# Talk:Parity of zero

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## Inscrutable graphic

Could someone please clarify what the graphic File:NuerkFigure4.svg is meant to convey? 173.75.159.152 (talk) 11:01, 13 April 2010 (UTC)

It's a scaling analysis of response times. See the footnote or the image description page for more details. Melchoir (talk) 19:59, 13 April 2010 (UTC)
Well, the image caption says that much. But what does it mean? The image description fails to indicate the significance of the lines and the apparently random distribution of numbers in the oval. 173.75.159.152 (talk) 23:02, 13 April 2010 (UTC)
Good point. I've now answered this question in additions to the caption footnote[1] and the Commons description[2]. Melchoir (talk) 03:26, 14 April 2010 (UTC)
Much better, thanks. 173.75.151.23 (talk) 00:37, 15 April 2010 (UTC)
No prob! Melchoir (talk) 01:27, 15 April 2010 (UTC)

## included middle

Recently the following interesting exchange took place at WPM, between editors Mdcq and PrimeHunter:

Two thirds of elementary teacher candidates thought zero was odd? Eek! Surely that can't be right? Dmcq (talk) 16:18, 19 April 2010 (UTC)
That is not quite what the article says. It says about two thirds answered "False" to "Zero is an even number." I guess most of them thought it was neither even nor odd. It's bad but I'm willing to believe it. I would have been less willing to believe that two thirds thought zero was odd. PrimeHunter (talk) 23:00, 19 April 2010 (UTC)

I thought it may be interesting to pursue this thread here. Are the statistics perhaps an indication that the law of excluded middle is not as obviously correct as one generally assumes it to be? Has this been discussed in the literature in connection with the parity of zero problem? Tkuvho (talk) 13:07, 20 April 2010 (UTC)

No, I don't recall ever seeing excluded middle mentioned in the literature. This is probably because it's an orthogonal issue. Classical logic is perfectly comfortable saying that 0 is neither positive nor negative; that 1 is neither prime nor composite; that 1/3 is neither even nor odd. Melchoir (talk) 19:09, 20 April 2010 (UTC)
I expect the teachers think 0 is not even because you cannot divide it into two SMALLER groups with the same number of elements. The way parity would be typically taught is in terms of a pile of candies that you can (or cannot) divvy up between two children, with each naturally getting less than the full pile. If that's the definition, then 0 is not even. What do you think? Tkuvho (talk) 10:14, 21 April 2010 (UTC)
There is one child in one of the articles, I think by Ball, who says something that leads the author to surmise that he is looking for a number N < 0 such that N + N = 0. There's no evidence that this belief is common, let alone universal. Melchoir (talk) 18:05, 21 April 2010 (UTC)

## Removed "The empty set"

I'm removing this as OR. It's all true, but it's still not defensible at FAC. I do intend for the basic idea to be represented in the "Basic explanations" section. Melchoir (talk) 19:16, 25 July 2010 (UTC)

### The empty set

A mirror partitions an empty collection of pencils; half (0) are real and half (0) are reflections.

One way to interpret 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).

The essentials of the above structure can be specified in more compact language: a finite set has even cardinality iff it supports an involution without fixed points or, equivalently, a free action by Z/2. The empty set, having zero elements, does support such an involution, namely the empty function.[1] Fixed-point-free involutions are mostly studied not on finite sets but on topological spaces, where the most important examples are the antipodal maps on n-dimensional spheres Sn.[2] The involution on the empty set is a base example: it is the antipodal map on S −1, the sphere of dimension negative one.[3] In general, for a space to support a free involution, it must have an even Euler characteristic.[4] The Euler characteristic of the empty set S −1 is the same as the Euler characteristic of any other odd-dimensional sphere, and it is an even number: zero.[5]

### References

1. ^ Dekker 1993, p. 298
2. ^ Conner & Floyd 1960, p. 416
3. ^ Livesay (1960, p. 603), writing about involutions on the 3-sphere, uses the −1-sphere to denote the empty set.
4. ^ For triangulable spaces, see Matoušek (2003, p. 136). In fact, if a closed, even dimensional manifold has an odd Euler characteristic, then one can put a lower bound on the dimension of a fixed set of an involution. This 1964 result is due to Conner and Floyd. See Stong (1974, p. 224).
5. ^ In fact, any oriented, odd-dimensional, closed manifold, not just spheres; see Guillemin & Pollack (1974, p. 116).
• Conner, P. E.; Floyd, E. E. (1960), "Fixed point free involutions and equivariant maps", Bulletin of the American Mathematical Society, 60 (6): 416–441, doi:10.1090/S0002-9904-1960-10492-2
• Dekker, J.C.E. (1993), A Bird's-Eye View of Twilight Combinatorics, Birkhäuser
• Guillemin, Victor; Pollack, Alan (1974), Differential Topology, Prentice-Hall, ISBN 0-13-212605-2
• Livesay, G. R. (November 1960), "Fixed point free involutions on the 3-sphere", Annals of Mathematics, 72 (3): 603–611, doi:10.2307/1970232
• Matoušek, Jiří (2003), Using the Borsuk-Ulam Theorem, Springer-Verlag, ISBN 3-540-00362-2
• Stong, R. E. (1974), "Semi-characteristics and free group actions", Compositio Mathematica, 29 (3): 223–248

## Leftover FAC line items

I am copying below the discussion left over from Wikipedia:Featured article candidates/Parity of zero/archive1, with the struck-out lines deleted. I will be making new replies inline, and asking other editors to do the same. I hope this doesn't confuse anyone! Melchoir (talk) 19:25, 25 July 2010 (UTC)

• Oppose—sorry. I know it is expected to include the title of the article in the first sentence but it is not working here at all. The introductory sentence from Parity is much better. There are other expressions that are difficult to understand. The second sentence, for example is tantamount to gobbledegook: "Such proofs follow immediately from the definition of the term "even number", whose applicability to zero is not arbitrary in the least; it can be further motivated by the familiar rules for sums and products of even numbers." What on earth does "further motivated" and "not arbitary (sic) in the least" mean? And, what are these "familiar rules"? What is meant by "On the human level"? Does this mean it is better understood by chimps? And who are we writing for? I get the impression that the article is written for teachers of mathematics, "Discussing the parity of zero in class can spark vigorous debates as students encounter basic principles of mathematical reasoning". The term "students" is used relentlessly throughout this article. I feel the article is not about the parity of zero at all—it is about how to teach it.Graham Colm (talk) 19:48, 4 April 2010 (UTC)
• I often omit the title of an article from the first sentence myself, when it feels unnatural or artificial. Back when this article was named "Evenness of zero", that phrase certainly wasn't in the first sentence. Since it's been moved to "Parity of zero", which works better, I thought I'd include it. Why do you think it's not working / worse than Parity (mathematics)? Melchoir (talk) 21:33, 4 April 2010 (UTC)
• Second sentence: Does this edit help? I've replaced those phrases. Melchoir (talk) 21:33, 4 April 2010 (UTC)
• "On the human level" is a transition phrase between paragraphs. The preceding paragraph is all about how zero is definitely even; the following paragraph is all about how it's not so simple in people's minds. I'll admit that these four words have a low information content, but if they make the prose easier to read, they're worth it. Melchoir (talk) 21:33, 4 April 2010 (UTC)
• The word "student" or "students" appears around 25 times in the first top-level section: "In education". It appears just once in the other four sections, which make up the bulk of the article: "Numerical cognition", "History", "Mathematical contexts", and "Everyday contexts". Isn't that what you'd expect? Melchoir (talk) 21:33, 4 April 2010 (UTC)
I have toned down my comments, they were a little over the top, sorry. Let's see what other reviewers have to say. Graham Colm (talk) 13:20, 5 April 2010 (UTC)
No prob! Please feel free to follow up on individual points as well. I'm willing to make big changes, but I'll want to have a conversation about them first. Melchoir (talk) 16:21, 5 April 2010 (UTC)
Update: The first two paragraphs have been rewritten to avoid these problems, and "Education" has been scoped down and moved lower in the section order. Melchoir (talk) 20:09, 25 July 2010 (UTC)
• Oppose. The structure of the article, and the length of the text under "in education", make this read like a classroom guide or education essay. The education section should not appear so early in the article - history at the very least should precede it; possibly so should the mathematical contexts. There appears to be too much detail on educational / developmental studies. The lead is not a summary of the whole article, but emphasises education and cognition at the expense of history and mathematical context (all should be there). BTW i didn't really understand the use of the expression "on the human level", so i would re-think that transition. hamiltonstone (talk) 23:18, 4 April 2010 (UTC)
• I have the "In education" section first because it has the "Explanations" subsection. This subsection is the most accessible part of the article, and it explains why zero is even. Wouldn't you agree that it's a high priority to get that in as early as possible? If "In education" were split up into two top-level sections, I could see "History" going in between them -- not "Mathematical contexts" though (it's too long). Melchoir (talk) 00:02, 5 April 2010 (UTC)
• "History" and "Mathematical contexts" are represented in the first paragraph of the lead. Is there some additional sub-topic from those sections that you would like to include? Melchoir (talk) 00:02, 5 April 2010 (UTC)
• For "on the human level": are you commenting on the transition itself, or the wording used to execute it? Melchoir (talk) 00:02, 5 April 2010 (UTC)
• Update: most of this has been taken care of. The section order is "Why zero is even", "Mathematical contexts", "History", "Education", and "Numerical cognition". I replaced "on the human level". The lead could probably still have more material from "Mathematical contexts" and "History". Melchoir (talk) 20:17, 25 July 2010 (UTC)

Review by Charles Edward

• General
• The opening sentence in the lead is somewhat confusing, and as a single sentence, it should be integrated into the following paragraph. Maybe something like "Zero is an even number. The evenness and oddness of a number is its parity." You need to define and link parity somehow in the lead. Many readers won't know what it is.
• That is much better! You could still omitt " In other words", the rest of that sentence stands well on its own. —Charles Edward (Talk | Contribs) 12:39, 8 April 2010 (UTC)
• I suppose so, but without some kind of connector, the relationship between the first two sentences would be unclear. We don't want to give the reader the impression that they say different things. Melchoir (talk) 01:39, 9 April 2010 (UTC)
• Maybe I misunderstand this subject, but it seems like the primary topic of the article should be the fact that zero is an even number. But throughout all the explanations, the article keeps coming back and explains how the concept effects education. It would be much better, in my opinion, if you made a couple straight up sections only talking about the parity of zero, and leave out all mention of students, teachers, etc. Focus on defining the topic. Then put all the education related stuff into separate sections.
You say that "the article keeps coming back" to education. I'm not sure what you mean by that, since most of the article has nothing to do with education and doesn't mention the issue. Could you please clarify? Melchoir (talk) 03:06, 8 April 2010 (UTC)
• The section concerning education should come later in the article, while the section laying out what the parity of zero is should come first. Right now the article jumps into the education aspect at first without fully defining the topic. Correct me if I am wrong, but the primary application of partiy of zero would be in mathematical contexts, so those sections should come first.
• The "In education" section is the section that explains the parity of zero. It begins with a simple proof that zero is even, and its first subsection is all about elementary explanations of that fact. I agree that this material should come first. In fact, "In education" is the first section precisely because I wanted to present that material as early as possible. Did you have another section in mind? Melchoir (talk) 03:06, 8 April 2010 (UTC)
• The above comment and this one go together. The mathematical sections also make a good explanation of what the parity of zero is, and how it is mathematically determined. It also gives a bit of history of it all. However in the education section, the article defines the parity of zero in a basic way in the first sentence, but throughout the section it ties it back to education, students and teachers. You could take that opening sentence out of the education section, move it to the head of the mathematical section, and use the mathematics sections as your opening section. That would give a very thorough overview of the parity of zero and its mathematical application before delving into its educational usefulness. —Charles Edward (Talk | Contribs) 12:39, 8 April 2010 (UTC)
• Hmm. I'll give it a try. My practical concerns are: (1) Presenting the explanations without commentary means that an Education section later on will have to refer to those explanations when it says "explanation X is suitable for audience Y". This seems like duplication of material, and the reader might be forced to scroll between the distant sections to understand what the latter is saying. (2) Much of the material currently in "Mathematical contexts" is very advanced. Most readers won't appreciate it, and in the worst case it will cause them to stop reading, so they'll miss the more accessible discussion of education and cognition.

I also have a theoretical concern, namely, all the material about explanations is taken from books and journal articles written by educators, for educators. The cited sources are discussing explanations with the assumption that the reader doesn't need them. If we just state the explanations, implying that the reader does need them... it's not exactly sticking to the sources, and it's a little condescending. Of course you could argue that my version is patronizing in its own way. We'll see how the execution works. Melchoir (talk) 01:59, 9 April 2010 (UTC)

• This has been done. Melchoir (talk) 19:43, 25 July 2010 (UTC)
• I agree with the above comments that the article reads much like and essay. It is very editorial-like in places, especially the education sections. Check out WP:TONE. Here are a few examples.
• "There are several ways to determine whether an integer is even or odd, all of which indicate that 0 is even:" How about "Each method used to determine whether an number is even or odd proves zero is even:"
• There's actually a subtle problem with both options: they suggest that it's necessary to consider all methods to determine the parity of zero, when any one suffices. I've tried another option. Melchoir (talk) 03:38, 8 April 2010 (UTC)
• "First of all, the concise definition of "even" is often not intuitive to children.", who says this is the first most important thing? Drop the "first of all"
• The "first of all ... moreover" construction helps to demarcate the two related problems, the first straightforward and the second more subtle. Anyway, I can see how it could be read as indicating importance. Changed. Melchoir (talk) 03:59, 8 April 2010 (UTC)
• Removed. Naively you'd think that the speed of computing a result and the speed of certifying a result would be well-correlated, so it's a surprise that one is faster and the other is slower when 0 gets involved, and it's not completely obvious which result is more important to mentally deciding if 0 is even. But it's perhaps not necessary to warn the reader that something tricky is going on. Melchoir (talk) 04:41, 8 April 2010 (UTC)
• "Some other mathematical contexts, where the presence of 0 in the even numbers can be felt, follow.", you could drop that whole sentence
• It establishes the relationship between that section and the following section. Melchoir (talk) 05:08, 8 April 2010 (UTC)
• Removed. I rearranged the content, and in its new place, the sentence was no longer necessary. Melchoir (talk) 19:43, 25 July 2010 (UTC)
• There is quite a few other examples, but hope this helps to identify them
• Um... not really? Perhaps if you provide feedback on the edits I've made above? Melchoir (talk) 05:18, 8 April 2010 (UTC)
• Citations needed
• paragraph beginning "Age-appropriate explanations that zero is even...."
• This paragraph has moved to "Implications for instruction", which still needs work. Melchoir (talk) 19:43, 25 July 2010 (UTC)
• paragraph beginning "Early in elementary school, numbers..."
• This paragraph has been split up. The first paragraph of "Basic explanations" does have a citation. Melchoir (talk) 19:43, 25 July 2010 (UTC)
• I'll see if I need to come back to these after trying different section orders. Melchoir (talk) 02:02, 9 April 2010 (UTC)
• "The claims about zero alone take many forms: Zero is not even or odd; Zero could be even; Zero is not odd; Zero has to be an even; Zero is not an even number; Zero is always going to be an even number; Zero is not always going to be an even number; Zero is even; Zero is special."
• "Data is also scarce for teachers' attitudes on students' attitudes."
• I have no citation for that assertion; it was just a way to introduce the topic of the paragraph. This edit should feel less like it needs a citation. Melchoir (talk) 02:21, 8 April 2010 (UTC)
• "Adults who do believe that zero is even can nevertheless feel unfamiliar or uncomfortable with the fact, enough to measurably slow them down in a reaction time experiment."
• "Repeated experiments have shown a delay at zero for subjects from a variety of national and linguistic backgrounds, representing both left to right and right to left writing systems; almost all right-handed; from 17–53 years of age; confronted with number names in numeral form, spelled out, and spelled in a mirror image."
• paragraph beginning "The precise definition of any mathematical term..."
• paragraph beginning "The above rules would therefore..."
• The first section in "Mathematical contexts" has no cites
• Yes, the text "Most of the intuitive reasons ... Some of these follow." is meant to briefly summarize the section. Do you think it's a policy issue? Melchoir (talk) 02:26, 9 April 2010 (UTC)
• The summary has since been replaced; that section shouldn't be a problem. Melchoir (talk) 19:28, 25 July 2010 (UTC)
• paragraph beginning "The observation that zero is not odd..."
• paragraph beginning "Zero is the starting point of the even natural numbers..."
• paragraph beginning "One way of interpreting the evenness..."
• Skimming through the existing citations doesn't reveal a source for this paragraph, and I can't find one on Google either. I'll have to read through all the sources to see what support I can find; I know Frobisher has some quotes that are related but not ideal. For reference, this paragraph was discussed at Talk:Parity of zero#Section on The empty set. Unfortunately the other editor has left Wikipedia... so no help on citations there. Melchoir (talk) 02:13, 9 April 2010 (UTC)
• This paragraph has been removed. Melchoir (talk) 19:26, 25 July 2010 (UTC)
• MOS
• The notes section should precede the reference section. See WP:CITEX
• The references each contain a short paragraph following them explaining them. Those descriptions should be removed and put into the body of the article if they matter, otherwise removed completely.
• The article mixes shortened refs with full refs. That is a little confusing, although not required by the MOS, I'd suggesting fully using the shortened citation method.
• I'll give it a try. The reason for the current references style is that there are really two distinct kinds of references being used. First, there's a handful of authors who provide "significant coverage" of the parity of zero in the sense of Wikipedia:Notability. These are the backbone of the article. Then, there's the rest: sources that mention the parity of zero only in passing. Calling out the significant sources in References, while leaving the rest to Notes, helps make this distinction for the reader who wants to do further research.

One drawback of the current scheme is that Ball has many entries, but only one is called out. Mostly for that reason, I'm willing to move to a more standard-looking format. But I would still want some text at the top of References that points to the most valuable sources. Melchoir (talk) 02:22, 9 April 2010 (UTC)

• This has all been done. Melchoir (talk) 19:29, 25 July 2010 (UTC)
• Images

### Semi-arbitrary subsection

Comment

• The article has a much better structure. The prose is at times idiosyncratic, though i have some difficulty pointing out exactly what the issues are. The most glaring examples was this: "One can then motivate the definition itself—and its applicability to zero." I have no idea what is meant by "motivate" in this sentence - it doesn't make any sense to me.
• Apart from prose, the one thing that jumped out at me was the initial proof offered: "A number is called even if it is an integer multiple of 2. Zero is an integer multiple of 2, namely 0 × 2, so zero is even." I am neither mathematician nor logician, but putting on my scientific method hat for a moment, I immeadiately thought of the counterfactual here; that is, Zero is an integer multiple of 1, namely 0 × 1, so zero is ... odd?? Since any number multiplied by zero is zero, this does not to me (as a lay person) appear to be a proof. All the subsequent explanations make sense to me - it is just this initial explanation that does not. So i found that awkward. But that may just be me!
• Referencing is OK, but there are still a few facts and paragraphs that have no citiation.

Good progress. hamiltonstone (talk) 00:14, 26 July 2010 (UTC)'

Thanks! For "motivate", that's good to know. I'm not consciously aware of which subculture uses the word in this way -- maybe it's just math textbooks? I'll try to think of an alternative. Referencing: right, that still needs to be finished.
0 × 1 is an idea that might be dealt with in the sources. I'll have to go back and check. But here's my attempt to explain it without the burden of WP:NOR:
Even and odd numbers behave differently, and in general you can't take a true statement about the even numbers, replace "even" with "odd", and expect to get another true statement. In the case of multiples, there are three related ideas that should be contrasted.
1. Any multiple of 2 is even. This is the stated definition of "even".
2. Any multiple of an even number is even. This is, in some sense, a natural generalization of (1). But it is a hypothesis which could, a priori, be either true or false, and must be either proven or disproven. It so happens that the statement is true. The proof is to combine (1) with an added ingredient: the observation that multiplication is associative, which is not necessarily obvious.
3. Any multiple of a number has the same parity as that number?? This is another hypothesis. Once we prove (2), we feel emboldened to guess the further generalization (3). But (3) has not been proven, and in fact it is false. A multiple of an odd number need not be odd.
Now here comes the really egregious OR. :-) It is possible that educators glide so subtly from (1) to (2) that they give the impression that mathematics operates by taking all generalizations to be true, and all syntactical symmetries to be faithful. Then applying this attitude, one slips all too easily into believing (3) without noticing that it's a new belief. Melchoir (talk) 09:18, 26 July 2010 (UTC)

## semi-arbitrary section

In connection with the suggestion to prove the even parity of zero by means of the fixed point-free involution of the empty set given by the empty function: has it occurred to anybody that the empty function can be shown to be an odd function? Namely, for every x in the domain, it satisfies f(-x)=-f(x). Truly odd :) Tkuvho (talk) 08:59, 26 July 2010 (UTC)

Heh. :) You know, in the course of researching this article, I've come to hate even and odd functions. They introduce so much noise into web search results! You get all these hits that look like a source is saying that 0 is both even and odd, or neither even nor odd. Then you follow the source, and no, it was talking about a function named theta, or capital-O, or the zero function on the real numbers. Bastards. Melchoir (talk) 09:25, 26 July 2010 (UTC)

## Mid-priority?

Well-written article, by the way. But I think that this article better fits the criteria for a low-priority article. I apologize if this is somehow addressed elsewhere, and I missed it. Leonxlin (talk) 23:05, 17 August 2011 (UTC)

It's sort of a weird case, isn't it? By Wikipedia:Version 1.0 Editorial Team/Release Version Criteria, Low means "Subject is mainly of specialist interest." And this topic is much more general-interest than specialist-interest; it's actually a topic that a significant fraction of readers have experience with. Mid means "Subject fills in more minor details", which seems to fit because the parity of zero is a minor detail when compared to zero, which is Top. On the other hand, Parity itself is only Mid, whereas I'd expect it to be High if not Top as well.
Anyway, it's not a big deal. :-) Feel free to change it if it still doesn't feel right. Melchoir (talk) 00:55, 18 August 2011 (UTC)

## Sum of an integer with itself

Just for the record, I've come to agree with removing this phrase. It's relevant and correct to describe 0 as "the sum of an integer with itself", but it's also unnecessary and misleading. Melchoir (talk) 05:06, 25 February 2012 (UTC)

## History Issues

For the history section, does anyone know beliefs about the parity of zero in the 1800s? I ask this because Froebel's Education of Man instructs schoolmasters to teach that 1 is not an even number nor an odd number. Two is discussed as even, then three as odd, etc. The current article makes it sound as though the issue of 1 not having parity was resolved after the Greeks adopted 1 as a number, but the concept obviously lingered in European minds through the 1800s.

This observation also makes me question the inability to date the parity of zero. I noticed Chase (1849) hidden in the references, and I think this information should be portrayed in the history content.

The statement about one historian's opinion about Greeks not knowing zero is misleading because even after ancient Greeks became aware of zero, they still had objections with the number 1 for various reasons.

Thelema418 (talk) 17:08, 18 June 2012 (UTC)

Chase (1849) was once in the main body text, but it was moved into the footnotes with this edit, in response to feedback at Wikipedia:Peer review/Evenness of zero/archive1. To avoid rehashing the reasoning, it would be great if you could take a look at that link, and then you can follow up here if you still think a change is warranted. Melchoir (talk) 07:17, 25 June 2012 (UTC)

After reading that Peer Review doc, I understand the hesitation of saying X person was the first, but I think it might be better to frame it in an anthropological perspective. ala: While Chase (1849) suggests that some European (?) mathematicians knew of the parity of zero in the 1800s, the 1850 American translation of Froebel's The Education of Man does not include zero in lessons on parity. Then include a footnote about the difficulty or mention the difficulties in that part of the document. Honestly, it was merely luck that I hit a link that took me to the footnote.

Historical educational resources and patents, rather than mathematical literature, may be more valuable here. I'll post a few I've come across later. Thelema418 (talk) 19:47, 27 June 2012 (UTC)

Here are a few things that I have since discovered that may be useful.

As far as textbooks and pedagogy

Hornbrook, A. M. (1898). A Primary Arithmetic: Number studies for the second, third, and fourth grades. American Book Co.: New York, NY. "These numbers that you have been giving, 2, 4, 6, 8, and so on, are called Even Numbers" (p. 36) One exercise problem would present a different answer if 0 was included as an even number: "Find the sum of all the even numbers in the first ten" (p. 99).

Brooks, E. (1877). The normal higher arithmetic. Christopher Sower Company: Philadelphia, PA. "An Even Number is one that is exactly divisible by 2; as, 2, 4, 6, etc." Note: Zero is never given as an example for even numbers.

Brooks, E. (1880). The philosophy of arithmetic as developed from the three fundamental processes of synthesis, analysis, and comparison. Normal Publishing Co.: Lancaster, PA. "[Even numbers arise] in counting by 2's, beginning with the duad. The even numbers are divided into the oddly even numbers, 2, 6, 10, 14, etc.; and the evenly even numbers, 4, 8, 12, 16, etc." (p. 375). Note: this claims that 2 is the first even number, not 0.

Mansford, C. (1802). Mental arithmetic for schools & training colleges. Hughes & Co.: London. "Find the sum of the even integers from 2 to ..." (p. 14). Note: doesn't consider 0 the first even integer; yet, this would not have changed the calculation strategy for the sum.

• Arnold, C.L. (1919, January). The Ohio Educational Monthly. 68. "Zero is an even number." p. 21. Note: this is quoted in the article about evenness of zero, but I think it is more significant that it shows some American teachers had recognized parity of zero, even though it seems that the textbooks (like those above), do not give zero as an example.
• Thomson, J. B. & Quimby, E. T. (1880) The collegiate algebra: Adapted to colleges and universities. Clark & Maynard: New York, NY. "It must not be forgotten that zero is an even number" (p. 253). NOTE: So far, this is the oldest textbook reference I could find in English texts that explicitly states the parity of zero.

As far as mathematical publications Hinkley, E. (1853). Tables of the prime numbers and prime factors of the composite numbers, from 1 to 100,000; with the methods of their construction and examples of their use. [Self-published]: Baltimore, MD. "From the definition of a composite number, it is evident that all the even numbers, 2, 4, 6, 8, 10, 12, &c., except the number 2 only, are composite" (p. 12). Note, if 0 is an even number, the author's statement is not true, since zero is not composite and not prime.

Patents U.S. Patent Office 3,356,990. Dec. 5, 1967. Well logging telemetry system including synchronizing a remote recorder and error detection of the transmitted data.

Note: "Zero" is an "even" number.


The use of quotation marks suggests that zero is only even in terms of a meaning to what zero is and the meaning of even, but also that other definitions of those terms might exist.

Other publications Westcott, W. W. (1850). Numbers: Their occult power and mystic virtue. Theosophical Publishing Society: London. "All even numbers also (except the duad--two-- which is simply two unities), may be divided into two equal parts, and also into two unequal parts, yet so that in neither division will either parity be mingled with imparity, nor imparity with parity: the binary number two cannot be divided into two unequal parts." (p. 6) Note: Zero cannot be divided into two unequal parts. Thelema418 (talk) 18:53, 1 July 2012 (UTC)

## Increasingly Even Numbers.

The article states "The powers of two—1, 2, 4, 8, ...—form a simple sequence of increasingly even numbers." I have never seen the phrase "increasingly even number" or "increasingly even integer". I did a Google search, and only this article and documents citing this article appeared.

I do think this should be reworded or removed from the document, esp. for the confusion of having 1 in the set.

Thelema418 (talk) 07:03, 20 June 2012 (UTC)

It's to do with the degree of evenness the number has, which might be better called the "2-order", symbolised as ${\displaystyle \mathrm {v} _{2}(n)}$ or ${\displaystyle \mathrm {ord} _{2}(n)}$, as in Singly and doubly even. ${\displaystyle \mathrm {v} _{2}(1)=0}$, ${\displaystyle \mathrm {v} _{2}(2)=1}$, ${\displaystyle \mathrm {v} _{2}(4)=2}$, and so on. As the 2-order increases, the number goes from being odd (not even at all) (${\displaystyle \mathrm {v} _{2}(n)=0}$) to being singly even (${\displaystyle \mathrm {v} _{2}(n)=1}$), doubly even (${\displaystyle \mathrm {v} _{2}(n)=2}$), and so forth, so it is natural to say that it is becoming increasingly even. Double sharp (talk) 07:47, 22 June 2012 (UTC)
Again, I am questioning the wording because the phrase "increasingly even number" only appears in this article and works that cite it. I also checked SpringerLink and other math research databases: I have not found any reference to this term. If it is a natural description and important content, it should be in other mathematical literature. I think the first three paragraphs are more important to the topic featured under the heading, and the paragraph containing this "increasingly even number" content is not necessary. This paragraph also contains the phrase "mathematically interesting" which is both an opinion and unsubstantiated. — Preceding unsigned comment added by Thelema418 (talkcontribs) 00:34, 25 June 2012 (UTC)
I changed the wording to avoid those two phrases; is that better?
For the record, I think that "increasingly even number" is a straightforward derivation from the ordinary meaning of "increasing" and the phrases "surpasses all numbers in 'evenness'" and "0 is the most 'even'" given in the citations, using only the rules of English grammar, so it doesn't count as new terminology. That said, we can just avoid the issue entirely. Melchoir (talk) 07:05, 25 June 2012 (UTC)
That wording definitely works for me. As a conclusion to the section, I think it sounds better. Thelema418 (talk) 07:18, 27 June 2012 (UTC)
"Degree of evenness"?  In what sense do two and four not possess the same exact degree of "evenness"?  Both, after all, are exactly evenly-dividable by two, no more nor less than the other, are they not?  allixpeeke (talk) 05:29, 11 March 2017 (UTC)
Two is divisible by two once, while four is divisible by two twice. So it makes sense to say that 4 is more even than 2, as you can halve it and still get an even number. Double sharp (talk) 00:02, 2 June 2017 (UTC)
"Powers of two"?  One-by-the-power-of-two isn't two, it's one, which isn't even even in the first place.  Sure, two-by-the-power-of=two is four, but four-by-the-power-of-two isn't eight; it's sixteen.  Indeed, there is no integer that yields eight when multiplied by itself.  allixpeeke (talk) 05:29, 11 March 2017 (UTC)
Powers of two are numbers of the form 2n for integer n, not those of the form n2. Double sharp (talk) 00:02, 2 June 2017 (UTC)

## Status of Article...

I've been asked to comment on the article before an effort is made to move it to Featured Article, however I personally don't think it qualifies as a good article. My reasons are the following.

The range of this article is *way way* too broad. There is no way that a discussion of beliefs of elementary school children belongs in the same article as one on p-adic numbers. At minimum, this article should be split into one on "Education of Children and general belief on the parity of Zero" (obviously a shorter title) and one on "effects/reasons in various parts of Mathematics caused by the fact that 0 is Even.", this could even be shortened in name to "parts of Mathematics that get screwed up if Zero is odd". In the "Degrees of Evenness section, *both* images need significantly more explanation or should be dropped. The one on the left, there is no indication how the image was created or what the meaning is of the other dots. The one on the right, there is no indication of why each line is the the length it is, the lengths are unclear enough that I can't tell if 48 has the same degree of 16, of 32 or of something else...

The entire "Everyday Contexts" section is a collection of statements that as much as they can be unified are simply the other half the education piece. Essentially "kids don't get it and here is the situation where adults don't get it either. Some of them have *no* apparent connection to the article such as "Linguist Joseph Grimes muses that asking "Is zero an even number?" to married couples is a good way to get them to disagree." and " One bookmaker offers a "cricket roulette" in which a batsman who is dismissed for a duck wins for the bank."

To finish up, this article spans from Hard Mathematics to Education and Sociology, At least two smaller and better articles should be made from this and if this is not done, then the GA rating should be reconsidered.Naraht (talk) 00:27, 30 June 2012 (UTC)

Addressing just the split issue, here are some factors to keep in mind:
• The "Why zero is even" section leads into both the "Mathematical contexts" section and the "Education" section. I'm not sure if it makes sense to separate them.
• I agree that this article wouldn't work in a journal or a textbook in either mathematics or education, but Wikipedia aims to be an encyclopedia. Our articles should be comprehensive.
• The article on 0.999... combines proofs, education, popular culture, and extensions, including the p-adic numbers, and it's already an FA.
Given that, I'm not totally sure what would motivate a split. Is there a particular practical benefit you're interested in? Melchoir (talk) 04:18, 30 June 2012 (UTC)
• The two questions are more properly "Why zero is even" and "Why is it more difficult to convince people that zero is even".
• There is such a thing as Too broad, and I think we've hit it.
• 0.999... both the education and popular culture sections are smaller and more organized. (I still find the p-adic graph/image there to be confusing).
• Benefit is coherence. This article lacks it.Naraht (talk) 12:41, 2 July 2012 (UTC)
Okay, I have some ideas about how to improve coherence. For historical context, at the time of the first (failed) FAC, the order of sections was different, and in particular, "Explanations" was a subsection of "In education". In response to feedback from hamiltonstone and Charles Edward, "Why zero is even" was broken out into its own section. Here's the last version before that change was made: Old revision of Parity_of_zero . In general, all three reviewers felt that the topic of education was too integrated throughout the article. Perhaps the pendulum has swung too far... I'll try reworking the "Implications for instruction" material, maybe combining it with some of the older language. The children's explanations are clearly related to the explanations in "Why zero is even", so that can be brought out as well.
I think it's also possible to reduce some of the methodological details in "Education" and "Everyday contexts", so that should cut down on their size a bit without losing the big ideas.
About the images, I just expanded the description of File:2adic12480.svg (as well as File:4adic 333.svg from 0.999...). I also have an idea for improving File:EvennessVert64.svg, which is hard to explain without just creating a new image... Melchoir (talk) 02:03, 8 July 2012 (UTC)
I will just express my unsolicited opinion here: I would have to respectfully disagree with Naraht here on the split issue. The parity of zero is already a tiny enough subject matter; an article just about why it doesn't make sense to have zero be even would be something someone might post on Facebook for laughs. Leonxlin (talk) 02:28, 8 July 2012 (UTC)

I'm also unsatisfied with the article as a reader, because it creates definitions of parity and shows that zero meets them, but does not explain why parity is not a trivial concept, just an alternation for convenience. It honestly comes across as preachy and self-important with the big focus on "kids are dumb and you are too if you do not acknowledge that zero is even." From a mathematics education standpoint it makes math seem like trivial number games. Applications in science and engineering that rely on the parity of zero would make the article seem less trivial and justify why this is important to learn rather than just a statement of mathematical dogma. 71.231.186.92 (talk) 01:01, 29 August 2013 (UTC)

71.231.186.92, how much of the article have you read? Melchoir (talk) 01:13, 29 August 2013 (UTC)
The section on 2-adic numbering includes a relevant example from computer science, though it's not explained very well. This is the only point I see in the article where there's any indication that calling it even is anything but a word game. The "everyday examples" section even outright supports the suggestion that it's a purely arbitrary decision, and the rest of the section relies on the tautology that "even by definition means even by definition" (i.e. defining the parity of 0 is convenient when talking about things explicitly described as even or odd). The roulette example specifically contradicts the article, because it is a case where 0 is not even. It's actually important that zero is not even or odd, because otherwise there would be a statistical advantage to betting on even instead of odd. The same applies to the example with navy vessels, where they specifically state that zero is neither even nor odd and there is no earth-shattering kaboom. 71.231.186.92 (talk) 01:29, 29 August 2013 (UTC)
Was that the only example from computer science that you saw? Melchoir (talk) 01:38, 29 August 2013 (UTC)
If there's another one, I missed it, and that's the point - the article is not well written because it fails to indicate the reason why we should care. I'm not saying it's not there, I'm saying that it's buried and sometimes unexplained. I wouldn't have a problem with this as a GA, but the prose is neither engaging nor brilliant. The lead, in particular in the second paragraph, is not the accessible language that would be expected - it immediately charges into blue links and assumes that the reader has read them all. For example, "Recursively defined" is WP:JARGON and has no place in a good lead section unless paired with an explanation. The article also editorializes without need, for example the line in the 2-adic section that starts "It is clear that..." These are just random examples - it's not that the article is wrong, it's just that it's not very well written. 71.231.186.92 (talk) 01:52, 29 August 2013 (UTC)
Okay, I've replaced the phrase "It is clear that...". I'm reluctant to try to explain recursive definition in the confines of the lead section, given that it would be a more significant change. We could say "recursively defined as alternating with odd numbers", but that's a little awkward. Feel free to suggest alternatives here! Melchoir (talk) 02:16, 29 August 2013 (UTC)

The entire second paragraph is very jargony, so it's hard to know where to start. Parity rules aren't defined here, and a lot of this comes across to a reader that's not following as potentially circular logic. My general approach when I've worked with WP:MED articles is to just leave the complex ideas out of the lead and stick to the simple things. For example, the following bit:

The parity rules of arithmetic, such as even − even = even, require 0 to be even.

What I think you're trying to say in the first line is:

Zero also behaves as an even number in all of the other ways that are expected. For example, even numbers subtracted from even numbers always produce even numbers, and zero has the same effect when used in subtraction.

The rest of this, I'll admit, I'm not even sure what the article's trying to say:

Zero is the additive identity element of the group of even integers, and it is the starting case from which other even natural numbers are recursively defined. Applications of this recursion from graph theory to computational geometry rely on zero being even.

I understand what's meant by an identity element after reading the other article, and that might be true and might be good for the full explanation on the even - even behavior, but I'm not sure why it's in the lead. I'm guessing that the "...it is the starting case..." bit refers to 0x2, 1x2, 2x2, etc... but that's just a wild assumption because I'm not following what's trying to be said. The next bit:

Not only is 0 divisible by 2, it is divisible by every integer.

It's stated as if it proves something, but I'm not clear what it proves. It states that 0's relationship to 2 is not special. This is relevant to point out, as it means that non-binary groupings have the same behavior (e.g. for our dates-and-fuel example, if there were five groups instead of two one group might have on x0/x1, someone else x2/x3, etc...). This is probably worth explaining in the article, but it's confusing for the lead, especially because of the way it's stated. 71.231.186.92 (talk) 04:00, 29 August 2013 (UTC)

Well, I don't know about most of this. It's infeasible to avoid the appearance of circular logic when dealing with matters of definition. After all, we define things in a certain way because we want certain statements to be true, and then we verify that the definition makes the statements true. That's a standard practice, and it would be misleading to pretend that we're doing otherwise. In fact, behold -- you're not an expert, and you picked up on this property of mathematical definitions by reading this article! Yes, you can redefine words and the world doesn't blow up. I'd say that's a success.
For the arithmetic part, that's not exactly the point made in the body of the article. It's not just that zero can be used on the left side of the equation; the stronger statement is that zero appears on the right side of the equation. This, and most of the other points, boil down to the question of the purpose of the lead section. It's just a preview of the body. In the body sections, the equations are expanded out, the recursive definition is given, etc. The one exception here is the phrase "divisible by every integer", which is supposed to lead into the following sentence. It's not supposed to prove anything. Melchoir (talk) 05:16, 29 August 2013 (UTC)
The purpose of the lead, however, is that it's supposed to stand alone as an accessible introduction to the topic. It's not a preview - it's a summary. Read WP:LEAD again. It also explicitly says "the lead should not "tease" the reader by hinting at content that follows" and that's exactly what the lead as written does. That I'm not an expert is *very* important - the lead in particular must be in language understandable to a non-expert. If the only people you can inform are people who already know what you're saying, it's not very educational. 71.231.186.92 (talk) 14:35, 29 August 2013 (UTC)
That's a good point about the purpose, but there are conflicting goals there. To avoid teasing the reader with hints, we have to mention groups and recursion. We can't fire off some generality like "Zero plays a central role in various mathematical structures formed by the even numbers." That really wouldn't mean anything by itself. On the other hand, it is difficult to describe those topics in language understandable to a non-expert within the space of a couple sentences. One possible compromise would be to include both the above sentence and then the specifics. But that's already the purpose of the "patterns" sentence, so it would rather redundant. Melchoir (talk) 15:04, 29 August 2013 (UTC)
The lead can afford to be a little longer if you need more space. Stating things precisely and technically is generally useless for the lead, because a non-technical audience will not understand them. It will undoubtedly lose some precision in the translation, but careful writing will ensure that what's said is not wrong or misleading. Writing a good lead isn't easy - it's a completely different skillset from expertise in the topic. Carl Sagan was famous for his ability to actually explain astronomy to the mere mortals - his colleagues may have been irritated because he was being imprecise but he was an excellent communicator, and he's the kind of guy you want writing an article lead. The lead doesn't have to summarize the entire article, just the "take home messages", and for this those are simple enough ("Even-ness" is called parity, zero is even because of the way parity is defined, zero is even in all of the ways that are used to test for parity, this is important because computer science takes advantage of parity because it's based on binary information, this is also important for some other reasons, this is commonly misunderstood). 71.231.186.92 (talk) 15:38, 29 August 2013 (UTC)
Well, the appearance of the parity of zero in subtle and powerful theories of higher mathematics is one of the take home messages, at least in my opinion. If there's a problem here, I'm sure the solution is to add explanations, rather than to subtract concepts. I also agree that the lead can be made a little longer... for certain values of "little"! Melchoir (talk) 16:59, 29 August 2013 (UTC)

I don't think little will cut it, unfortunately. Explanations will help, but if the explanations need explanations they don't fix the problem. Regardless, I have problems with this as a FA because of WP:LEAD. 166.147.88.45 (talk) 17:34, 29 August 2013 (UTC)

I think this is a wonderful article, excellent work. It is valuable to math education at the elementary and middle school level. The reaction time experiment and the everyday contexts also give insights into the human brain that are fascinating, especially from a math education point of view. The article is important because there are many people that teach elementary math. Teachers can use the balance scale image to convince doubting students of evenness of 0 and the beauty of math. The high-level math sections (abstract algebra, symmetric group, zeroes of polynomials, fast Fourier transform, axioms in higher algebra) do not add to the value of the article because they can only be understood at a math-level where the evenness of 0 is a non-issue. In summary, most of this article is wonderful, but the high-level math sections can be (partially) deleted. MvH (talk) 13:46, 12 February 2014 (UTC)MvH

## Arguments for oddness of zero?

Excellent page.

Would it be feasible to have another section covering the arguments stating that Zero is odd? Clearly there has been some historical debate, so for the sake of completeness, it would be illuminating to have that viewpoint covered in a section of its own. RedTomato (talk) 18:11, 7 February 2013 (UTC)

In principle, sure! For example, I think that it's pretty common for people to think that zero is odd because it's a multiple of 1. It also seems common for people to believe that since zero is the only number that's neither positive nor negative, it's reasonable to extrapolate that it's an exception to other dichotomies, and so neither odd nor even. But that's just my experience; I can't cite any research that would suggest that these beliefs are widespread or historically significant.
So the trick is finding sources. There aren't any secondary sources that claim that zero is odd, but there is some research on misconceptions held by students. You'll find a couple quotes in the "Students' knowledge" section already. If those sources go into more detail about the students' arguments, we could add such details to the article. I don't remember off the top of my head if there's anything to add; you might want to track down the sources yourself. Melchoir (talk) 04:24, 8 February 2013 (UTC)
Personally I never understood why anyone would use the odd-because-multiple-of-1 argument: after all, it's pretty obvious that 2 = 2 × 1, but 2 isn't odd. So what I would really want to see is the justifications for that argument offered by people who use it... Double sharp (talk) 06:43, 12 August 2013 (UTC)

Great article and nice to see it featured on the front page. As a historian of a branch of medieval mathematics I do think that the article should have a history section which does cloud the issue somewhat since some prominent late medieval mathematicians took no stance on the parity of zero even after it was established as a full number. Best. -- Michael Scott Cuthbert (talk) 04:54, 29 August 2013 (UTC)

Well, the article used to have a small history section, but it was very weak, so I axed it a few months ago. Here's the last version of the article that had it: Old revision of Parity_of_zero
The problem is simply a research issue: no reliable sources have been found that describe the history of the parity of zero! So I'm curious about these prominent late medieval mathematicians you speak of. Could you maybe expand on what they said and what sources we might use? Cheers, Melchoir (talk) 05:29, 29 August 2013 (UTC)

I think a lot of people would consider zero to be a pretty odd number regardless of what mathematicians think ... (Apologies to Steven Wright, who did a better take on this joke). Daniel Case (talk) 06:20, 29 August 2013 (UTC)

Excellent page? Odd page (pun intended). Even (pun intended) I wonder if a page as strange as this would be FA if the subject were anything other than mathematical. Reads like a load of obvious hyperbole. AfD AFAIAC;)1812ahill (talk) 00:40, 30 August 2013 (UTC)

I'm moving this content here from the Further Reading section of the article. The following sources have something to say about the parity of zero, but they couldn't be worked into the article for one reason or another. Melchoir (talk) 04:47, 31 July 2013 (UTC)

• Ball, Deborah Loewenberg (1989), Breaking with experience in learning to teach mathematics: the role of a preservice methods course, East Lansing, MI: National Center for Research on Teacher Education
• Ball, Deborah Loewenberg (August 1992), Implementing the NCTM Standards: Hopes and Hurdles. Issue Paper 92–2, National Center for Research on Teacher Learning, pp. 1–25, retrieved 2007-10-13
• Ball, Deborah Loewenberg (March 1993), "With an Eye on the Mathematical Horizon: Dilemmas of Teaching Elementary School Mathematics", The Elementary School Journal, 93 (4): 373, JSTOR 1002018, doi:10.1086/461730
• Ball, Deborah Loewenberg (1997), "What do students know? Facing challenges of distance, context, and desire in trying to hear children", in B. Biddle, T. Good, & I. Goodson, International handbook on teachers and teaching (PDF), 2, Dordrecht, Netherlands: Kluwer Press, pp. 679–718, retrieved 23 August 2009
• Ball, Deborah Loewenberg (2003), "Using Content Knowledge in Teaching: What Do Teachers Have to Do, and Therefore Have to Learn?", Archive of the Third Annual Conference on Sustainability of Systemic Reform, retrieved 2007-10-01
• Ball, Deborah Loewenberg; Bass, Hyman (2000), "Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics", in J. Boaler, Multiple perspectives on the teaching and learning of mathematics (PDF), Westport, CT: Ablex, pp. 83–104, retrieved 23 August 2009
• Ball, Deborah Loewenberg; Bass, Hyman (2000), "Making believe: The collective construction of public mathematical knowledge in the elementary classroom", in D. Phillips, Yearbook of the National Society for the Study of Education, Constructivism in Education (PDF), Chicago: University of Chicago Press, pp. 193–224, retrieved 23 August 2009
• Barnett, S. (1990), Matrices: methods and applications, Oxford Applied Mathematics and Computing Science Series, p. 115
• Cameron, Peter Jephson (1994), Combinatorics: topics, techniques, algorithms, p. 167
• Dickerson, David (2006), Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A.(Eds), ed., Aspects of preservice teachers' understandings of the purposes of mathematical proof, Mérida, Mexico: Universidad Pedagógica Nacional, pp. 710–716, ISBN 970-702-202-7
• Dulski, Thomas R. (1999), Trace elemental analysis of metals: methods and techniques, p. 549
• Evans, James Robert; Minieka, Edward (1992), Optimization algorithms for networks and graphs (2nd ed.), p. 283
• Fias, Wim (November 2001), "Two routes for the processing of verbal numbers: evidence from the SNARC effect", Psychological Research, 65 (4): 250–259, PMID 11789429, doi:10.1007/s004260100065
• Golomb, Solomon Wolf (1994), Polyominoes: Puzzles, Patterns, Problems, and Packings, Princeton: Princeton University Press, p. 119, ISBN 0-691-02444-8
• Kalsi, P. S. (2004), Spectroscopy of Organic Compounds (6th ed.), New Age International Publishers, p. 437
• Koshy, Thomas (2004), Discrete mathematics with applications, Elsevier, p. 765
• Leinhardt, Gaea; Steele, Michael D. (2005), "Seeing the Complexity of Standing to the Side: Instructional Dialogues", Cognition and Instruction, Lawrence Erlbaum, 23 (1): 87–163, doi:10.1207/s1532690xci2301_4, ERIC # EJ724916
• Lengyel, Tamas (1994), "Characterizing the 2-adic order of the logarithm" (PDF), The Fibonacci Quarterly, 32: 397–401, retrieved 23 August 2009
• Merttens, Ruth; Vass, Jeff (1990), Sharing maths cultures: IMPACT, Inventing Maths for Parents and Children and Teachers, The Falmer Press, p. 70
• Peterson, Penelope L. (1994), "Research Studies as Texts: Sites for Exploring the Beliefs and Learning of Researchers and Teachers", in Ruth Garner and Patricia A. Alexander, Beliefs about Text and Instruction with Text (1st ed.), pp. 93–120
• Schoenfeld, Alan H. (2002), "A highly interactive discourse structure", in J. Brophy, Social constructivist teaching: Its affordances and constraints (PDF), Advances in Research on Teaching, 9, Elsevier, p. 131, ISBN 0-7623-0873-7, doi:10.1016/S1479-3687(02)80007-4, retrieved 23 August 2009
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• United States Senate Committee on Energy and Natural Resources, Subcommittee on Energy Conservation and Supply (1980), Energy Management Partnership Act of 1979: hearing before the Subcommittee on Energy Conservation and Supply of the Committee on Energy and Natural Resources, United States Senate, Ninety-sixth Congress, first session on S. 1280, Washington, D.C.: U.S. G.P.O., LCCN 80602009

## I think that zero is odd.

It is odd. I mean, what other number is the sum of itself with itself? What other number is both positive and negative? What other integer number has the same factorial as its successor?

Since this makes zero odd, it makes it the only number that is both even and odd, which in turn makes it even odder.

David Olivier (talk) 10:18, 29 August 2013 (UTC)

How do any of those three things you've stated make it odd? To my knowledge, no other odd number has any of those characteristics. OrganicsLRO 10:55, 29 August 2013 (UTC)
Well, that's true! That only makes it odder still! David Olivier (talk) 11:02, 29 August 2013 (UTC)
That's wordplay, isn't it. --benlisquareTCE 11:20, 29 August 2013 (UTC)
That's indeed what it appears to be.  (For what it's worth, what I find most odd about the number zero is that any human ever in the history of humanity failed to innately realise it was even, always even, and nothing but even.  That, to me, is supremely odd.)  allixpeeke (talk) 05:36, 11 March 2017 (UTC)

## Is zero a number?

Just a musing, but is the article missing a section asking if zero is a number? Before anyone can answer if zero is an even number, they must first determine if it is a number. That's not so straight forward with zero.

Few everyday numbers have a "history" or "invention" as a concept. Nobody really invented "one" or "two" (as amounts, at least). But "zero" (as common as it is today) is a relatively new concept compared to other numbers that may be called "even".

Suppose Manchester United and Liverpool are playing football. The game ends 2 : 0 to Manchester. In football parlance this is referred to as two - nil and with good reason. How many goals did Liverpool score? None or zero? If they scored zero goals, then they scored a number of goals: exactly zero. But if they scored a number of goals, how can you then say they scored no goals? They either scored a number of goals or they didn't.

So, is zero a number? Depends on one's perspective on what is a number.

Is there anything in the body of knowledge about zero's even-ness that deals with the question of whether zero is a number? It might explain why so many people hesitate at the question. --RA () 12:01, 29 August 2013 (UTC)

The issue of zero being considered as a number or not has a long history which is detailed at 0 (number). Maybe a sentence or two could be inserted in here with a link. Staecker (talk) 13:59, 29 August 2013 (UTC)
Hmm, what would you recommend, and where in the article? Melchoir (talk) 14:13, 29 August 2013 (UTC)
I'm not aware of an article that says quite what you're expecting. Levenson, Tsamir & Tirosh (2007) note on page 84 that many students don't consider zero to be a number, and they spend the next three paragraphs explaining how destructive the related "zero is nothing" analogy is. The students they interview do call zero a number. One of them treats zero as a number which is still interchangeable with nothing. So Levenson et al. don't claim that the question of whether zero is a number, in itself, is a cause for confusion. The issue is more the incomplete manner in which number-hood has been extended to zero in some students' concept images.
There's a quotation in Frobisher (1999) where "Richard" says that zero is "not a number", which is currently included in the "Students' knowledge" section. Unfortunately, there isn't any further discussion of that belief in the article.
Generally, none of the secondary sources treat the question as necessary to decide whether zero is even. There is, however, plenty of background material at the article 0 (number), including the history of the concept.
(By the way, the number "one" absolutely had to be invented. This whole discussion could have been carried out with "one" in place of "zero" in ancient Greece. If a team scores but a single goal, have they really scored a number of goals? And so on.) Melchoir (talk) 14:07, 29 August 2013 (UTC)
"So Levenson et al. don't claim that the question of whether zero is a number, in itself, is a cause for confusion." That's exactly the sort of thing, I'm getting at. (Though these Levenson and pals sound a little dogmatic that zero is a number and the problem is that other people don't understand what a number is :-))
"There's a quotation in Frobisher (1999) where "Richard" says that zero is "not a number",..." That sounds good too. I didn't see it. Thanks, --RA () 18:44, 29 August 2013 (UTC)
The plural situation is a matter of language. The problem can be got around by changing the example from a football match to a fishing competition (Fred catches no fish, Eddie catches 20 fish, Brian catches 1 fish). But the point around 1 (or indeed any number) is taken and I'm reminded of the headline Harvard Beats Yale 29-29, --RA () 18:35, 29 August 2013 (UTC)
Ah, sports are a great source of non-standard number usage. In basketball, "1+1" is different from 2. In soccer, "45+2" is different from 47. Of course, such examples do not prove that arithmetic is flawed! They prove that natural language is richer than the language of arithmetic. Is there a useful distinction to be made between 3 fish and a brace of fish, 2 fish and a pair of fish, 1 fish and a fish, 0 fish and no fish? Maybe there is... if you're a poet.
Anyway, I meant what I said about the Greeks. I think it was the Pythagoreans who didn't consider the monad (1) to be a number, because it doesn't represent a multitude. Imagine getting in a time machine and talking to someone who insists that the smallest number is 2! Melchoir (talk) 20:30, 29 August 2013 (UTC)
On natural language (none, a single, a pair, etc.), you may find it interesting that in Gaelic people are counted using a different set of number words to other objects. And, I've just remembered, counting (non-people objects) starts at náid (nought) which is not the same as nialas (zero).
I can see where the Pythagoreans would have been coming from. Like 0, 1 has some very unique qualities. The three most useful numbers for a programmer are probably 0 and the two 1's (1 and -1) just because they can do very powerful things to other numbers (obliterate them, leave them unchanged, or reverse their sign). They are number like no other. But we're off topic. Regards, --RA () 23:28, 29 August 2013 (UTC)
"But if they scored a number of goals, how can you then say they scored no goals?" - because that number is zero. I don't question your general point, but what problem is this sentence trying to highlight? MartinPoulter (talk) 16:33, 29 August 2013 (UTC)
I mean that not all numbers are the same. Zero is not a counting number. You cannot score goals (0) and score no goals.
I wondered if in the research on why people have difficulty answering the question, "Is zero an even number?", is there any investigation asking if they hesitated on the question of whether zero is a number at all. I would.
To put it another way, maybe the given mathematical proofs for "evenness" are unnatural. For many, "evenness" means you can divide a given number of items equally between two people. Theoretically you can divide 0 by 2. In practice, you can't because you don't have anything to divide between you.
I am tempted to make a joke about "handwaving" on this one. (After all, what would it look like to share no pencils between two people?) Double sharp (talk) 14:02, 17 October 2015 (UTC)
So, is there anything on this kind of problem with the "numberness" of zero in the literature on the "eveness" of zero? --RA () 18:35, 29 August 2013 (UTC)
Depending on which book you use, zero is, or is not, a natural number. The ancient Greeks did not consider 0 or 1 to be numbers, because they reasoned that in order to count objects, you need at least two of them. Even in the modern world, many people are still uncomfortable with 0 and sometimes 1 as well (consider: can you take a sum of an empty set? What about a sum of a set with just 1 element?). We have to teach students that math logic is not the same as ordinary reasoning. Can you ride every elevator in a building that has no elevators? In ordinary language, that's a weird question. Math students should learn that in math logic, the answer is yes. MvH (talk) 14:06, 12 February 2014 (UTC)MvH

"But if they scored a number of goals, how can you then say they scored no goals?"

Easy: there is a difference between scoring goals and scoring a "number" of goals.  Even those who score no goals score a "number" of goals (viz., zero).  Indeed, all human on Earth are constantly scoring a "number" of goals, even if they are not scoring any goals.

"How many goals did Liverpool score? None or zero?"

The answer to that question is 'yes.'

To ask "how many" is to request a quantity.  Both "none" and "zero" express the same quantity.

(As an aside, one may notice an interesting similarity between the words none and one.  That's because it was derived from Old English meaning not one.)

"I mean that not all numbers are the same."

True.  Only same numbers are the same.  Different numbers of different.

"Zero is not a counting number."

I am suddenly reminded of the term null hypothesis, which is the 'default' or 'starting' hypothesis that the hypothesis is false.  One has to start at some default, after all.

If zero were not the null counting number, then what would be?  One?  Then, if I were to count the number of apples in my apartment, I would start with the default of one, and add whatever apples I find to that default.  In the process, I would have to conclude that I have one apple in my apartment, since I have no apples in my apartment and therefore had no additional apples to add to the null counting number.  But, obviously, that'd be absurd.  If I have no apples in the apartment, then I do not have one apple in the apartment; I have zero.  Zero, therefore, and not one, is the null counting number (and quite literally!).

"You cannot score goals (0) and score no goals."

Not only can I, I cannot do one without doing the other, and I also cannot not do one without not doing the other.

(And as it so happens, I'm not not doing both right now.)

"Theoretically you can divide 0 by 2. In practice, you can't because you don't have anything to divide between you."

You're confusing the number of the thing being counted with the thing being counted itself.  One can hold a single apple, but one cannot hold the number one itself, nor the number two itself, nor any number.  If you have zero apples, then sure, you cannot cut them in half because there is nothing there for you to cut, but you also have nothing there to cut when we're dealing with negative-two apples (i.e., a debt of two apples).  But just because we lack anything to cut does not mean we cannot divide negative-two in half qua number, so the same is true of zero.  Even if we have no apples to divvy up amongst the two of us, we can still agree that we possess exactly the same amount of apples, that the apples are divided evenly amongst us.  I just said "same amount," but I could have just as easily said "same number" and clarified that that number was zero.

Remember, zero is a quantity of the thing being lacked, not the thing being lacked itself.  That's why, even though nothing is not an element within the set of something, zero somethings is an element within the set of amounts of something.

"After all, what would it look like to share no pencils between two people?"

Whenever two people are totally lacking in pencils, do these two people have a different number of pencils from one another?  If not, then they have the same number of pencils as one another, and since they have zero pencils, that makes zero a number.

"Can you ride every elevator in a building that has no elevators?"

Incontrovertibly. :)

allixpeeke (talk) 06:39, 11 March 2017 (UTC)

## Divisibility by 2 (of zero)

In the introduction of this article it is stated "zero shares all the properties that characterize even numbers: 0 is divisible by 2..." and " Not only is 0 divisible by 2, it is divisible by every integer. "

This seemed highly dubious to me, and after some research, the Wikipedia article "Division by zero" states that " In ordinary (real number) arithmetic, there is no number which, multiplied by 0, gives a (a≠0), and so division by zero is undefined". " Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value to a /0 is contained in George Berkeley's criticism of infinitesimal calculus in The Analyst ("ghosts of departed quantities").[citation needed]

There is a clear fallacy in one of these statements / articles and must be corrected at once. --DragonFly31 (talk) 16:14, 29 August 2013 (UTC)

Whether it is possible to divide by zero is irrelevant to the question of whether zero can be divided by other numbers. It seems like you are drawing a link between two unconnected things. MartinPoulter (talk) 16:28, 29 August 2013 (UTC) There is no "clear fallacy that must be corrected at once" until you show how these two things are contradictory: it seems that they are clearly not. MartinPoulter (talk) 16:29, 29 August 2013 (UTC) I see what DragonFly is getting at, now. MartinPoulter (talk) 21:46, 29 August 2013 (UTC)
I changed it to "divisible by every positive integer." Pedantry satisfied. Ntsimp (talk) 16:38, 29 August 2013 (UTC)
Is there such a thing as pedantry in mathematics? I guess another debate for another dayDragonFly31 (talk) 13:48, 3 September 2013 (UTC)

DragonFly31, division by zero is undefined, but division of zero is defined (except when it is division both of and by zero).  Thus, your conclusion that "There is a clear fallacy in one of these statements / articles" is itself incorrect.

MartinPoulter, I do not.  It still appears to me that everything you struck out is accurate.

Ntsimp, by which negative integer is zero not divisible?

allixpeeke (talk) 06:53, 11 March 2017 (UTC)

## Sum of an integer with itself

Regarding these two edits [3] [4], would it work to replace "sum of an integer (0) with itself" with "sum of an integer with itself (0 + 0)" or similar? Melchoir (talk) 16:48, 29 August 2013 (UTC)

## Is zero positive?

This Wikipedia article says it is positive, but the "Zero" Wikipedia page states that zero is neither positive nor negative.  ?? — Preceding unsigned comment added by 64.211.112.98 (talk) 17:45, 29 August 2013 (UTC)

Er, which part of this article are you referring to? A quotation would help. Melchoir (talk) 17:47, 29 August 2013 (UTC)
When grafted, zero is the point at which negativity meets positivity.  allixpeeke (talk) 06:55, 11 March 2017 (UTC)

## Calendar date

Ib the section 'Everyday Contexts", I see this: "Around the year 2000, media outlets noted a pair of unusual milestones: "1999/11/19" was the last calendar date composed of all odd digits that would occur for a very long time..." That doesn't make sense. This must be a typo for "1999/11/29" and I am correcting it.. 70.114.146.227 (talk) 20:19, 29 August 2013 (UTC) Eric

The 2 in "29" is an even digit, not an odd digit. Melchoir (talk) 20:32, 29 August 2013 (UTC)
Twenty-nine is a number, not a digit.  allixpeeke (talk) 06:57, 11 March 2017 (UTC)

## Why Zero is Even

"zero is an integer multiple of 2, namely 0 × 2, so zero is even." A classic example of an unstated assumption. This is only true if zero is assumed to be an integer which, luckily, it is.67.170.238.175 (talk) 21:52, 29 August 2013 (UTC)

## References

It's impressive that a statement of the trivial and obvious can have 75 references. Next time someone complains about the difficulty of finding references for their article, I shall refer them here. Maproom (talk) 21:58, 29 August 2013 (UTC)

The Monty Hall problem is trivial too, and yet, lots of people get it wrong. The same is true for the evenness of 0. That means that these are useful problems for math education. MvH (talk) 14:10, 12 February 2014 (UTC) MvH

## Is specifically better than namely?

I'm not sure that I understand this edit. Replacing the word "namely" with "specifically" makes the sentence harder to read. What's the point? Melchoir (talk) 21:59, 29 August 2013 (UTC)

"Namely" is wrong - it is not followed by its name. Maproom (talk) 22:12, 29 August 2013 (UTC)
Hmm, that's not quite a slam-dunk case. A formal multiple of 2 is precisely an arithmetic expression of the form N × 2, which is a name that happens to signify the number formed by multiplying N by 2. It's not the canonical name of that number, but it's a valid one. Either way, you were right to call it pedantic. :-P Is the distinction all that important in this context? Melchoir (talk) 22:30, 29 August 2013 (UTC)
...Actually, the more I think about this, the less sure I am. Does anyone else have an opinion? Melchoir (talk) 22:41, 29 August 2013 (UTC)
Nothing much is at stake, but wouldn't "since" work perfectly well, and enhance readability because it has one syllable rather than five? MartinPoulter (talk) 21:20, 4 September 2013 (UTC)
How would that work? Replacing just the one word would read, "it is an integer multiple of 2, since 0 × 2", which doesn't make sense. I suppose you could say "since it is", but that's awkward in the context. Melchoir (talk) 23:34, 4 September 2013 (UTC)
Yes, I was mistaken. I was thinking "since 0 x 2 = 0". "specifically 0 x 2" is the most correct and elegant. MartinPoulter (talk) 08:00, 5 September 2013 (UTC)

Maproom is wrong. You do not have to follow "namely" with a name. This is basic English. As a foreigner this is baby step English. "Namely" as an adverb is defined as "That is to say; to be specific." It is elegant and concise and acts as a "gentle" sequitur. The current usage of the word "specifically" is ugly in the text as written. If you cannot switch back to the correct usage of the word "namely" because you feel intimidated, then simply use "i.e., 0 x 2". I hope this helps. Cheers 213.66.81.80 (talk) 08:48, 13 September 2013 (UTC)

## Could we include some different opinion regarding zero parity?

1. A pair consists of two elements.
2. Dividing a number by two, we find the number of pairs.
3. Zero elements, resulting zero pairs.

The number Zero is neither odd nor even. Zero parity --Gvitalie (talk) 07:40, 17 October 2015 (UTC)

This is not a matter of opinion. You're just wrong. Double sharp (talk) 14:28, 11 January 2017 (UTC)

You lament the creation of zero pairs while ignoring the simultaneous creation of a pair of zeros, treating 2 × 0 as though it did not = 0 × 2.

Zero and zero are both elements within the set of integers.  The fact that there is no quantitative difference between zero and the sum of a pair of zeros does not change the fact that a pair of zeros (both of which are elements by virtue of the fact that both of elements within the set of integers) results when we divide the number zero by two.

The root of your error, I suspect, is that you are confusing the number of things being divided with the thing being divided itself.  Yeah, if I have zero apples, then I have no apples to physically divide; yet, I can still divide the amount or number of apples (i.e., "zero") even though I cannot divide the apples themselves (as they, unlike the number, do not exist).

allixpeeke (talk) 07:22, 11 March 2017 (UTC)

## Roulette tables

A recent attempt to say that zero is neither odd nor even comes from a misinterpretation from the gambling world. The 0 that appears on a roulette table is just a symbol, it is not a number. Casino owners need to be very clear on this point since their profits depend upon it. If someone bets on evens and a 0 comes up, the casino owners will not give a payoff. This is justified by the statement that 0 is a transnumeral, that is, it is beyond numbers, i.e., it is not a number. Since it is not a number, it can not be even (or odd for that matter). Similarly we have 00 as a transnumeral on Vegas tables. In the same way, 0 is not red or black (usually it is placed on a green background), but this doesn't really cause any problems because this is a visual property and few would argue that there should be a payoff for 0. --Bill Cherowitzo (talk) 22:24, 31 May 2017 (UTC)