TITLE: Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?
QUESTION [70 upvotes]: In the book "Zero: The Biography of a Dangerous Idea", author Charles Seife claims that a dart thrown at the real number line would never hit a rational number. He doesn't say that it's only "unlikely" or that the probability approaches zero or anything like that. He says that it will never happen because the irrationals take up all the space on the number line and the rationals take up no space. This idea almost makes sense to me, but I can't wrap my head around why it should be impossible to get really lucky and hit, say, 0, dead on. Presumably we're talking about a magic super sharp dart that makes contact with the number line in exactly one point. Why couldn't that point be a rational? A point takes up no space, but it almost sounds like he's saying the points don't even exist somehow. Does anybody else buy this? I found one academic paper online which ridiculed the comment, but offered no explanation. Here's the original quote:

"How big are the rational numbers? They take up no space at all. It's a tough concept to swallow, but it's true. Even though there are rational numbers everywhere on the number line, they take up no space at all. If we were to throw a dart at the number line, it would never hit a rational number. Never. And though the rationals are tiny, the irrationals aren't, since we can't make a seating chart and cover them one by one; there will always be uncovered irrationals left over. Kronecker hated the irrationals, but they take up all the space in the number line. The infinity of the rationals is nothing more than a zero."

REPLY [4 votes]: One very useful way to think about probability is in terms of betting.  Suppose someone offers you a payoff of 1 dollar if event X happens, and 0 dollars if event X does not happen.  What's the largest amount of money that you're willing to pay to play this game?  That amount is the probability of X happening.  (Probably I need to be a bit more careful, but this is roughly the idea.)
So what does it mean to say that an event has probability zero?  It doesn't mean that it can't happen, it just means that you wouldn't be willing to play that game for 1 cent, or a tenth of a cent, or any actual non-zero amount of money.
If you want to read more about this way of thinking about probability, you can search for "Dutch book."