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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?

Pixelomo
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    Your answer is correct: zero is no less even than any other even number. – Zach Effman Jan 21 '15 at 16:18
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    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
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    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
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    @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
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    elementary number theory - Is zero odd or even? - Mathematics Stack Exchange –  Sep 24 '15 at 06:39
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    Aren't odd numbers integer multiples of 2 too? Just with 1 added. 0 is an integer multiple of every single number there is... – Deji Aug 18 '16 at 12:58
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    Number is Odd if and only if the remainder division by two is not zero. Number is Even if and only if the whole division by two is not zero and the remainder is zero. Therefore number Zero is neither Odd nor Even. Odd or Even apply only for Something, Nothing is always Nothing. –  Jan 02 '21 at 12:37

2 Answers2

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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 $x = -x$.

amWhy
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"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."

goblin GONE
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    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 GONE Jan 21 '15 at 16:37
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    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 GONE Jan 21 '15 at 16:45
  • I can think of such a context where $1$ is odd is not good easily. Even typically means divisble by $2$ and odd just not even, whence in a domain where $2$ is invertible everything is even. – quid Jan 21 '15 at 16:49
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    @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 GONE Jan 21 '15 at 17:14
  • I agree that a different naming could avoid some issues. – quid Jan 21 '15 at 17:39
  • @quid, I've actually changed my mind slightly; the term "odd" is actually pretty useful, because the odd numbers are closed under multiplication. This is because: since $2$ is prime, hence $2\mathbb{Z}$ is a prime ideal, hence $(2\mathbb{Z})^c$ is closed under multiplication. Another wacky thing about $(2\mathbb{Z})^c$ is that the sum of any three elements is itself an element. So its kind of a "generalized ring", in the sense that we have a ternary operator for addition (which is associative in the appropriate sense), rather than a binary operator. – goblin GONE Feb 07 '16 at 15:25
  • @quid, another neat observation is the following: writing $\mathbb{O}$ for the odd numbers, we can define a binary operation $+$ on $\mathbb{O}$ as follows: $a+\mathbb{O} b$ is obtained by first computing $a+{\mathbb{Z}} b$, then deleting all factors of $2$. e.g. $1+\mathbb{O} 3 = 1$. In this language, we can rehash the Collatz conjecture as follows: if you repeatedly apply the function $n \in \mathbb{O} \mapsto 3n+\mathbb{O} 1$ to an odd number, you'll eventually get $1$. – goblin GONE Feb 07 '16 at 15:34
  • Number is Odd if and only if the remainder division by two is not zero. Number is Even if and only if the whole division by two is not zero and the remainder is zero. Therefore number Zero is neither Odd nor Even. Odd or Even apply only for Something, Nothing is always Nothing. –  Jan 02 '21 at 12:37