# Zero. Integer. Natural. Even?

Discussion in 'Mathematics' started by pipipi, Jun 26, 2012.

1. ### pipipiNew commenter

Just trying to check about zero.

It is an integer.

It is even.

It is not positive.

It is a natural number.

Thanks.

2. ### KarvolOccasional commenter

I'll get my coat.

3. ### BillyBobJoeEstablished commenter

Not positive, not negative.
Integer
Not Natural
Even is arguable but I would say yes.

4. ### AnotherMathsHoD

Neither Positive nor Negative, but it is non-negative and non-positive

Definitely an Integer

Lots of disagreement here! Natural or Not Natural depending on definition - The IB use the definition with 0 as Natural!

Definitely Even

5. ### Vicky496

I reckon... 0 is an Integer, Rational, Real, Imaginary, Complex etc. But not Natural (some exam boards disagree!), positive/negative, or odd/even.

6. ### likeglue2

From (ahem, cough) Wikipedia:
Several set-theoretical definitions of natural numbers were developed in the 19th century. With these definitions it was convenient to include 0 (corresponding to the empty set) as a natural number. Including 0 is now the common convention among set theorists, logicians, and computer scientists. Many other mathematicians also include 0, although some have kept the older tradition and take 1 to be the first natural number.[4] Sometimes the set of natural numbers with 0 included is called the set of whole numbers or counting numbers. On the other hand, integer being Latin for whole, the integers usually stand for the negative and positive whole numbers (and zero) altogether.

7. ### cyolba

As for the is it even question, I think that depends upon your definition of even.
I have always assumed even numbers were multiples of 2, thus rendering 0 and alkl negative numbers not even. I could be wrong, and I probably am. In the same way, the first odd number is 1, and the definituion (for me, at least) is that odd numbers are multiples of 1 that are not multiples of 2.

8. ### Polecat

2 = 1 +1; 4 = 2 +2; 0 = 0 + 0; -2 = -1 + -1; -4 = -2 + -2 ...
Do I detect an emerging definition of an even number.
Perhaps the other ones are odd
Who knows?

9. ### cyolba

So your definition, poley, is that an even number is the sum of two integers? Fair enough, a fine definition. But which is the generally accepted one?
cyolba, now becoming a little curious

10. ### algebraist

Like so many definitional questions, this one is getting confused by a failure to say of *which* previously known set we're defining a subset. Does anyone else remember being taught basic set theory in early primary school, using large laminated cards on the floor and sets of plastic triangles, squares, circles that could be thick or thin, red yellow or blue, large or small? Does anywhere still do that? I enjoyed it...
Anyway, a standard definition is: an even number is an integer e such that there exists an integer k such that e = 2k. By that definition, 0 is even because 0 = 2 x 0.
If you defined even numbers to be a subset of the natural numbers, instead of a subset of the integers, *and* you defined the natural numbers in such a way that 0 wasn't one, then 0 wouldn't be an even number (because it wouldn't be a natural number at all!). I can't think of any other natural definition that excludes it. To my mind this is a sufficient reason to prefer to define 0 to be a natural number, at least if you're going to be talking to people who don't know about negative integers yet!

11. ### florapost

shouldn't even numbers be wholly divisible by 2 - which would then include negative numbers and zero, and be akin to polecat's defintion
as for natural numbers, let us all throw our hands up in the air in the manner of wolfram:
The term "natural number" refers either to a member of the set of positive integers 1, 2, 3, ... (Sloane's A000027)
or to the set of nonnegative integers 0, 1,
2, 3, ... (Sloane's A001477; e.g., Bourbaki
1968, Halmos 1974). Regrettably, there seems to be no general agreement about whether
to include 0 in the set of natural numbers. In fact, Ribenboim (1996) states "Let
be a set of natural numbers; whenever convenient,
it may be assumed that ."

nice to see a mention of dear old bourbaki there

12. ### florapost

will this post?

13. ### florapost

ok - that's weird - because a long, technical answer i doubt i can be bothered to repeat has just been pre-moderated
is bourbaki that controversial?

14. ### pipipiNew commenter

Bour what baki? Is that something to stick in my pipe and smoke.

Thanks to everyone for the replies, I'm not sure it's completely cleared everything up, but I've been reassured, so thanks.

15. ### algebraist

Probably! On this topic the Bourbaki connection I know about is briefly discussed here:
http://mathforum.org/kb/message.jspa?messageID=448053
(hmm, wonder if a link is OK?) - it's a pity Prof. Kiselman's paper is in Swedish.

16. ### florapost

this gets weirder - i tried again and the forum did the same thing - i don't know what's in the post that's so horrific
i am getting well paranoid here

17. ### florapost

ah - i arrived
and my posts are now totally dis-ordered