Hacker News Reader

Part of being a great teacher, I think, understanding where your students trip up and their wrong thought processes. Next time, you teach the course, perhaps you might want to ask that person their train of thought.

I think it is based on mistaken beliefs like these :

* 0 is not a number, it's the absence of a number. Therefore, standard number properties like parity does not apply.

* All even numbers are twice another number, which is not true of 0.

Also, very early on, people learn that 0 is "weird", e.g. you cannot divide by it. Also there's something that irks the common psyche about treating the statement "this set has zero elements" similarly to "this set has 5 elements".

This doesn't show the stupidity of the general public (or some of your students), it's just that 0 is among those abstract mathematical concepts (like infinity) that you just have to get used to. Remember that it was invented much later than natural numbers.

P.S. People also see other weird conventions (e.g. 1 is not a prime, 0!=1) and may think that this is another one.

Who says infinity is a number?

http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n...

http://nrich.maths.org/2756

http://mathforum.org/dr.math/faq/faq.large.numbers.html

http://www.kuro5hin.org/story/2003/6/3/95744/71866

Once we are confident that infinity is not a number, why would we expect it to have parity?

> Who says infinity is a number?

Not I:

> Oh, or is it maybe a confusion with infinity, which, if we were to try to classify it, would have some claim to being both?

… although the old maxim “What's a number?” “Whatever you treat as a number” (which I encountered in Linderholm's wonderful book (http://www.amazon.com/Mathematics-made-difficult-Carl-Linder...) ) means that the various flavours of infinity, considered as ordinals or cardinals, have some claim to being numbers; and it is often helpful in measure theory, and my own work, to treat even plain old infinity as being a number on at least as good a footing as any other (positive real).

> Once we are confident that infinity is not a number, why would we expect it to have parity?

Why not?

Parity is a property of numbers and sets only. Or do you think infinity is a set?

> Parity is a property of numbers and sets only.

Why? Seriously, though: what do you mean by parity (EDIT: I mean even-ness) of a set? Almost assuredly, something like “it can be partitioned into two disjoint subsets that can be put into bijection”; but then you've just defined even-ness of cardinals (which are sets after all), and all infinite cardinals are even (in fact, satisfy 2κ = κ; EDIT: and odd; they also satisfy 2κ + 1 = κ).

You read my mind for "parity of a set". So Q.E.D.: parity is only applicable to reason about numbers (and infinity is not a number). See also surreal numbers.

> Parity is a property of numbers and sets only.

It's a property of _integers_, or any system with a natural homorphism into the integers mod 2 under addition. It's certainly not a property of numbers in general. Is pi even or odd? How about sqrt(2) + i*e?

Infinity is certainly not a real number, nor a complex number. There are plenty of systems (such as various compactions of the real or complex numbers, like the Riemann sphere, or the surreal numbers) for which infinity is a member in good standing, and then there is no reason not to consider it a number.

Although it's always dangerous to say to a mathematician something like “But there's no sense in which that's true!”, still I would be reluctant to refer to elements of the Riemann sphere as numbers; the sphere seems fundamentally a geometric rather than algebraic object (of course, until you start thinking of it as the complex projective ‘line’, at which point it becomes both :-) ).

With that said, as I have argued elsethread, I certainly do think that there are occasions when infinity can be considered a number; the surreal numbers are an elegant example.

I think the confusion is simple. They believe that an even number is 2n, but think about n over the naturals. Often they'll also be confused about -1.

This wouldn't be a problem if people would stop thinking that the natural numbers start at 1.

That's probably the most popular way it is taught. And whole numbers starting at 0. Although oddly some people will learn in the complete opposite way.

We could just throw away these short hands and just integers, positive integers, negative integers.

Which actually led me to another potential reason for the confusion. Is zero positive or negative? OK, now is zero even or odd? I can imagine some just see zero with weird properties.

    We could just throw away these short hands and just [use]
    integers, positive integers, negative integers.

This depends on who you consider "we" to be. If we're talking about computer programmers, in most languages integers are all you get: natural numbers aren't available (although the Haskell community want them [1]).

On the other hand, from the perspective of mathematics, the natural numbers are very different to the integers. For one thing, they're wellordered whereas the integers are not. Obviously a set-theoretic definition of the natural numbers (as the finite ordinals) will start at 0 since it's just identified with ø. Set-theoretic definitions of the integers are typically in terms of equivalence classes of pairs of natural numbers (or arbitrary elements from these equivalence classes if you want to work in ACA0) [2].

[1] http://hackage.haskell.org/trac/haskell-prime/wiki/Natural

[2] See SoSOA p.10 for details. http://www.math.psu.edu/simpson/sosoa/

> Is zero positive or negative? OK, now is zero even or odd? I can imagine some just see zero with weird properties.

I think that's the best explanation I've heard. Thanks! (Also you can't divide by it (although (http://www.bbc.co.uk/berkshire/content/articles/2006/12/06/d...) …), so who knows what trickery it's up to?)

> Often they'll also be confused about -1.

Hmm, I've never encountered that—which is not to say you're wrong. I'll bet you'd have a hard time finding someone post-secondary school who wouldn't answer instantly that -2 was even, though.

You must not be from the US. Ask random adult on the street if 2,229 is even or odd. I think you might be surprised at how many people get that wrong. Not that I've ever done it, but I suspect that you'll get at least a couple in ten who don't know.

Indeed I am from the US. I guess I would be surprised to hear people get that one wrong, but I'd be truly shocked to see someone get the classification of -2 wrong.

(A disclaimer: The only people with whom I've had occasion to have this conversation have been students in (my) college mathematics classes, so I guess I should clarify that these are the people of whom I'm really thinking.)

Perhaps they learn that "even" is equivalent to "divisible by 2", and then they go on to conjecture that "odd" is equivalent to "divisible by 3", or maybe "divisible by 1" or "divisible by at least one odd prime"?

Without meaning to be unduly cynical, I suspect that's attributing to the average person altogether too much mathematical inquisitiveness and willingness to conjecture; but I'd love it if your explanation, especially the last, were the correct one.

EDIT: Incidentally, this reminds me of one of my favourite pedagogical techniques, which I was heartbroken to discover recently is not original to me. After discussing even and odd numbers, as a very basic introduction to modular arithmetic, I like to introduce and discuss ‘threeven’ and ‘throdd’ numbers, and discover that they don't behave quite as nicely—what to do, what to do ….

If you can write is a 2n for some integer n, then it is even. If you can write it as 2n + 1 for some integer n, then it is odd. Therefore, zero is even and not odd.