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The zero

  1. Aug 27, 2009 #1
    is '0' an odd or even number,i mean in my maths book it was
    given it is neither odd nor even
     
  2. jcsd
  3. Aug 27, 2009 #2
    0 is even.
     
  4. Aug 28, 2009 #3
    why is it even?
     
  5. Aug 28, 2009 #4

    uart

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    "why is it even?"

    It can be divided by 2 without remainder.
     
  6. Aug 28, 2009 #5
    Of course, it can also be divided by 3 (or any number) without remainder, so your book may be trying to confer the notion that 0 is not strictly even, although saying it is neither doesn't sound right.

    I wouldn't say that 0 "is" even, but rather it has the attribute of evenness.
     
  7. Aug 28, 2009 #6
    No it's just even. The entire structure of math is axiom, definition, proof, theorem. Even is a property DEFINED as being divisible by 2 without remainder. 0 has this property. It's even plane and simple.
     
  8. Aug 28, 2009 #7

    HallsofIvy

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    I don't know why you, or the text book, would not say that 0 is "strictly" even. It certainly is. An "even" number is defined as a number of the form "2n" for some integer n, a "multiple of 2". 0= 2(0). Yes, n can also be divided by 3: 0= 3(0) so 0 is also a multiple of 3. In fact it is a multiple of every number.

    Monty37, please quote exactly what your textbook says.
     
  9. Aug 28, 2009 #8
    Furthermore, the definition of an odd number is a number that is not evenly divisible by two. Since zero is evenly divisible by two, it can't be odd.
     
  10. Aug 28, 2009 #9
    Definitions in math often have cases which don't fit the general definition. In most cases, the decision to include or exclude a number from a class of numbers is simply arbitrary. In most cases, a certain symmetry follows, making common problems easier to solve.

    Some examples:

    Zero is even (why? because if you extend the definition to the set of all integers, not just positive integers, the sequence goes even-odd-even-odd from +infinity to -infinity)

    One is NOT a prime (why? otherwise, almost every proof in number theory would begin with "let p be a prime number greater than 1)

    0! = 1. (why? because otherwise, [tex]e = 1 + \Sigma_{k=1} \frac{1}{k!}[/tex], not simply [tex]e = \Sigma_{k=0} \frac{1}{k!}[/tex]

    0^0 = 1.... sometimes. In number theory, this definition is convenient, because things like taylor series work out very nicely at x=0. But in analysis, 0^0 is left undefined, because then exponentiation becomes a continuous function.
     
  11. Aug 28, 2009 #10

    symbolipoint

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    Why is this a problem for anyone? If you know that 2 (or 4) is even then if you subtract exactly 2 from it, the result is even. A good reassuring idea about evenness is that if you subtract any integral multiple of 2 from an even number, the result is also an even number.
     
  12. Aug 28, 2009 #11
    Or, more conveniently, because Gamma(1)=1
     
  13. Aug 28, 2009 #12
    Would you say 6 is "just even"? I wouldn't, because it's also threeish (evenly divisible by 3).

    Anyway, we're not talking geometry.
     
  14. Aug 28, 2009 #13
    What properties do threeish imply exactly? Is 22 elevenish? These are still even.
     
  15. Aug 29, 2009 #14
    even number: 2k,

    odd number: 2k-1,

    where k is integer.

    Let suppose, 2k=0 and k=0

    If we let 2k-1=0, 2k=1, k=1/2 (which is not integer, and it is contradiction).

    Now we proved that 0 is even. :smile:
     
  16. Aug 29, 2009 #15

    HallsofIvy

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    He meant "just even" in response to the assertion that 0 is "not even but has an attribute of evenness", not that it was not divisible by other numbers. If some one said "6 is not even but has an attribute of evenness", I might well respond, "no it is just even". The fact that it is also divisible by 3 doesn't change the fact that it is even.

    Anyway, why the reference to geometry? Surely you understand that all of mathematics "axiom, definition, proof, theorem"?
     
  17. Aug 29, 2009 #16

    Mark44

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    The attributes of evenness and oddness are special in that they refer to being exactly divisible by 2 or not. I'm not aware of any other words, at least in English, that describe whether a number is divisible by something other than 2. A number can be even and also divisible by primes other than 2, such as 6 in your example, but again, there aren't any special words that relate to this.
     
  18. Aug 29, 2009 #17
    That's why I just made up the word "threeish". But if "even" means divisible by two, what exactly does "just even" mean? I would interpret that to mean divisible by two and not divisible by other numbers and that certainly doesn't apply to 0, does it?
     
  19. Aug 29, 2009 #18
    Well, I didn't assert that 0 is "not even but has an attribute of evenness", what I asserted is that "even" is inadequet description, not that it's wrong. Calling 6 "even" is not wrong, but it doesn't give you the full picture. And I agree that 0 is even, but there's more to it than that. My guess is the book in question is trying to convey something like this but doing a poor job at it.

    Anyway, the reference to geometry is plane if you carefully read the post I was replying to.
     
  20. Aug 29, 2009 #19
    No number is just divisible by 2. Every number is divisible by 1. If you don't allow 1 as a divisor, then 2 is the only number that's only divisible by 2.
     
  21. Aug 29, 2009 #20
    Any given number has tons of titles. It could be a perfect square, prime (of which there are dozens of types, balanced, carol, chen, cousin, etc.), perfect number, etc. I was saying "just even" in response to the OP's suggestion that there was something "fishy" about the "evenness" of zero. It is 100%, no subtlties or confusions, an even number. I was by no means saying that zero doesn't have other properties. To the contrary, zero is chock full of special/unusual properties.
     
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