|WikiProject Mathematics||(Rated C-class, Mid-priority)|
- 1 Redirection
- 2 Integers only
- 3 Music theory
- 4 Evenness (or non-evenness?) of 0
- 5 Parity
- 6 Number Theory
- 7 Arithmetic rules of parity
- 8 Prime Connection
- 9 Parity in culture?
- 10 Title
- 11 Parity of any rational number can be determined
- 12 Are there more even numbers than odd, more odd numbers than even, an equal number of odd and even numbers, or is it unknown which is greater?
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.
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 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 ?188.8.131.52 (talk) 18:51, 17 June 2012 (UTC)
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
"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?
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 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)
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)
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 184.108.40.206 (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
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)
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?
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 (talk • contribs) 08:37, 28 June 2011 (UTC)
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
http://math.stackexchange.com/questions/92451/can-decimal-numbers-be-considered-even-or-odd 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.
- 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)
Are there more even numbers than odd, more odd numbers than even, an equal number of odd and even numbers, or is it unknown which is greater?
I once conjectured that not all infinities were equal, and asked a mathematician friend to see if my conjecture was correct. He confirmed it was, pointing out to me that the set of all even numbers was not as great as the set of all integers, which in turn is not as great as the set of all real numbers, despite all three of these sets being infinite. This meant that the latter infinities were larger than the former.
That made sense.
Unfortunately, one thing this article does not discuss is whether the set of all even numbers is infinitesimally larger than, infinitesimally smaller than, or equal to the set of all odd numbers. Here's why this appears to me a conundrum:
Obviously, zero is even, always even, and nothing but even. This is completely obvious. Nothing in the history of humankind, in fact, has ever been so obvious. But, unlike every other even number, zero lacks a negative counterpart. In other words, +0 and −0 are the same number, viz., zero; the value they represent is identical.
If there were no zero, then we could easily conclude that there were an even number of even and odd numbers, exactly the same infinite number of each, with exactly half of each being on the positive side and the other half of each being on the negative side. But the fact that zero exists throws what appears at least to me to be a wrench in our count.
At first, the solution appears simple enough. Since there is a negative one and a positive one, both of which are negative, and a negative two and a positive two, both of which are positive, if there were no zero, we might conclude that, starting at the ones and moving out to the twos, we are even, with exactly two evens and two odds. If we move forward a couple more steps, to include both the positive and negative threes and fours, we're still even, just as we are if we extend out to the tens, the quadrillions, or infinity—again, if there were no zero. And, since there is a zero, and since it is even, we would then conclude that the set of even numbers is infinitesimally larger than the set of odd numbers.
At first, that solution appears pretty convincing. But, then I got to thinking, that's because I'm starting with the ones and adding the zero at the end. But, I get a very different result if I acknowledge and start with the zero from the start.
Remember, negative zero is not a different number than zero; they're the same number, by which I mean, they occupy the same exact point on any graph, not two separate points. They are but one number.
So, if I start with the zero, I can either throw it in with my count of the positive evens or with my count of the negative evens, but I can't throw it into both, for by doing so, I treat it as two separate points, when it is only a single point.
I throw it in with the positive evens because, well, why not? I've got to throw it in with one side, and both sides are just as good as the other. Certainly, the count isn't going to be effected regardless of on which side I throw it; either way, it's a single even number, not two even numbers.
So, zero is an even number, while both negative one and positive one are odd numbers. Right now, odds and beating even two-to-one. If we move forward a couple more steps, to include both the positive and negative twos and threes, we still have one more odd number than even, just as we have if we extend out by another ten steps, another quadrillion steps, or another infinite steps. This way of looking at it would lead us to concluding that the set of odd numbers is infinitesimally larger than the set of even numbers.
And, yet, there is yet a third way to look at this pattern. We start with an even integer, and add to it two odd integers (viz., one and negative one), making odd lead two-to-one. We add to that two positive integers (viz., two and negative two), making even lead three-to-two. Next, odd leads four-to-three, then even five-to-four, odd six-to-five, and so on, creating a wave-pattern in which the number in the lead constantly fluctuates back and fourth across the median. In this way of thinking about it, either the odds nor the evens have a lead at infinity, which might lead us to thinking that there are ultimately an equal number of even and odd numbers, that the two infinities are equal in size.
Unfortunately, I see nothing about the ratio of odd to even numbers being discussed in this article. If there is an answer to this, consider improving the article by adding the information and its proof.
- Not sure that you heard your mathematician friend correctly or whether they just made a mistake, but the number (in the sense of cardinality) of even integers is the same as that of all the integers. The proof of this is given informally in the introductory section of countable set. The rest of that statement (that the number of reals is larger than the number of integers) is correct. The type of counting that you discuss above is a never ending process and there is no way to make a conclusion about what happens when it's over, because it is never over! A different way of thinking about this issue is needed and this is discussed in cardinality. Also, when talking about infinite sets, it is not a good idea to refer to the "number" of elements ... that is a concept that works fine for finite sets but does not extend very well to infinite sets. The term "cardinality" is used for both finite and infinite sets, agrees with the idea of "number" for finite sets and makes sense for infinite sets.--Bill Cherowitzo (talk) 23:40, 11 March 2017 (UTC)