Is Zero a Square Number?

In my faculty we have an unofficial lunchtime meeting every Thursday. What do you think happens when you put several cleverer-than-thou maths enthusiasts together in one room for forty minutes, along with some "normal" people who sit back and observe detatchedly, for fear that their views will be immediately shot down as they have not used phrases like "corollarlarily" or "without loss of generality" in expressing their argument?

You get some pretty heavy meta-mathematical debates which question the very core of mathematics.

You also get some pretty pointless debates, like: is zero a square number?

Now obviously the answer to this depends on how you define a square number in the first place. Here are some of the definitions my department discussed:

*A number, n, that can be written in the form n = a x a, where a is a natural number (slight variation, where a is an integer).

*A number n for which there exists a square with area n square units, and whose length is a whole/natural number.

*A number whose square root is a whole number.

None of these definitions is particularly satisfying. When I'm teaching my students about square numbers, I tell them that if you times a number by itself the result is a square number. The implication is that by "number" I mean natural number. When I list the square numbers, I never include zero. However, I feel pretty strongly that zero is a square number.

Take definition number one. Does zero satisfy 0 = a x a where a is natural? I say yes. Other members of my faculty say no. I believe that zero is a natural number. They do not. Why do I believe this? Well my professors at uni, for the most part, included zero. I know this because when they didn't want to include zero they'd put a little + next to the blackboard-bold N. If you define natural numbers as a "counting number", ie a number that could be the cardinality of a set, then zero is without a doubt a natural number. How many cows are there in this classroom? Zero. I have counted the cows in the room and the answer is zero.

On to the next definition: does there exist a square with area zero and natural-numbered sides? Leaving aside the "is zero natural?" debate, we first have to deal with: can there exist a square with area zero? My immediate response was, "Yes! There's one right here! In fact, I've got twelve of them!" (I was a bit hysterical at this point. You must bare in mind this was the last day of term). But it's an interesting question. It's almost like, "What is the sound of one hand clapping?", a Zen koan that for some reason doesn't interest me at all (hey monks, listen up: the answer is NO SOUND. Done.)

And the third definition. This one was offered by probably the most down-to-earth of the faculty. despite having a maths degree, she's somehow managed to maintain a grip on reality other maths grads signed away during Freshers' Week. If you square root zero, do you get a whole number? Yes. You do. Zero is a whole number. But using "whole number" didn't feel precise enough. Which then led to another debate: integer versus natural versus positive natural. Is i a whole number? If it is, then -1 would also be a square number. Can you have a square with an area of -1? Why not?

What do you think? Do you have another definition of square number that would settle this argument? Do you think zero is a natural number?

Extension:
Why is 1 a triangle number? You can't draw a triangle with one dot. Are the triangle numbers defined by the picture or by the formula (1/2)n(n+1)? And hence would zero be a triangle number?

Enjoy the Easter break!

Emma x x x