Is 0 Odd or Even?

http://en.wikipedia.org/wiki/Evenness_of_zero

by yarapavan • 17 years ago

18 28 17 years ago

en.wikipedia.org

kinghajj• 17 years ago

My computer science teacher has a saying, "Mathematicians are always right." Why is zero even? Well, because it's defined to be even. If you try to bring personal intuition into mathematics, you'll just repeatedly shoot yourself in the foot; there's no use arguing against it.

I had a similar problem with accepting that 1/inf = 0; to me, there could always be a smaller number, and therefore would never reach zero. Calculus made me realize I was wrong, and now I don't even bother trying to reason why mathematicians decided these things.

EDIT: Man, I knew as I wrote it that someone would get smart-ass with the sentence "Well, because it's defined to be even," but I didn't expect so many. Yes, you're right, zero isn't defined to be even, but zero meets the definition of evenness. Happy?

patio11• 17 years ago

Well, because it's defined to be even.

This implies that defining it to be odd would be an acceptable choice. Defining zero as odd would break almost every generalization you can make about even and odd numbers.

And even number plus an even number is even. An even number plus an odd number is odd. And odd number plus an odd number is even.

Oh, whoops, zero was odd? Sorry, let me rephrase: "And even number plus an even number is even. An even number plus an odd number is odd. And odd number plus an odd number is even. SPECIAL CASE: IGNORE ALL THAT IF ONE OF THE NUMBERS IS ZERO."

Numbers alternate odd, even, odd, even, odd, even. EXCEPT AROUND ZERO.

Even numbers can be written as 2 * k, for k is some integer. Odd numbers can be written as 2 * k + 1, for k is some integer. EXCEPT FOR ZERO.

Even numbers can be divided into two equal parts by two. Odd numbers can not. EXCEPT FOR ZERO.

gloob• 17 years ago

Which is why it was defined as even. Similar reasons apply for why one is not considered a prime number.

dnaquin• 17 years ago

Gotta pick what you're defining and what you're deriving.

teeja• 17 years ago

Arguing along those lines has some interesting results. How can -2 be a number? If you have two apples, and I have none, you can give me two apples. <b>You can't give me -2 apples.</b>

You want zero diamonds? I can SHOW you zero diamonds. You can take zero diamonds from me anytime you want. I'll just make more!

There's something special about 0. And it's right next to 1, which is the mother of all other integers. And zero looks way more pregnant than 1.

Zero is more like the absence of a number (which is why it was invented to be a placeholder). In physics there are electron holes, etc., they're called quasi-particles. There are no magnetic monopoles (yet) but there are quasi magnetic monopoles ... that carry current.

So maybe the day will come when zero becomes a quasi-number. Or better still, a DWARF number. And some people's mathematical kids will write threatening letters to the New York Times.

RevRal• 17 years ago

> "Mathematicians are always right."

A math professor once complained to me that all the cars in line should just go when the light turns green. I could tell this had been bugging him for a while....

Besides that example (that wasn't math related), do take the simple route and agree with the mathematicians, unless you wish to continue studying more and more nuanced math.

(posting here mainly to share the car story)

tjmc• 17 years ago

A math professor once complained to me that all the cars in line should just go when the light turns green.

That happens in Sydney

rms• 17 years ago

>I had a similar problem with accepting that 1/inf = 0; to me, there could always be a smaller number

The issue here (the same issue as the .9999=1 debate) is that people intuitively perceive the hyperreal numbers. It is absolutely not wrong to intuitively perceive the hyperreal numbers; it's not like one number system is more correct than others. http://en.wikipedia.org/wiki/Hyperreal_number

http://theassayer.org/cgi-bin/asbook.cgi?book=770 is a textbook of the calculus constructed rigorously using infinitesimals.

rkts• 17 years ago

Zero is not defined to be even. A number is even if it's twice some integer. Zero is twice an integer (namely 0), so it's even.

If you try to bring personal intuition into mathematics, you'll just repeatedly shoot yourself in the foot

I disagree. Intuition should always be foremost in mathematical understanding. You may have gotten a different impression from your math classes, and that IMO is a consequence of bad teaching.

I had a similar problem with accepting that 1/inf = 0

As you should; the statement is false. The limit of 1/x as x->infinity is zero, but there is no x at which the quotient "reaches" zero. In non-standard analysis, if x is positive infinite, then 1/x is positive infinitesimal, meaning it is infinitely close to, but still greater than zero.

redcap• 17 years ago

"A number is even if it's twice some integer"

You could argue that is the mathematician's definition - I think this is covered in the article to an extent.

A layman's definition would be along the lines of that if you keep taking 2 objects from a collection of objects (e.g. apples), the smallest number you end up with is 2.

The number zero is a very handy mathematical trick that the Romans didn't grasp, I'm no surprised that concepts around it aren't easy to understand as I posit that it's =not= intuitive.

I'd also guess that there are maths teachers in schools who aren't mathematicians or aren't as well educated in maths as they could be.

nrr3• 17 years ago

Be careful with your language here. Zero isn't defined to be even; rather, it has the property of being even by way of the definition of an even number. Usually, this process involves making a conjecture and then proving or disproving it.

On the other hand, definitions are, well, whatever we want them to be. We can define a sheep to be woolly, and we can define a collection of crows to be a murder, but we have to justify conjectures that all sheep are black or that all corvids are crows.

Think about it this way. In Scheme/Lisp, consider the notions of functions and macros. The mathematical analogue to a definition is a function; they're defined in very much the same manner, and the idea is effectively the same. Likewise, the analogue to a theorem/corollary/lemma is a macro. A theorem allows one to traverse a logical chain from P to Q very easily, and a macro allows the Schemer/Lisp hacker to abstract dull, repetitive tasks into short, succinct statements.

The humdinger: the proof to a theorem is analogous to the macro expansion. :)

Now, I'm hoping I'm not being too terribly nitpicky, but 1/inf does not equal zero. It's still undefined. (See also: 1/0 does not equal infinity.) This mess hinges on the idea that infinity is not a number, and in general, it holds true for the kind of real analysis one would typically learn as an undergrad.

kwamenum86• 17 years ago

When I was in school I heard something similar about math definitions. I didn't hear "Mathematicians are always right" or "because it is defined to be even"; Rather, in response to questions of "why is [this] defined to be [that]" one professor simply said "because it is useful." Usually this meant "because it means this property of this system holds true if [this] is defined to be [that]." For example, the first two Fibonacci numbers, fib(0) and fib(1), are defined as 0 and 1 because it makes the Fibonacci sequence work. They are 0 and 1 because it is useful as far as calculation of the sequence. I am pretty sure that in practical applications (por ejemplo, examining the fib sequence in nature) fib(0) has no use.

eggnet• 17 years ago

The limit of 1/x as x approaches infinity is 0.

1/inf != 0

jrockway• 17 years ago

Is this surprising? An integer is even if it is congruent to 0 mod 2. 0 is congruent to 0 mod 2.

nearestneighbor• 17 years ago

I can't believe there's a Wikipedia article about this. The best part:

    [Prospective elementary school teachers] found it to be
    a "tricky question", and about two thirds answered "False"

jacquesm3• 17 years ago

Why does that surprise you ?

I've had a fair argument with my sons school when he was between the ripe old ages of 7 and 10 about teaching him arithmetic using a calculator instead of learning how to do it by himself.

After the third school issued calculator came back to school nicely chopped in half they stopped trying and I managed to teach him what he needed to know, but the really interesting bit here is that once, during all this I quizzed the teacher on his arithmetic skills, hard stuff like 76 and 72/9 and he instantly whipped out his* calculator.

It completely floored me, here was someone with a teaching certificate for all the 'basics' not knowing the basics.

And I was being accused of being a 'Luddite' for not allowing my child to use modern tools.

nrr3• 17 years ago

Whoah! I was actually kept inside for recess in grade school if I tried to use a calculator for arithmetic, and here, your son is being issued a calculator by his teacher.

That doesn't mean that using a calculator at his age is necessarily a bad thing, but with the current math curricula worldwide being what they are, I can't think of anything more pedagogically unsound than what you're describing.

EDIT: (See also: "A Mathematician's Lament" by Paul Lockhart http://www.maa.org/devlin/LockhartsLament.pdf)

greendestiny• 17 years ago

Not just elementary school teachers. I got laughed at in an undergraduate maths class once for asking 'what about zero' in an example that was suppose to be for all even numbers. I actually assumed I was wrong until this submission.

ez77• 17 years ago

Good luck suggesting that 2 is prime and 1 isn't...

Dove• 17 years ago

I don't understand why people have trouble with this. Perhaps because they are so used to zero being an exception to things? Perhaps because the sheer act of raising the question makes them nervous?

The everyday definition of even -- divisible by two -- applies just fine to zero, nor are there properties of evenness or oddness that should make one doubt whether the definition is appropriate. Questioning whether zero should be even is like questioning whether six should be even. It meets the definition, and there are no issues, so what motivates the concern?

Of course, one could always make another definition, but what you'd really be doing at that point is appealing to the mathematicians around you that you have a better definition of even. To them, that means one that makes things simpler. And it's unlikely that you have one.

I once had the following conversation with my abstract algebra teacher, while in grad school:

Him: If you study the history of this particular field, as time goes on, the definitions get more complex, while the theorems and proofs get simpler.

Me: Why is that, do you suppose?

Him: It's progress!

Is 0 Odd or Even?

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