Talk:Parity (mathematics)

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Maybe merge & redirect to Parity? -- Tarquin 23:30 Jan 16, 2003 (UTC)

It was rather ridiculous that [[Even]], [[Even integer]], and [[Even number]] all redirected to [[Odd number]]. So, I've expanded the content and moved it to the generic "Even and odd numbers". -- Minesweeper 01:10 3 Jun 2003 (UTC)

I second your decision. Sometimes it is meaingless to have two seprate articles for some pair of concepts.

Integers only[edit]

Anyway, I know this is too detailed but the opening setence says:

any integer can be either even or odd. A number is called an even number if it can be evenly divided by two.

The readers might wonder so what's exactly difference between integer and number? Any integer can be even or odd and a number dividable by 2 is an even number. What about integers? -- Taku 03:37 3 Jun 2003 (UTC)

I wonder can a real number be even or odd? -- Taku 04:10 3 Jun 2003 (UTC)

Well, if the real number in question happens to be an integer, it can. ;) -- Oliver P. 04:13 3 Jun 2003 (UTC)

Whata about 1.2? It's not an integer but a real number and even. -- Taku 12:44 3 Jun 2003 (UTC)

Eh? How 1.2 an even number? No fraction can be an even number by definition.

Right. non-integers are neither even nor odd by definition. -- Taku

But it can be evenly divided by 2: 1.2/2 = 0.6. ^_^

How long has 0.6 been an integer?

-- Toby Bartels 05:37 12 Jun 2003 (UTC)

You're a wild and crazy guy. -- Cimon Avaro on a pogo stick 05:43 12 Jun 2003 (UTC)

By the definition that mathematicians come up with, only is an integer an even number. But it may be wrong in practice. Because to me, 1.2 or 0.6 look like an even number while 1.3 may be odd. -- Taku

I guess that the point is that "evenly divided" is ambiguous until you specify what the quotient is allowed to be. But we say "multiple", which works fine. As for Taku's perceptions, 1.2 is an even multiple of 0.1, while 1.3 is an odd multiple ... but 1.3 = 1.30 is an even multiple of 0.01! -- Toby Bartels 05:53 12 Jun 2003 (UTC)

And if you write 1.2 in binary instead, you get 1.0011001100110011001... which suddenly makes it look not at all even any more... :) -- Oliver P. 06:08 12 Jun 2003 (UTC)

Rather depends where you stop writing it out (it's an infinitely recurring binary fraction; the number you wrote out is not equal to 1.2). mfc 17:03, 2 Dec 2004 (UTC)

I still think we should move this to parity and open with something like: "the parity of an integer is whether it is even or odd ... " -- Tarquin 09:13 12 Jun 2003 (UTC)

I don't want to think about combinging those, but I ceratinly won't stop you if you want to do it. -- Toby Bartels 09:37 12 Jun 2003 (UTC)

Is the odd / even parity of numbers natural or man made ? Eg. one / many is a "natural" distinction and I would argue, so is prime / non-prime. But parity depends on choosing 2 at the start - why not 3 or 5 ? (talk) 18:51, 17 June 2012 (UTC)

Music theory[edit]

Seriously though, could someone who knows wind instruments please clarify what fundamental means in that context. Thanks. (and do it by editing the article) -- Cimon Avaro on a pogo stick 05:49 12 Jun 2003 (UTC)

Evenness (or non-evenness?) of 0[edit]

"The number zero is even, because it is equal to two multiplied by zero."

But wouldn't zero then also be considered an odd number since zero is also equal to one multiplied by zero?

n * 0 = 0. Thus, regardless of what number n is, the result is always 0? —The preceding unsigned comment was added by Kindragon (talkcontribs) .

One multiplied by zero does equal zero, but one multiplied by 987348792834 equals 987348792834 and 987348792834 is even. Multiplying something by one does not make it odd. Catch my drift? Rsercher (talk) 19:32, 20 June 2011 (UTC)

As the article says, an odd number is 2*n+1, where n is an integer. You can't write 0 that way, so it's not odd. "Odd" does NOT mean "2 times another odd number" —dcclark (talk) 06:57, 27 December 2005 (UTC)
Zero is an even number, as Dr. Math (a.k.a. Dr. Rick) states. – TTD Mocha! Bark! (pawprints) 06:50, 8 August 2006 (UTC)
Zero is an even number because two divides 0.--Jersey Devil 00:28, 6 May 2007 (UTC)
See Evenness of zero. Melchoir 06:56, 14 September 2007 (UTC)


It would be nice to see the word "parity" used in a few sentences to get a sense of its usage. I've seen that numbers are said to have "even parity", the quality of being even, or "odd parity", the quality of being odd. What about numbers that aren't even or odd -- do they have "no parity", "neither parity", or "imparity"? Would one say that integers "have parity" but non-integers "do not have parity"? Would one say that "the parity of 53 is odd" or is that an invalid grammatical construction? Zeroparallax 20:44, 25 March 2007 (UTC)

That is better suited for a dictionary, not an encyclopedia. --Cheeser1 08:58, 23 August 2007 (UTC)

Number Theory[edit]

In number theory, Nielsen proved the following. An odd positive number N such that : ( n,d ∈ N * ) and ω(N)= k is less than . —Preceding unsigned comment added by (talk) 19:41, 16 September 2007 (UTC)

In case anyone here is not familiar with this case: Wikipedia:Requests for comment/WAREL; Category:Suspected Wikipedia sockpuppets of WAREL. —David Eppstein 19:48, 16 September 2007 (UTC)

Arithmetic rules of parity[edit]

I removed from the subsection on multiplicative parity the following remark:

These rules only hold because 2 is a prime number; the analogous rules for divisibility by a composite number would be more complex.

I removed it because this is not an adequate explanation of parity rules. The rules for primes larger than 2 are also more complex. 2 and parity occupy a special place, not generalized directly to other numbers. Zaslav (talk) 21:16, 24 February 2008 (UTC)

Prime Connection[edit]

I think it would be very germane to mention the connection to prime numbers via Goldbach's. thus, odd+odd = even [via GC =prime + prime] and
odd+odd = even [via GC =prime + prime] I understand it is not "proven", but is widely believed to hold, and really drives home the connections between all of the positive integers.--Billymac00 (talk) 21:31, 6 March 2008 (UTC)

Parity in culture?[edit]

Is it worth to mention that in many cultures even numbers were associtated with bad luck, and odd numbers were supposed to bring luck? — Preceding unsigned comment added by Ilya-42 (talkcontribs) 08:37, 28 June 2011 (UTC)


It is weird that there is no article named Even number and Odd number on English Wikipedia (they are only redirect pages), even MathWorld has these. [1] [2]

I think the current title "Parity (mathematics)", which refers to the evenness or oddness, is too technical and unfamiliar for non-mathematicians. So I suggest moving this article to "Even and odd numbers", which is also avoids a parenthetical disambiguation. (I think it is unnecessary to divide it into Even number and Odd number.) --Minawa Yukino (talk) 09:06, 22 September 2012 (UTC)

I would actually support this suggested rename as a better-known description of the concept described, and as being less ambigiuous. The concept of parity (as described in the last section of the article) is a common use of the term, but is a distinct meaning: the hamming weight of a bit string modulo-2; it is only indirectly related to the topic of this article. As per WP:NAD, we should not be covering distinct uses of a term in one article. In a sense, the topic is really oddness and evenness of numbers, but this makes for a clumsy title. —Quondum 23:49, 1 May 2015 (UTC)

Parity of any rational number can be determined[edit] What I gathered from Patrick Da Silva's answer is such: Any rational number a can be represented as a fraction n/d. If n is even, a is even. If n is odd, a is odd.

I found this after wondering about the function y=x^(1/a), which is an easy way to determine whether a number is odd or even. If a is an even number, such as 2, the graph includes only real values of y when x>=0. If a is an odd number, such as 3, the graph of the function includes real values of y when x<0. Here we find that any rational number follows the same rules, not just integers. When a is a number such as 2/5, or 0.4, the graph will include only real values of y when x>=0. When a is a number such as 3/5, or 0.6, the graph will include real values of y when x<0

To summarize: A fraction is even if the numerator is, therefore all rational numbers have parity.

For another explanation, with visual effects, visit Zekava (talk) 23:08, 3 August 2015 (UTC)

While you can define the parity of fractions in this way (or in any other way that pleases you), it is problematic. In particular, consider the case 1/8 + 1/8 = 1/4. Shouldn't we have a rule that adding an odd number to itself produces an even one? To my mind, the correct generalization is to go from binary values (is it odd or even) to integer values (what is the smallest integer k such that 2k times the given number is an integer), or in fancier terms the 2-adic valuation. —David Eppstein (talk) 23:43, 3 August 2015 (UTC)
PS If you stick to rational numbers n/d for which d is odd, the rule you describe for calling them even or odd produces results that are much more consistent with integer arithmetic. One way to explain this is that in 2-adic arithmetic, these numbers are actually (2-adic) integers. —David Eppstein (talk) 08:01, 4 August 2015 (UTC)