# Zero - even, odd, or neither?

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!

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Well, what dispute would that be?

freerangequark
Well, what dispute would that be?

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.

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

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

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

Another dispute I read is whether zero is a number at all and therefore eligible to receive an even or odd tag.
Why shouldn't 0 be a number?

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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.

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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.

_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.

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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..)

Howers
I'd say even.

Even = 2n
Odd = 2n - 1

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

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If n=1/2..

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

Howers
If n=1/2..

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

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

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

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

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

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

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

LukeD
Depending on your definition of natural, yes? Most analysts take them as {0, 1, 2, ...}.
Lies!
... Wait, do they? I don't really know what the most popular conventions are for each field.

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.
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.

Howers
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:

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?

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

No, zero is not natural. At least that's 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.

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

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

Not anymore :)

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No, zero is not natural. At least that's 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.

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

Howers
Cursed zero. Makes life so hard.

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If $(1 + \sqrt{17}) / 2$ is even, why isn't 3 seeing as we can write 3/2?
The rationale I used would say that 3/2 isn't even -- but 2/3 is. I chose this particular example for maximum 'absurdity': to most dramatically emphasize that different contexts define their words differently.

The relevant fact here is that $(1 + \sqrt{17}) / 2$ is actually an algebraic integer, and it's prime factorization in (the algebraic integers of) $\mathbb{Q}{\sqrt{17}}$ has a prime factor lying over 2. When working in this context, I might want to consider such a number even.

Another one is that, both possible values for that expression are 2-adic integers, whose binary expansions are:
...10101,
...01100,
and we see the second such value is divisible by 4.

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After glancing through several basic algebra and "precalculus" books, they all define "natural numbers" to be the same as the "counting" numbers: 1, 2, 3, ..., not including 0. The "whole numbers" then include 0. The "integers" includes all negatives.

Apparently, it used to be common to include 0 in the natural numbers: Peano's axioms, in particular, included 0 originally.

However, any definition of "even" number that I have ever seen included all integers: an integer, n, is "even" if and only if it is of the form n= 2m where m is some integer. 0 qualifies for that since 0 is an integer and 0= 2(0).

DrBoom
Zero isn't natural ?
I was pretty sure that it is...or..was...at least when i learned the natural numbers...and i do remember of N* which are the natural numbers excluding zero...so did anything changed recently or i actually got that wrong for all this time ?

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0!+0!=2

That's just weird.

I never really liked zero as a number. I preferred to think of it as an absence of number.

I had zero cars and multiplied them by 5. I still had zero cars.

I had five cars and got greedy and multiplied them by zero, I lost all my cars.

Dang that zero. :(

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LukeD
Zero isn't natural ?
I was pretty sure that it is...or..was...at least when i learned the natural numbers...and i do remember of N* which are the natural numbers excluding zero...so did anything changed recently or i actually got that wrong for all this time ?

The assistant head of the math department and a well liked assistant computer science professor at my university have long debated whether or not 0 is a natural number (the asst. head of the math department contends that it is not while the asst. computer science professor claims that it is). The asst. CS professor has challenged the other to a game of Wii Tennis to determine the answer.

As you can see, some people take the question very seriously.

(That last statement may or may not have been a non sequitur)

DrBoom
As far as i got it...there is no "answer"...or...no "correct answer"...only opinions

And making a connection between the tennis game and the affirmation "natural numbers are counting numbers"...consider a tennis game...or any game it really doesn't matter...in which one team ends with 0 (zero) points...if someone asks you "how many points did each team have at the end of the game" will you say "one team had X points, but i couldn't count the other team's points.." (considering 0 is not a "counting" number)

Maybe what i said was pretty stupid...but anyway i tought it's weird..interesting..or whatever

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Apparently, it used to be common to include 0 in the natural numbers: Peano's axioms, in particular, included 0 originally.

I thought Peano's original axioms didn't even include 0 as an element...?

Santa1
I'm used to seeing $$\mathbb{N}$$ as the positive integers, and $$\mathbb{N}_0$$ as the non-negative integers, but that's in number theory. (Not to be taken as fact by the way, as I'm sure you are aware.)

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