# Is zero an even number? [duplicate]

This question already has an answer here:

A quick google returns the answer on the parity of zero:

Zero is an even number. In other words, its parity—the quality of an integer being even or odd—is even. The simplest way to prove that zero is even is to check that it fits the definition of "even": it is an integer multiple of 2, specifically 0 × 2.

So I know I've answered my own question but I still wanted to ask whether in some respects zero is the only number that is neither even nor odd?

## marked as duplicate by quid♦, Matthew Towers, Ali Caglayan, Najib Idrissi, Lord_FarinJan 21 '15 at 18:08

• Your answer is correct: zero is no less even than any other even number. – Zach Effman Jan 21 '15 at 16:18
• What you quote says clearly that 0 is even. I do not understand how you phease the question. This is surely a duplicate though. – quid Jan 21 '15 at 16:18
• There is no respect in which zero is odd. There is no respect in which zero is not even. Some of these sorts of questions are genuinely controversial. For example, whether 1 is considered prime has changed over time. However, the parity of 0 is not controversial. – MJD Jan 21 '15 at 16:20
• @MJD zero can be pretty odd. Just as some saying goes, all primes are odd, indeed two is the oddest of all. :-) – quid Jan 21 '15 at 16:35
• elementary number theory - Is zero odd or even? - Mathematics Stack Exchange – user103028 Sep 24 '15 at 6:39

Zero is even, as you argue. There's no circumstances where zero is taken to be odd, nor can it be taken to be neither odd nor even.

What is true is that $0$ is the only real number that is neither negative nor positive, alternatively the only real number $x$ such that $x = -x$.

• There are some that think zero is the only number that is positive and negative. – quid Jan 21 '15 at 16:19
• @quid: In French it is. – Henning Makholm Jan 21 '15 at 16:20
• I guess that's my point that while it might be possible to prove it's even, zero is still an unusual number, different from all others – Pixelomo Jan 21 '15 at 16:21
• @AlanSutherland: Every small integer is an unusual number, different from all others. – Henning Makholm Jan 21 '15 at 16:22
• @AlanSutherland The parity of zero is only unusual in that many people are miseducated about it from childhood and seem to retain a sense of confusion and wonder even though the objective evidence is clear and straightforward :). – Erick Wong Jan 21 '15 at 16:35

"So I know I've answered my own question but I still wanted to ask whether in some respects zero is the only number that is neither even nor odd?"

There is a respect in which $0$ is neither even nor odd; since $\mathbb{Z}$ is an integral domain, it often makes sense to study the multiplicative structure of $\mathbb{Z} \setminus \{0\}$, and nevermind its additive structure. So we could, if we wanted to, define that an even number is an $x \in \mathbb{Z} \setminus \{0\}$ such that $2 \mid x$, in which case the statement "$0$ is even" is not true.

But, rather than redefining "even", its probably easier to just say: "$0$ is the only even number that isn't regular."

• If we wanted to we could also define a number to be even if $x^2$ is divisible by $20$. – quid Jan 21 '15 at 16:37
• @quid, but that would be an ill-motivated thing to do. – goblin Jan 21 '15 at 16:37
• Yes, sure. And sorry for the somehwhat stupid remark. My point is though that this is still a bit far fetched. When studying questions of divisibility in integral domains one also often excludes invertible elements, so would you propose $1$ might not be odd because of this. – quid Jan 21 '15 at 16:39
• @quid, probably not. Units have a unique (up to unit-multiples!) factorization into irreducibles just like all the other non-zero numbers, so I cannot think of a context where it would be optimal for $1$ not to be odd. Of course, that is not the nature of the question; we're asked to ask ourselves if there might ever be a context where the statement "$2$ is even" is no longer an optimal convention. – goblin Jan 21 '15 at 16:45
• @quid, ah yes, true. On a related note, I think the word "odd" should be purged from our vocabulary, and the same goes for "irrational" and "transcendental." Better to say "non-even", "non-rational", and "non-algebraic." The reason is that while the sets of even, rational and alegbraic numbers are closed under addition and multiplication, the sets of non-even, non-rational and non-algebraic numbers are not. So the latter are very non-fundamental sets, and I do not think they deserve their own special names. – goblin Jan 21 '15 at 17:14