Zero: Even or Odd

Zero is Even.

1) No remainer :P
2) 0 Even, 1 Odd, 2 Even, 3 Odd, 4 Even, 5 Odd....so on....

Dr. Math's answer.

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Original post by JonWoyame Has it been unequivocally proven that zero is an even number? I know that people say it is even because 0 / 2 has no remainder, but is there any proof that it HAS to be even? I think I found one... I wanted to check with you guys to make sure it wasn't already done. Will this make me rich and famous? Will schoolgirls around the world read my proof in math books and lust after me? Jon

Look up the definition of what an even integer is: any integer of the form 2k where k is an integer. You will note that 0 meets that definition since 2*0 = 0. It cannot possibly be odd since an odd integer is defined as: any integer of the form 2k + 1 where k is an integer. 2 * k + 1 = 0 => 2 * k = -1 => k = -1/2. Clearly since k is not an integer, then 0 cannot be odd.

Zero is even.

A number n is said to be even if there exists a q such that n = 2q. In the case of n = 0, we have 0 = 2(0).

Edit: Meh, beaten to the punch.

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Original post by Promit Dr. Math's answer.

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From Link Above I'm surprised that some teachers believe 0 to be neither even nor odd....

I'm surprised too.

But the zero FUNCTION is both even and odd.

Even function: f(-x) = f(x)
Odd function: -f(x) = f(-x)

Even I can't argue against 0 being an even number though. Unless 0 = 1 of course!

That stinks.

Anyway, my proof used some of Georg Cantor's set theory:

You have the set E, containing all even numbers, and O containing all odd. Both are equal in size, as proved by Cantor. Therefore for each element in E you have an element in O. The relationship can be formed as y = x + 1, where y is any element of O and x its related element in E. For 11 you have x = 10; for -53, x = -54; for -1, x = -2; and for 1, 0.

If 0 is not even then 1 does not have an even complement, thus contradicting the fact that the length of E = length of O.

Q.E.D.


Yeehaw.

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Original post by JonWoyame That stinks. Anyway, my proof used some of Georg Cantor's set theory: You have the set E, containing all even numbers, and O containing all odd. Both are equal in size, as proved by Cantor. Therefore for each element in E you have an element in O. The relationship can be formed as y = x + 1, where y is any element of O and x its related element in E. For 11 you have x = 10; for -53, x = -54; for -1, x = -2; and for 1, 0. If 0 is not even then 1 does not have an even complement, thus contradicting the fact that the length of E = length of O. Q.E.D. Yeehaw.

I do believe that you could make them correspond as such, though:

...-3, -1,  1,  3,  5,  7......-6, -4, -2,  2,  4,  6...

Sure, it's not intuitive to skip zero, but I don't see why it doesn't work.

From MathWorld:

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An even number is a number of the form n == 2k, where k is an integer. The even numbers are therefore ..., -4, -2, 0, 2, 4, ...

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Original post by cowsarenotevil Sure, it's not intuitive to skip zero, but I don't see why it doesn't work.

Good point... I'll have to think about this more.


Thanks for destroying me dreams everyone.

0 = 0dd

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Original post by cowsarenotevil [...]Sure, it's not intuitive to skip zero, but I don't see why it doesn't work.

It's less elegant because such a mapping must be a two-part(when x>0 ..., when x<0 ...) function or otherwise be more complex than the simple "y = x + 1". Elegance and simplicty are preferrable as long as correctness is maintained.

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Original post by JonWoyame You have the set E, containing all even numbers, and O containing all odd. Both are equal in size, as proved by Cantor. ... If 0 is not even then 1 does not have an even complement, thus contradicting the fact that the length of E = length of O.

I don't buy it. Being lazy, I can't be bothered to find & read through some academic's wonderfully esoteric proof, except to say that I'd be interested to see the E=O part(in a nutshell) - so did he count all the way to infinity + 1 to find that out? I'm guessing it was some kind of proof by induction but these have a tendency to just piss me off due to sheer gayness.

[wink]

I go with the zero is neither even nor odd idea, taking it from the perspective of real life & not the world of maths. Or in laymans' terms (my favourite kind) having no oranges does not mean you posess an odd or even number of oranges, it just means you're gonna get scurvy.

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Original post by Extrarius Quote: Original post by cowsarenotevil [...]Sure, it's not intuitive to skip zero, but I don't see why it doesn't work. It's less elegant because such a mapping must be a two-part(when x>0 ..., when x<0 ...) function or otherwise be more complex than the simple "y = x + 1". Elegance and simplicty are preferrable as long as correctness is maintained.

I realize all this, but I am reasonably certain that correctness is indeed maintained. Are you saying there's a "better" way of putting the two sets into correspondence? I'm not really sure what you're saying.

I don't think 0/2 is unequivocally defined. Thus, Zero is not even.

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Original post by owl I don't think 0/2 is unequivocally defined.

Of course it is. 0/2 = 0. What do you think it should be?

It's dividing anything by zero that's not defined. Dividing zero by anything is zero.

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Original post by martiandragon It's dividing anything by zero that's not defined. Dividing zero by anything is zero.

Except zero.

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Original post by Sneftel Quote: Original post by owl I don't think 0/2 is unequivocally defined. Of course it is. 0/2 = 0. What do you think it should be?

LOL. I meant 0/0. Any odd or even number divided by itself is 1. How can we say something about zero if it doesn't follow the same rules that the other numbers?