Is 0 Odd or Even?
en.wikipedia.org
en.wikipedia.org
[Prospective elementary school teachers] found it to be
a "tricky question", and about two thirds answered "False"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.
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)
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.
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!
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?
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.
The set of odd integers is 2Z+1 = {2k+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.
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.
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.
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)
That happens in Sydney
For unsuspecting tourists traffic in Italy can be quite taxing.
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.
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.
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.
Ummm I get 0. It surely depends on whether the layman makes a logic choice about his ability to remove the final 2 apples.
EDIT: by the way the other flaw in that example is you need a definition of "even" to start with; otherwise you could begin with 9 apples. :D
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.
1/inf != 0