0.3333.... an infinite number of times is exactly one third.
0.6666.... an infinite number of times is exactly two thirds.
0.9999.... an infinite number of times is exactly three thirds. But three thirds is equal to one.
With me so far? Good.
Someone wrote about this in the Guardian, saying his daughter's maths teacher had explained this to him. The comments section is worth reading if you want an understanding of the way people argue about things they don't really understand, especially online.
Several people build straw men. One attempts to prove to us all that 0 followed by a finite number of 9s is not equal to 1:
How about trying your number games with money. 0.9 pounds (90p) is equal to x. y is equal to 1 pound. Think of a number greater than x but less than y. Easy, 99p. i.e. 0.99 pounds. Even better, 0.9 million pounds. is it equal to 1 million pounds? if you think so, then I will gladly write you a cheque for 0.9 million pounds in exchange for your cheque drawn up to the sum of 1 million pounds.One person appears not to realise that (y-x) can equal zero in
"Can you think of a number that is higher than x but less than y? No, you can't." Yes, I can. x + (y-x). Teachers eh?Another person attempts a stumbling explanation of why he believes that the numbers are different, and ends up by explaining:
All a bit of bollocks really - as usual, when you get what looks like a paradox, it's because your starting conditions are incorrect - in this case it's because infinitely recurring numbers don't actually exist. (my emphasis)Yes, folks, you've never divided anything into thirds, because the number one-third doesn't actually exist. And then there's someone who writes in support of that comment using the age-old argument of questioning the sexuality of anyone who disagrees with him:
I for one will take [this user]'s common sense mathematics over any amount of pointy headed claptrap from some four-eyes who’s probably working on a thesis about “lesbian algebra” (grant aided no doubt).