# Zero - even, odd, or neither?

1. Apr 27, 2008

### freerangequark

Is Zero an even or odd number, or neither?

The common answer I have found seems to focus on the fact that you can evenly divide by two. However there is a bit of dispute whether this is accurate or not.

Can anyone offer clarification on this?

Thanks!

2. Apr 27, 2008

### arildno

Well, what dispute would that be?

3. Apr 27, 2008

### freerangequark

The dispute was along the same lines as the question of whether 1 is a prime number or not.

Another dispute I read is whether zero is a number at all and therefore eligible to receive an even or odd tag.

Last edited: Apr 27, 2008
4. Apr 27, 2008

### D H

Staff Emeritus
One is neither prime nor composite. It is its own special thing. Zero is similarly neither positive or negative. It, too, is its own special thing. This speciality does not apply to the concept of even or odd. Zero is even because zero modulo two is zero. That is the definition of evenness. There is nothing special about zero with regard to even/odd nature of a number.

Zero is a number, just as is negative one, or pi, or the square root of negative one, or some even more esoteric beasts you probably have not yet encountered. You are mistaking the counting numbers (1,2,3,...) as the the only things that qualify as numbers.

5. Apr 27, 2008

### arildno

Just pick your favourite definition of it.
Using the conventional one (when 1 is not regarded a prime number), however, we may state that every composite number has a unique prime factorization (permutations ignored).

If you choose to let 1 be a prime number, that theorem, and practically every other theorems become invalid, unless you modify them by tagging statements like "all prime numbers" with the tail "EXCEPT 1".

Why shouldn't 0 be a number?

Last edited: Apr 27, 2008
6. Apr 27, 2008

### LukeD

The only people I've ever seen dispute whether or not 0 is even were English and Computer Science teachers. I've never seen any Math people question that (and I've certainly never seen any math people question whether or not 0 is a number, least not in the past few hundred years)

An integer is even if it is evenly divisible by 2. 0 is divisible by every integer. Therefore, it is certainly even. (Note that I said integer; therefore, the number -4 is even for example)

As to whether or not 0 is a Natural Number is in fact debated, but this is just a matter of notation, not definition or question of properties. 0 has many different properties from the positive integers, so most number theorists do not include it among the natural numbers. However, many people who study combinatorics like to include 0 because they use it very often as do many computer scientists, set theorists, etc. But this is really just a question of convention.

---

As to why 1 is not a prime number; there are a lot of reasons for this. One reason is that if you divide an integer by a prime number that evenly divides it, your result is always smaller than the original number. If you divide a number by 1, your result is always the same. If you learn some more math, you might learn some of the other reasons why 1 is not prime.

7. Apr 27, 2008

### Hurkyl

Staff Emeritus
People sometimes fall into the philosophical trap of thinking that there's "one true number system" which everyone must use, and any other putative number systems are just some sort of magic trick. This is part of a more general problem of thinking there's more to a mathematical definition than simply being a definition.

8. Apr 27, 2008

### _Mayday_

What is $\frac{0}{0}$?

It could be one, as any number divided by itself is 1, or is it 0 as any number divided by 0 is 0. Maybe there is another way in defining numbers divided by themselves and by 0.

9. Apr 27, 2008

### arildno

People are used to chatter back and forth, and this capacity for effective social interaction is most likely the primary evolutionary reason for the development of the "higher intellect" in the first place.

An abstract definition, an essentially arbitrarily chosen "starting point for discussion" is quite correctly a wholly different type of "discourse".

Most people are unfamiliar with that type of discourse, and wish maths, and science, to be familiarly chatty subjects.

Since this is a wholly unsuitable mode for conducting maths and science, laymen are put off by it as something vaguely threating, filled with arrogance, coldness and authoritarianism.

(Just chatting here, I know..)

10. Apr 27, 2008

### Howers

I'd say even.

Even = 2n
Odd = 2n - 1

If you define it that way, there is no way 0 can be odd.

11. Apr 27, 2008

### arildno

If n=1/2..

12. Apr 27, 2008

### uman

_Mayday_, $$\frac{0}{0}$$ is undefined and thus is neither equal to zero nor to one.

13. Apr 27, 2008

### Howers

Where n is an element of the naturals. I assumed everyone understood that notation :)

14. Apr 27, 2008

### Hurkyl

Staff Emeritus
Oh, so -2 is neither even nor odd? And what about $(1 + \sqrt{17}) / 2$? I very much think that should be an even number. :tongue:

Last edited: Apr 27, 2008
15. Apr 27, 2008

### arildno

Besides, restricting n to the naturals would make 0 NEITHER even or odd..

(Okay, I'll stop being evil..)

16. Apr 27, 2008

### CRGreathouse

Depending on your definition of natural, yes? Most analysts take them as {0, 1, 2, ...}.

17. Apr 28, 2008

### LukeD

Lies!!!
... Wait, do they? I don't really know what the most popular conventions are for each field.

Very true. Fortunately, this seems to wear off as one gets further in his mathematical training; for instance, I don't believe that anyone in my current math course believes that there is any philosophical problem with the Banach-Tarski Paradox, but many people in my earlier courses couldn't accept the idea that the positive even integers are equinumerous with the integers. They are both just consequences of the definitions and axioms being used. If the definitions and axioms are useful and apply to the situation, then use them. Otherwise, use something else that is applicable.

There is a chance I may be TAing for the intro level proof course at my university over the summer (first time teaching math, so I'm excited, and I do really hope I get to TA for that as opposed to Calculus I or II), so I've been thinking lately about how I would have to explain this to the students.

18. Apr 29, 2008

### Howers

Fine let n be an integer hehe. No, I think odd and even only applies to integer.

If $(1 + \sqrt{17}) / 2$ is even, why isn't 3 seeing as we can write 3/2?

No, zero is not natural. At least thats what I learned. Whole numbers include zero, natural numbers don't. Integers include negatives. But I think at one point you stop making the distinction.

Not anymore :)

Last edited: Apr 29, 2008
19. Apr 29, 2008

### CRGreathouse

I was just saying that there are different definitions of natural numbers, one including and the other excluding zero.

20. Apr 29, 2008

### Howers

Cursed zero. Makes life so hard.