Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?

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."

Solution 1:

Mathematicians are strange in that we distinguish between "impossible" and "happens with probability zero." If you throw a magical super sharp dart at the number line, you'll hit a rational number with probability zero, but it isn't impossible in the sense that there do exist rational numbers. What is impossible is, for example, throwing a dart at the real number line and hitting $i$ (which isn't even on the line!).

This is formalized in measure theory. The standard measure on the real line is Lebesgue measure, and the formal statement Seife is trying to state informally is that the rationals have measure zero with respect to this measure. This may seem strange, but lots of things in mathematics seem strange at first glance.

A simpler version of this distinction might be more palatable: flip a coin infinitely many times. The probability that you flip heads every time is zero, but it isn't impossible (at least, it isn't more impossible than flipping a coin infinitely many times to begin with!).

Solution 2:

Note that if you randomly (i.e. uniformly) choose a real number in the interval $[0,1]$ then for every number there is a zero probability that you will pick this number. This does not mean that you did not pick any number at all.

Similarly with the rationals, while infinite, and dense and all that, they are very very sparse in the aspect of measure and probability. It is perfectly possible that if you throw countably many darts at the real line you will hit exactly all the rationals and every rational exactly once. This scenario is highly unlikely, because the rational numbers is a measure zero set.

Probability deals with "what are the odds of that happening?" a priori, not a posteriori. So we are interested in measuring a certain structure a set has, in modern aspects of probability and measure, the rationals have size zero and this means zero probability.

I will leave you with some food for thought: if you ask an arbitrary mathematician to choose any real number from the interval $[0,10]$ there is a good chance they will choose an integer, a slightly worse chance it will be a rational, an even slimmer chance this is going to be an algebraic number, and even less likely an transcendental number. In some aspect this is strongly against measure-theoretic models of a uniform probability on $[0,10]$, but that's just how life is.

Solution 3:

I think the author is exaggerating a bit in order to convey the idea. This is more clearly noticed with the phrase "the infinity of the rationals is nothing more than a zero", which is certainly not true when taken literally. What does happen, as Qiaochu says, is that the Lebesgue measure of the set of rational numbers is zero, because it's a countable set, and the probability of getting a rational number when picking a random number on the real line is indeed zero. However, that doesn't mean it's not possible to get a rational number; you can get "really lucky" and pick any of the infinite rational numbers. However, it's very unlikely, in a specific sense that you will learn from measure theory and probability theory.

Solution 4:

No.
The probability of hitting a specific number is 0, whether it's rational or not. However, when we throw the dart, we'll inevitably hit a specific number. Thus hitting this specific number was not impossible.