure" is misleading... zero is even because dividing the integers by two yields a zero coset closed over addition, not the other way around... if you want to somehow assign causality here in the first place. Potatoswatter 09:37, 19 September 2007 (UTC)

Breaking replies out of place...

The mathematics education literature is extensive, perhaps surprisingly so for those who haven't seen it, and who therefore still hold the naive expectation that "understanding" is a binary state and that everything true is "obvious". Well, as long as the researchers keep getting funding and publishing articles, there is little cost for us to spend a tiny fraction of their words to reflect and point to that research.

Your original claim of unanimity was, shall we say, an incautious extrapolation from the evidence. This article can certainly do a better job of explaining why the evenness of zero is a challenge and how widespread the misconceptions are, but we must not allow it to speculate about either of these questions; no information is perferred to false information. The journal article currently linked in the "Further reading" section answers some of these questions, and it will eventually be incorporated into this article. Meanwhile, try to keep an open mind.

Everyone is welcome to provide more mathematical or historical "depth" to this article, to Zero, and/or to any other related article. In my own research for this article, I've turned up a couple gems that properly belong in more general articles like Even number. Perhaps if this talk page demanded less energy, I could have added those gems already.

No, I don't want to assign causality to the group concept, and if you read closely, I didn't. That section's strategy is to tour through the various topics in mathematics where an writer might need to point out that zero is even and how that fact fits into the topic at hand. To get from one such topic to the next, I typically need a sentence or two to segue and introduce definitions, but it is important not to stray too far from the evenness of zero. (The moment I did, someone would leap in here and complain that that material belongs elsewhere.) It's a delicate balancing act that necessarily introduces a higher-than-usual density of jargon.

Your suggestion about zero being a more even number than others inspired me to track down and include the tidbit now under "Generalizations". I assure you that I am well aware that zero is divisible by everything, and I already intended to address this as part of a strategy for finding integer zeros of polynomials that may be motivated by the evenness of zero. This is covered by the "Other uses in mathematics" bullet point at the top of this talk page. You know, the part labeled "To do"? As in not bloody well done yet? Melchoir 17:09, 19 September 2007 (UTC)

Potatoswatter, I suggest you calm down and let the article be developed. These concerns have been raised, and can be revisited when the article has been filled out. There is room for a possible article here, per WP:N, since there has been study devoted to the evenness of zero (perhaps the mathematics here belong in group theory or something, but the evenness of zero spans several contexts and may merit its own article). Give it some time before raining down judgment on things. And try to a bit more calm. This should not instantly become a heated debate. --Cheeser1 17:15, 19 September 2007 (UTC)

Thank you! I could probably do better to stay calm myself, but it's so very frustrating when someone won't extend you the benefit of the doubt. I think it is probably too easy for a Wikipedian to get caught up in revert patrol and AfD, and to begin thinking of all strangers as self-promoting, thoughtless incompetents. Someone who starts a new article with the stated intention of Featuring it is a rarity, and it is understandable to wonder if I have the best intentions for the project as a whole, and indeed if I have any idea what I am doing. Well, what can I say? I try to indicate that I have good reasons to be optimistic about this article, but ultimately we'll just have to wait and see. Helpful suggestions will decrease the gestation period; whining will increase it. Melchoir 18:24, 19 September 2007 (UTC)

Open questions

• I still don't know who first called 0 even.
• Origins of the Chinese Zodiac claims that the fact that a Snake has an even number of legs (0) is significant to its place in the zodiac order. No reference found.
• House numbering claims that houses numbered 0 tend to be found on the even side of the street. No reference found.
• Morra (game) claims that in variants where 0 fingers are allowed, a total of 0 is usually counted as a draw. No reference found.
• It's conceivable that the Michigan study published detailed results somewhere, which would give an idea of the percentage of teachers who know that zero is even. It would be a valuable addition, but I doubt it.

Melchoir (talk) 22:14, 22 November 2007 (UTC)

Strange image

−3 + 3, 2 − 2, 4 × 0, and selected nearby evens

What is this image meant to show? It looks like a load of random dots and arrows to me... I think the dots are meant to represent numbers, so shouldn't they at least be labelled? --Tango (talk) 22:34, 22 March 2008 (UTC)

Yeah, I was experimenting there to see if I could get away without labeling things. The caption was supposed to be enough. Perhaps not? Melchoir (talk) 22:46, 22 March 2008 (UTC)

Well, I certainly have no idea what the image is meant to show, and I'm studying maths at uni. What chance does a layman have? --Tango (talk) 22:56, 22 March 2008 (UTC)

I think I just figured it out - it's an attempt to graphically show how 0×4 = 0, -3+3 = 0, and 2-2 = 0. I don't think it's particularly helpful (at best, it's unhelpful). --Cheeser1 (talk) 23:01, 22 March 2008 (UTC)

Yeah, I can tell from the caption that that's what it's trying to do, but I can't see it. --Tango (talk) 23:10, 22 March 2008 (UTC)

The bottom black line/dots represents the number line, which is then scaled by four - the zero point stays fixed (note the bigger arrow on that one). The vertical lines originate at -3 and at 2, and the arrows from each of them come back down to zero, representing the addition of 3 and subtraction of 2 (respectively). Red and blue represent parity. Understandably confusing - far too complex/unintuitive. --Cheeser1 (talk) 23:17, 22 March 2008 (UTC)

If I may, the original intention was not 0 × 4 but 4 × 0, if that makes sense, and the "selected nearby evens" are important. That is, the image attempts to demonstrate that -3 +/- (odd) = (even), 2 +/- (even) = (even), and 4 x (any) = (even), and to do so in such a way that the results equal to 0 are emphasized, the point being that these operations are actively identifying 0 as an even number and a peer to other even numbers.

I think perhaps a few more visual cues are needed. I'll try to put these in, and think of a better caption. I sympathize with the problem of making the image too complex, which is one of the reasons why I'm trying to label as few things as possible. Melchoir (talk) 23:32, 22 March 2008 (UTC)

To be honest, I'd scrap the image entirely. It's a nice idea, but I just don't think it works. I think explaining it in words is much easier - an image just doesn't add anything in this case. --Tango (talk) 23:39, 22 March 2008 (UTC)

I'm willing to scrap it, but not for those reasons alone. There's already accompanying text. And surely it adds something for someone? I can't be the only person who would rather interpret a rule like (even) +/- (even) = (even) in a more natural setting than an addition table. Melchoir (talk) 23:54, 22 March 2008 (UTC)

The problem is, it takes longer to interpret the image than it would to just read the text and work it out for yourself. A visual aid would be good, but only if it's immediately obvious what it's showing. Perhaps something involving sharing beans between two bowls? A number line isn't a very good way of depicting numbers unless you're trying to count and counting isn't really involved in evenness - division by 2 is. --Tango (talk) 00:27, 23 March 2008 (UTC)

Um, the "sharing" image is already at the top of the article. But the section we're talking about here concerns arithmetic operations. The image we're talking about concerns those same arithmetic operations. It's not supposed to be a catch-all depiction for whatever is "really involved in evenness". There are many interpretations, involving division and counting, most of which are not relevant to the section. Melchoir (talk) 00:42, 23 March 2008 (UTC)

I don't mean a general image, I mean an image based on sharing to show arithmetic operations. For example, you can show even+even=even by having two piles of beans, each shared into two equal sub-piles, and then combine the main piles by combining the sub-piles, and you end up with the sum shared into two. (Does that make sense without me actually drawing the pictures? Art never was my strong suit...) --Tango (talk) 00:55, 23 March 2008 (UTC)

I see what you mean, and it would be a good image for Parity (mathematics), but I don't know how you can work 0 into the pattern. Melchoir (talk) 01:10, 23 March 2008 (UTC)

(By the way, I suck at drawing myself, but with a tool like Illustrator it doesn't matter!) Melchoir (talk) 01:13, 23 March 2008 (UTC)

True, introducing 0 really requires subtraction and that's rather more difficult to show than addition... I think it's probably easiest just to stick with words. --Tango (talk) 01:17, 23 March 2008 (UTC)

Hmm. Well, I'll take it out of the article, at least for now. It's not like the image is going anywhere. Melchoir (talk) 06:49, 23 March 2008 (UTC)

GA Review

Not too much to say here. The article is excellently written, and meets the GA criteria quite handily. Good work. --erachima talk 14:32, 22 September 2008 (UTC)

Thanks! Melchoir (talk) 20:14, 22 September 2008 (UTC)

Should be Split

Without getting into whether the article is encyclopedic, I think it should be split. The education sections have good things to say, but should be treated separately from the mathematical information. Of course, there should be adequate treatment where crossover is necessary.--Unimath (talk) 07:15, 18 February 2009 (UTC)

Is there any particular reason why? Melchoir (talk) 08:51, 18 February 2009 (UTC)

0 is not a natural number

there is a line in the intro paragraph:

"...and it is the starting case from which all other even natural numbers are recursively generated."

However, 0 is not a natural number (natural numbers are defined as all integers >= 1) as this statement implies. I am not familiar enough with the method of generation that it is talking about, so I can't change this myself without risk of making it more incorrect. Thanks. Minirogue (talk) 20:48, 14 April 2009 (UTC)

Mathematicians sometimes include 0 as a natural number. See the natural number article. Jim (talk) 02:41, 15 April 2009 (UTC)

No they don't. There are 'natural numbers including 0', there are 'whole numbers', and there are 'natural numbers' which does never includes zero. Does Alice have an even number of apples? Well, if she has none, then no; Alice have no apples. 82.131.119.251 (talk) 19:42, 13 July 2010 (UTC)

I've removed this recently added material from the article: "On the other hand, if you work with natural numbers, then no natural number multiplied with 2 will provide the result of zero. In fact zero is not even a natural number and therefor not an even one either."

As Jim points out, 0 is sometimes counted as a natural number. Moreover, the argument that 0 is not a positive integer, therefore not an even number, is not found in the literature AFAIK. If you have a reference, please provide it. Melchoir (talk) 19:48, 13 July 2010 (UTC)

Yes, they do. Some mathematicians use "natural number" to mean "positive integer" and some use it to mean "non-negative integer". --Tango (talk) 20:44, 13 July 2010 (UTC)

Mathematical texts typically have to define what the term natural number means because there is disagreement and potential for confusion. That said, when natural number is used in a text it should be defined clearly. I think it is always better for an article to give a definition of 'natural number' when the term must be used.

I disagree with how natural number appears in the introduction of the article: "it is the starting case from which other even natural numbers are recursively generated." I wasn't certain what this meant until I read further into the article, section 'Even and Odd Alteration'. The method cited in that section does not generate the set of even numbers, but indicates the parity of the number. I.E. put 5 in the function and it will tell you it is not even. A recursive function that generates the set of even numbers would be something like $a_{0}=0$ and $a_{n+1}=a_{n}+2$ to generate a set of positive integers. That said, recursive functions can be defined any number of ways, for example, there might be a progression that goes {2, 0, 6, 4, 10, 8 ...} and this set would contain all the positive even numbers, just not sequentially. But this also means, such a recursive function does not have to start from 0.

Thelema418 (talk) 05:38, 20 June 2012 (UTC)

n even if n / m = 2?

If one defines a natural number n to be even if and only if there exists a natural number m so that n divided by m equals 2 then zero is excluded from the group of even numbers. How do we know which definition to use without being biased? —Preceding unsigned comment added by 82.95.156.242 (talk • contribs) 12:35, 3 April 2009

For Wikipedia's purposes, I don't think any published research actually uses that definition, so we don't have to worry about it.

For mathematical purposes, I would hope that this question is answered in the section Evenness of zero#Motivating modern definitions. If I may editorialize... There are many equivalent definitions of "even positive integer" which give inequivalent results when applied to the rest of the integers. Some include 0, and some don't; some include negative numers, and some don't. One feels that "most" of these definitions, including the "natural" ones, result in the currently accepted definition of "even integer" which includes 0, but it's hard to count definitions and it's hard to motivate a priori which definitions are most natural. You have to look at the consequences. In short, you have to decide whether it is more convenient to call 0 even, and if it is, you make sure that the definition includes it. This isn't "bias" so much as it's doing mathematics. Melchoir (talk) 17:42, 9 August 2009 (UTC)

The standard definition of "even number" is "integer that is a multiple of 2". Your definition is equivalent to that definition for all integers other than zero, for zero you end up with 0/0=2. 0/0 is usually left undefined because it is "indeterminate", which means (roughly) it can consistently be defined to equal anything at all, including 2, so 0/0=2 isn't really false, it's just not very meaningful. --Tango (talk) 17:56, 9 August 2009 (UTC)

Alt text viewer link

http://toolserver.org/%7Edispenser/cgi-bin/altviewer.py?page=Evenness_of_zero Melchoir (talk) 19:09, 9 August 2009 (UTC)

Did you have a point? --Tango (talk) 19:27, 9 August 2009 (UTC)

Nope, just putting it here for future reference. Melchoir (talk) 19:31, 9 August 2009 (UTC)

Displaced material

This is for holding material that doesn't really have a home in the article, but which might be reintroduced someday. Melchoir (talk) 00:32, 16 August 2009 (UTC)

In a proof by Solomon W. Golomb that a 10 × 10 torus cannot be covered with 1 × 4 tiles, the author reminds the reader that zero is an even number. The proof relies on the fact that a sum of many even numbers, which may include 0, is always even. Golomb's argument does not end with even numbers. It also generalizes to divisors other than 2, resulting in a weak relative of de Bruijn's theorem on box packing. The key is that 0 is not just a multiple of 2; it is a multiple of every other number.

diff

The connection of pairing with multiplication can be made more explicit by depicting elements in two rows. This picture has the advantage of simultaneously showcasing the properties of parity under addition: placing blocks next to each other, one sees that the sum of two even numbers is even, and the sum of an even and an odd is odd. Taking zero to be even preserves these two patterns.

diff

this isnt true

it says that all primary and elemntary teachers are prone to a misconception that zero is neither even nor odd...well im in year 7 rite now in australia and throughout my entire schooling ive been taught that zero is even so yea i think this needs abit of work...Adozenlies97 (talk) 23:17, 21 November 2009 (UTC)

It says they are prone to the misconception, not that they all make it. --Tango (talk) 23:25, 21 November 2009 (UTC)

Move?

The following discussion is an archived discussion of a requested move. Please do not modify it. Subsequent comments should be made in a new section on the talk page. No further edits should be made to this section.

The result of the move request was: page moved. Vegaswikian (talk) 02:39, 16 March 2010 (UTC)

---

Evenness of zero → Parity of zero

• The "Evenness of zero" title is actually my fault. When I started the article, I thought that title would look better in search engine results, and it would more clearly preview the content when people post links to the article on message boards. But as time passed, I've come to cringe at its awkwardness. "Parity of zero" is IMO a more natural noun phrase. Also note that Levenson et al's article is named "Neither even nor odd: Sixth grade students’ dilemmas regarding the parity of zero". So I'm a bit conflicted myself -- and hoping for more opinions. :-) Melchoir (talk) 20:09, 8 March 2010 (UTC)
• Move. Is "evenness" even a real word (no pun intended)? Adrianwn (talk) 06:38, 9 March 2010 (UTC)
• Kind of... it's certainly not common. Melchoir (talk) 19:50, 9 March 2010 (UTC)
• A Google search for "evenness" (excluding pages containing "wikipedia") got "about 500,000" results. Anthony Appleyard (talk) 16:09, 14 March 2010 (UTC)
• Yeah, I don't think most of those are about even and odd integers though. Melchoir (talk) 22:01, 14 March 2010 (UTC)

The above discussion is preserved as an archive of a requested move. Please do not modify it. Subsequent comments should be made in a new section on this talk page. No further edits should be made to this section.

Archive 1