I suddenly recall something interesting a professor of mine pointed out a couple years ago while on a tangent during lecture. It has to do with the nature of infinity and how accepting something perfectly reasonable as true leads to less intuitive, but equally true conclusions.
The following expression is true and most people would not argue otherwise:
1/3 = .3333333333 . . .
Assume, of course, that there is an infinite number of 3s trailing the decimal point. The following expression is also true, and even fewer people would argue otherwise:
1/3 + 1/3 + 1/3 = 1
This may seem obvious, but what may be less obvious is what follows logically from the two expressions above:
.333333 . . . .333333 . . . + .333333 . . . _______________________ .999999 . . . = 1
You may be reluctant to accept that the third expression is true, but if you accept the first two expressions, there is no avoiding it. Most people with a background in computer science or mathematics probably won’t be blown away by this, but it is fun, nerdy thing to point out to your friends in the humanities department.
Actually, I don’t quite agree with the first expression, 1/3 = .3333333333…
I say this because if there were a number of infinite threes repeating themselves after a decimal point, it would still never quite equal 1/3, as 1 would always come up short by that .000001. Thus 1/3 cannot be accurately represented in decimal form, IMO.
If it were, it would have to be something like .3333333…. with a 1/3 fraction at the end.
I think it helps if you think of it in terms of limits. 1/3 – .3 is some value greater than 0. The same goes for 1/3 – .33 and 1/3 – .33333. As the number of 3s after the decimal increases the difference between that number and one third approaches zero. So the limit of that expression as the number of 3s goes to infinity is zero.