Is Zero Even?

Date: 03/28/2001 at 02:59:56
From: John Matousek
Subject: Zero odd/even

At numerous sites across the Internet the answer to the question 
whether zero is odd or even seems to be totally subjective, and the 
proofs used to justify 'even' (zero can be divided by two, therefore 
it is even), sound reasonable. But zero can't really be divided by two 
since the result is zero - neither a positive or negative integer.

   Q: How many times does 2 go into 0?
   A: Zero times. Or to rephrase, two doesn't go into zero.

The question arose when a retired math teacher stated "2/20/2000, the 
first day ever with seven even numerals in its date." Of course he is 
wrong, 2/20/2000 BC being the most obvious example - if you accept 
zero as even. But there are also thousands of dates from the 
astronomical, Hebrew, Chinese, Hindu lunar, old Hindu solar and lunar 
calendars where zero would not even need to be considered. 'Ever' is 
such a big word.

Thanks.

Date: 03/28/2001 at 09:18:35
From: Doctor Rick
Subject: Re: Zero odd/even

Hi, John. Thanks for writing! I hope I can clear up some confusion.

Our archives sometimes say that zero is neither positive nor negative, 
not that it is neither even nor odd. That's very different.

The question of evenness or oddness is based on definitions. There may 
be variations on how "even" is defined, just as there are on how 
"natural numbers" are defined; but once you have established your 
definition, the question can be answered objectively based on that 
definition.

An even number, as our archive pages say, is defined as one that is 
divisible by 2. Divisibility by 2 is defined as giving an integer 
quotient when divided by 2. The only matter open to debate is whether 
this last statement should say "integer," "natural number," or "whole 
number."

If integer, then the even numbers are ..., -6, -4, -2, 0, 2, 4, 6, ...

If natural number (0, 1, 2, 3, ...), then the even numbers are 0, 2, 
4, 6, ...

If whole number (1, 2, 3, ...), then the even numbers are 2, 4, 6, ...

There is no reason to be restrictive in our definition of 
divisibility: the definition introduces no contradictions or special 
cases when it is extended to all integers. If you have found places on 
the Web where a restrictive definition is used, I'd like to see them.

You state that the problem you have with zero being even is that zero 
can't really be divided by 2, because the quotient is 0, which is 
neither positive nor negative. Putting this in my terms, you are 
defining "divisible" as meaning "giving a quotient that is a positive 
or negative (that is, non-zero) integer." I could accept one of the 
alternative definitions I gave above before I would accept yours. 

If we say that zero cannot be divided by anything, then this 
introduces lots of special cases to our mathematical properties. For 
example, the sum of two even numbers is even. You are telling me that 
the 4 and -4 are even, but that the sum of 4 and -4 is *not* even. 
We'd need to change the rule to "The sum of even numbers is even, 
UNLESS it is zero." It's so much simpler to define our terms in a way 
that does not require such special cases. Defining evenness and 
divisibility as we do does not introduce special cases.

Before zero was introduced to our number system, negative quantities 
were treated as an entirely separate kind of entity from positive 
numbers. Different rules were needed for lots of different cases, 
depending on whether a quantity was added (positive) or subtracted 
(negative). The history of quadratic equations illustrates this. A big 
part of the genius of introducing zero in the first place was that it 
unified all these special cases into one. I am asking you now to see 
that you do not need to treat zero as special; and when you treat it 
like every other integer, it follows that zero is even.

For related answers in our archives, see:

Zero is even:

   Are these numbers odd or even?
   http://mathforum.org/library/drmath/view/57062.html   

   Is Zero Even, Odd, or Neither?
   http://mathforum.org/library/drmath/view/57104.html   

   Is Zero Odd or Even?
   http://mathforum.org/library/drmath/view/57132.html   

   FAQ: Integers, Rational and Irrational Numbers
   http://mathforum.org/dr.math/faq/faq.integers.html   

Even and odd numbers enumerated, start with 1:

   Infinity, Zero
   http://mathforum.org/library/drmath/view/52400.html   

Neither positive nor negative:

   Why Zero is Neither Positive nor Negative
   http://mathforum.org/library/drmath/view/58735.html   

   What is 0?
   http://mathforum.org/library/drmath/view/58743.html   

   Is Zero Positive or Negative?
   http://mathforum.org/library/drmath/view/60300.html   

- Doctor Rick, The Math Forum
  http://mathforum.org/dr.math/