The other day, I was suddenly struck by the question, “Are there an even number of even numbers?”. I mean, is it something that is known?, can it be calculated?, and then I went on to worry about the rather obvious follow-up question involving odd numbers.
There has actually been a lot of confusion about this over the years. The mathematics of evenness and oddness is called Parity. And it was really confused by the Greeks. The rule is pretty simple: if you pick a number and divide it by 2, and there is no remainder, it is even. So 4 is even because 4 divided 2 is 2. 3 is not even (and therefore odd) because 3 divided by 2 is 1 and a half. 2 divided by 2 is 1 and so 2 is even…
‘Whoops, hold on’, said the Greeks. ‘1 isn’t a number. So 2 can’t be even.’
Why didn’t the Greeks think that 1 was a number? Because they hadn’t discovered the concept of zero yet.
Basically, because the Greeks didn’t understand the concept of zero, they had to make 1 do all kinds of complicated stuff to try and get around the fact that zero didn’t exist. So they figured the simplest way to get around this was to declare 1 not a number. Therefore the Greeks believed that neither 1 or 2 were odd or even.
But then some smartypants in India started saying zero was a number for sure, and they had the maths to prove it. So if zero was a number then 1 was definitely a number, so suddenly 2 became even and 1 became odd.
But what about zero? Presumably, some of the previous concern about whether zero was a number led to many thinking that it was neither odd nor even. But a lot of people still believe, for some hard wired reason, (perhaps because it is lower than 2) that it must be odd. But it isn’t, it is even. If you take 0 and divide it by 2 then you get no remainder, you get zero.
Also, it helps with the symmetry. If you line the numbers up including minus numbers you will see that it’s really good to have an even number there between -1 and 1.
So now we know that, we can answer the question. Between 1 and infinity and between minus 1 and minus infinity there are the same number of even and odd numbers. There must be. So everything is in pairs (or parity). But we know there is 1 extra number, the number zero. And zero is even.
This means that there are an odd number of even numbers and an even number of odd numbers. Weird huh?