Oop. I thought I was getting in at the beginning of the thread -- didn't realize that there were already s few dozen responses. I'm sure my point has already been raised and then some.

 

I agree with studiot (and others).

First, zero is a special case that has to be considered separately. This is comparable to the question of whether 1 is a prime number. If we just consider the simplistic definition that a (natural number) is prime if the only factors are 1 and itself, this 1 would have to be classified as a prime number since it clearly satisfied that definition. But if you allow that, then many other things become messy. For instance, the claim is that the prime factorization of any natural number is unique. But if 1 is a prime number, then any natural number has an infinite number of prime factorizations since all we have to do is increase the exponent of the prime factor 1. So you have to make a special rule about the factorization being unique modulo the exponent of the prime factor 1. But if you simply define 1 as being not a prime number, then all of those other special rules are no longer needed and you haven't caused any major problems in making this definition, other than you can no longer say that all natural numbers are either prime or composite, because 1 is clearly does not have two or more prime factors (remember, 1 is already not prime and so it doesn't count). So now you define that 1 is neither prime nor composite and you modify your other rule to say that all natural numbers, other than one, are either prime or composite.

The same would apply to zero. You can't just use a simple definition of even/odd, positive/negative, etc. and stop there. You have to look at the implications for the system as a whole. In the case of zero, calling it either positive or negative causes problems, so it is generally defined as being neither -- hence the distinction between "positive integers" and "non-negative integers". You have to do the same for allowing it to be even or odd and you could define it to be neither if that was what was needed in order to make the overall system more coherent. In this case, it doesn't cause any problems and so letting it be even per the definition of an even integer is simpler and doesn't require that we modify that rule to specifically exclude zero (like we had to do with the prime number rule).

 

So... if k = 11 and 2k = 22, an even number, proves that 11 is an even number?

2k is an even number for ANY integer k, even, odd, or zero.

Thanks WB. Yes, you're absolutely right. I can't believe I missed that one. As the saying goes, you live and learn!

 

We all make those kinds of goofs -- I throw 'em into the catchall category called, "Brain Farts".