Is Zero Even, Odd, Both or Neither?

Here’s another “debate” from the intense arena of mathematics. I use debate in quotes because there is one answer that is universally correct, and it’s just a matter of seeing the proof and choosing to accept it. It’s great that in math, most things are either 100% right or 100% wrong. This is one of those things.

Consider the following multiple choice question:

Zero is…
a.) even
b.) odd
c.) all of the above
d.) none of the above

I’ll give you time to answer.

Okay, time’s up. Congratulations to all of you who said “A”. That is the right answer.

What you say? You want proof. Luckily I’m here to give you some proof. I wrote about 0.999… = 1 earlier, and then I linked to Wikipedia on the topic of zero in this post. And that Wikipedia article is fairly conclusive in the beginning assertion it makes in bold: The number 0 is even. And this appears occasionally on the ACT and SAT, so you may want to pay attention, or at least memorize the answer.

For this experiment, let’s pretend we don’t have access to bold assertions contained in Wikipedia or the rest of the internet and we wanted to avoid appeal to authority fallacies.

How could we tell if zero is even? Maybe we could analyze all the properties of even numbers, and see if zero fits in to any of them. Then, maybe we could analyze all the properties of odd numbers and see if zero fits into any of them. If zero fits into every property, it’s both even and odd. If it fits into none of the properties, it’s neither. If it fits into only the even properties, it’s even. Same for the odd properties.

Let’s go!

The easiest definition in the book
We all know that even numbers are divisible by two, and thus return integers when divided by two. Odd numbers do not.

For example, 14 (an even number) divided by two is seven, which is an integer.
And 29 (an odd number) divided by two is 14.5, which is not a integer.

Zero divided by two is zero. Zero is an integer. This means zero is even.

Properties of Addition and Subtraction

  • even number – even number = even number
  • odd number – odd number = even
  • even – odd = odd
  • even + even = even
  • odd + odd = even
  • even + odd = odd

Well, for zero:

  • 2 (even) – 2 (even) = 0
  • 3 (odd) – 3 (odd) = 0
  • -4 (even) + 4 (even) = 0
  • -27 (odd) + 27 (odd) = 0

Therefore, zero fits exactly within the properties for even numbers. Additionally, it is impossible to construct a problem such that “even + odd = zero” or “even – odd = zero”. Therefore, zero doesn’t match any of the properties for odd numbers, either.

Properties of Multiplication

  • even × even = even
  • even × odd = even
  • odd × odd = odd

Manipulating the properties of multiplication are far less conclusive, but it does make it clear that zero cannot be odd and odd alone.

If you define zero to be even, the problems are always true.

  • 420 (even) × 0 (even) = 0 (even)
  • 0 (even) × 613 (odd) = 0 (even)

If you define zero to be odd, the problems can’t prove zero to be odd alone.

  • 110 (even) × 0 (odd) = 0 (even) Zero is both odd and even here.
  • 15 (odd) × 0 (odd) = 0 (odd) This works.
  • However, if you’re going to define zero as “odd” or “both” (as done in the above example) then the previous properties, like even + odd = odd become false. 42 (even) + 0 (as an odd) = 42, which is not odd.

    Properties of Splitting in Two
    When an odd number is split into two groups, there is always one remaining.

    Even numbers have no “odd one out”.

    Similarly, zero split into two groups is hard to see. However, there is not a single dot that does not fit into one of the two groups. There is no “odd one out”, thus zero fits in the definition of an even number.

    As you can see, there are two even groups on either side of the line.

    In Conclusion
    Zero is even. Have fun on your next standardized test.

    -

    I now blog at EverydayUtilitarian.com. I hope you'll join me at my new blog! This page has been left as an archive.

    On 6 Apr 2009 in All, Mathematics. 9 Comments.

    9 Comments

    1. #1 Simon says:
      18 May 2009, 9:23 am  

      … i thought zero was neither odd or even…
      is there any way (mathematical illusions etc.) to prove zero is odd or that zero could be neither odd or even?

    2. #2 Peter says:
      18 May 2009, 8:00 pm  

      Some teachers redefine the properties of addition to define zero as neither odd or even by giving them exceptions like follows:

      * even number – even number = even number (except when it gives 0)
      * odd number – odd number = even (except when it gives 0)
      * even + even = even (except when it gives 0)
      * odd + odd = even (except when it gives 0)

      But the exceptions prove that without them being there, 0 would be even. Additionally, 0 is considered even by the main mathematical community.

      As for proving that zero is odd or both using math tricks, you can show these either of these two properties alone:

      * If even × odd = even and 110 (even) × 0 (odd) = 0 (even), then zero is both odd and even.
      * Or if odd × odd = odd, and 15 (odd) × 0 (odd) = 0 (odd), then zero is odd.

      But both of those defy all the remaining properties, in addition to also working by replacing 0 as even. Additionally, the separation line would stop making sense.

    3. #3 Brian S. says:
      6 Jul 2009, 11:40 pm  

      I was taught that any number (n) is even if there exists some integer(m),such that 2m =n . examples would be 2(3) =6 or 2(1) =2. Clearly 2(0) = 0. For odd numbers you take n+1, if n+1 is even then n must be odd.

    4. #4 w says:
      3 Feb 2010, 6:08 pm  

      I didn’t know.

    5. #5 benny says:
      5 Aug 2010, 11:38 pm  

      as if we follow this logic then 0 is the multiple for every number since mod(0,n)always 0 (no remainder when divide by any number!) 3/3 , 6/3 and 0/3 all have no remainder so they are multiples!

    6. #6 Ananda says:
      11 Jun 2012, 7:44 am  

      odd number – odd number = even

      zero – zero = zero ….

      same for odd number – odd number = even

      zero + zero = zero ….Zero is both odd and even.

    7. #7 Peter Hurford (author) says:
      11 Jun 2012, 10:17 pm  

      ….Zero is both odd and even.

      A key take away is that even though zero fits some of the properties of odd numbers, it doesn’t have all of them. On the other hand, it does have all the properties of even numbers.

    8. #8 pundikala.veeresha says:
      6 Mar 2013, 8:12 am  

      Zero is even

    9. #9 abhi says:
      13 Jun 2013, 9:13 am  

      zero is even.

    Leave a Reply

    You must be logged in to post a comment.