@mike4 wrote:
I think she has it exactly backward. You can only hit rational numbers on the number line.
Consider 1/2, that is the ratio of 1 to 2, not one divided by two. We can hit that with a dart every time, if you mean by "hitting with a dart" "retrieving the number's value".
It's the irrationals that you can't represent exactly, except symbolically. You can find a rational number as close as you please to an irrational number, but it will be only be the sum of a finite part of the infinite sequence in every case.
Some rational numbers (e.g. 1/3) must be represented as an infinite sequence of repeating digits in base 10, but that is irrelevant. 1/3 is a finite concept. Any number of infinite sequences will sum to 1/3, but 1/3 remains simple and finite.
