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Some people say that 0 is neither even nor odd. I say that 0 is even.

Is there a simple way to convince people that 0 is even and the statement that "0 is neither even nor odd" is false.

user172675
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  • Most people who practice mathematics regularly take $0$ to be an even integer, since it is divisible by $2$: $2 \times 0 = 0$. Furthermore, taking $2$ to be even makes a lot of statements a lot more concise. Call it odd, call it even if'n ya wanna, but I vote for $2$ *even*!!! Cheers! – Robert Lewis Apr 04 '15 at 04:20
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    http://en.wikipedia.org/wiki/Parity_of_zero – Todd Wilcox Apr 04 '15 at 04:21
  • "Believe"? This is mathematics, not some touchy-feely subject like poetry. – Jonathan Hebert Apr 04 '15 at 04:22
  • @JonathanHebert: Sorry, "think" is better? – user172675 Apr 04 '15 at 04:24
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    @JonathanHebert That is hardly necessary here. – Daniel W. Farlow Apr 04 '15 at 04:24
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    @JonathanHebert: how in heaven's name did you come with the notion that poetry is a "touchy-feely" subject? – Robert Lewis Apr 04 '15 at 05:47
  • $$\frac 7 2 = 3~remainder~1$$ 7 is odd, $$\frac 6 2 = 3~remainder~0$$ 6 - is even, then $$\frac 1 2 = 0~remainder~1$$ 1 - is odd, then, $$\frac 0 2 = 0~remainder~0$$ zero neither odd nor even –  Aug 02 '15 at 16:18

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An integer, $x$, is defined to be even whenever it can be written in the form $x=2k$ where $k$ is some integer.

Examples: $6=2\cdot 3,~~ 10 = 2\cdot 5,~~ 2218 = 2\cdot 1109,~~ -4 = 2\cdot (-2)$

An integer, $x$, is defined to be odd whenever it can be written in the form $x=2k+1$ where $k$ is some integer.

Examples: $-3 = 2\cdot (-2) + 1,~~~ 9 = 2\cdot 4 + 1,~~~ 1001 = 2\cdot 500 + 1$

Remember that $0$ is itself an integer, and that $0 = 2\cdot \color{red}{0}$, which is in the form $0=2k$ with $k = \color{red}{0}$, therefore $0$ is even.

JMoravitz
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  • Same as my proof. Thanks you. – user172675 Apr 04 '15 at 04:23
  • @user172675 This is hardly a proof--if someone accepts that $0$ is actually a number, furthermore an integer, then the fact that $0$ is even is a trivial conclusion. Your question is far more philosophical than you may think, for the concept of zero goes way way back. – Daniel W. Farlow Apr 04 '15 at 04:28