

Is 0 Odd or Even? - yarapavan
http://en.wikipedia.org/wiki/Evenness_of_zero

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nearestneighbor
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"

~~~
jacquesm
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 7 _6 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.

~~~
nrr
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>)

~~~
jacquesm
It seriously pissed me off, especially the fact that this guy was allowed to
teach a subject he didn't know anything about.

Calculators have their uses, simple sums are not what they were made for. Fwiw
this whole thing played out in rural Canada, Algoma district. I think they
must have thought me to be an exceptionally cruel parent for wanting to teach
my child how to do sums in his head or on paper for slightly more complicated
ones. Fantastic piece by the way, that pdf, you should post that separately.

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Dove
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!

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

~~~
jfarmer
Not really. The set of even integers is 2Z = {2 _k | k is an integer}. 0 is in
that set.

The set of odd integers is 2Z+1 = {2_k+1 | k is an integer}. 0 is not in that
set.

The reason we don't like 1 being prime is because it makes stating lots of
theorems that involve primes (in particular, the fundamental theorem of
arithmetic) really tedious. Instead of saying "For every prime p" or "Let p be
a prime" you'd have to say "For every non-unitary prime," etc.

Zero being even is a consequence of the above definition of even, while one
not being prime is a choice for our sanity.

~~~
kwantam
_The reason we don't like 1 being prime is because it makes stating lots of
theorems that involve primes (in particular, the fundamental theorem of
arithmetic) really tedious._

No, it makes the FTA wrong! If one were prime, then every number would have
infinitely many prime factorizations instead of one unique one.

 _you'd have to say "For every non-unitary prime," etc._

In the even/odd case, one could just as easily say "for every non-zero odd
number." This isn't somehow an argument that applies only to the question of
one's primality.

 _Zero being even is a consequence of the above definition of even_

...as is one not being prime a consequence of the definition of primes. A
prime is a number with exactly two distinct factors. One only has one distinct
factor, therefore it is not prime.

If you don't buy this definition, or if you claim that I'm gerrymandering the
definition to suit my preference that one not be prime, be aware that a
reasonable person could make the same claim about the definition of even
numbers, viz., that it was chosen for convenience. While it's might be
entertaining to argue about such chicken-and-egg problems, at the end of the
day the evidence is pretty clearly in favor of the properties proceeding from
the definition in both cases.

~~~
nrr
Hmm, I wouldn't say that defining a prime number as a number n with prime
factors 1 and a with 1 and a not necessarily distinct instantly breaks the
FTA, but it sure does make it a lot harder to state while still conveying the
same idea otherwise.

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

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ez77
Good luck suggesting that 2 is prime and 1 isn't...

