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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."

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    @ThomasE.: Well, then it's pointless to discuss a more complicated example.2019-04-22

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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!).

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    @Xnyyrznaa: you are welcome to write your own answer if you feel there is something lacking in the existing answers. I'm not sure how to respond to your accusation that I am trying to accrue upvotes. I certainly don't think this answer is worth 48 upvotes, but in answering questions on SE sites I think a philosophy of "rapid prototyping" makes more sense than anything else. Can you be more specific about what you find confusing about this answer?2012-06-09
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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.

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    (cont.) You could argue that I did not specifically name any number at all, but then again... what is a name of a number? I could augment the language by a constant $m$ and add the countable set of axioms which say that $m$ is not any of the numbers definable by the aforementioned functions, now $m$ itself is definable but in a richer language, and not in the original. So we're still fine! :-)2012-07-03
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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.

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    @MichaelBoratko It has everything to do with the reason you can't hit 0 with a dart. Obviously, your dart could hit 0 to within a millimeter, or a nanometer, or a picometer. In fact, the dart could hit within any arbitrary precision you specify, and if that's the best you can measure, you'd say that the dart hit 0. But if you could measure more precisely still—adding more and more digits—eventually you'd find that you had not hit 0 after all. (But this cannot happen in the real world, because of Heisenberg uncertainty. So in fact, in the real world, you will ALWAYS hit rationals.)2013-04-14
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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.

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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."

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    If only we where that rational, then the stock markets would not have crashed. But the probability of throwing a dart in to a trading pit at the stock market (or your pension providers office), and hitting a rational is also zero.2012-06-08
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It's possible, but it would take you an infinite amount of time to verify that you actually hit a rational number because you would have to keep "zooming in" forever.

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    @Thomas: okay, but it would also take an infinite amount of time to verify that you hit an _irrational number_ in this framework, and the probability of that happening is still $1$. This issue is unrelated to the issue the OP is asking about.2012-06-08