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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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If the probability that an event occurs is 0, that (counterintuitively) does not mean that it is impossible. This Wikipedia article may help. –  Zev Chonoles Jun 7 '12 at 14:34
food for thought : if you take a random real number between 0 and 1, the probability to get that number was 0. and yet you got it ! that should explain intuitively why we need to give zero probability to some events that can actually happen –  Glougloubarbaki Jun 7 '12 at 14:43
Hitting any individual number has probability zero. So with this logic we will never hit any point when throwing a dart. Can you see the flaw in this? Eventhough rationals cumulate zero mass on the real line, this doesn't mean that they don't exist. –  Thomas E. Jun 7 '12 at 14:45
Ok, so what happends if you throw uncountable many darts? –  user1708 Jun 7 '12 at 14:53
I think a valid answer to all those questions is : measure theory is just a mathematical theory, it's not perfect and while it's useful, you can't pretend to have an acurate description of physical reality. actually, the mere concept of "is going to happen" or "cannot happen" is already a heavy philosphical assumption on the nature of the future and the possible... –  Glougloubarbaki Jun 7 '12 at 15:17

6 Answers 6

up vote 63 down vote accepted

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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@regularmike: well, one can interpret this as a little artistic license on the author's part, and in any case the English language is not precise enough to make the kind of distinctions that mathematicians make (it is hardly precise enough to make the kind of distinctions that non-mathematicians make!). It depends on whether you interpret "never" as "impossible" or "probability zero." –  Qiaochu Yuan Jun 7 '12 at 14:48
@regularmike: yes. For any finite number of throws, the probability that any of those throws hits a rational number is still zero. We could fill the universe with dart-throwing nanites and force them to throw darts at a number line a Planck length away at the speed of light until the heat death of the universe and the probability would still be zero. (This is all, again, under the "magical super sharp dart" hypothesis.) –  Qiaochu Yuan Jun 7 '12 at 15:07
@regularmike: there's no way to actually find the "exact center" of a physical dart. Once you zoom in too far Heisenberg's uncertainty principle kicks in and you can't have complete knowledge of the position of all of the atoms in the dart without having zero knowledge of their momenta... more generally, exact positions don't really have physical meaning. –  Qiaochu Yuan Jun 7 '12 at 15:46
I’ve gotten the feeling that physicists too don’t understand the difference between “probability zero” and “impossible”. –  Lubin Jun 7 '12 at 21:06
@Lubin: well, for physics the distinction is pretty much irrelevant. –  Qiaochu Yuan Jun 7 '12 at 21:15

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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Dear Noah, "smallest amount" should probably read "largest amount", and I guess this is correct when the probability of $X$ is small. Regards, –  Matt E Jun 8 '12 at 2:39
Thanks for the correction. Not sure where small would come in... I think the technical bit that I don't have totally right is that it's best phrased as "there's no way for you to beat the house betting on either side" rather than what you would be willing to play. –  Noah Snyder Jun 8 '12 at 3:13
Dear Noah, I think I'm just wrong regarding "small" (I got confused), so ignore that! Cheers, –  Matt E Jun 8 '12 at 3:20
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. –  richard Jun 8 '12 at 10:11

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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True. Though I daresay if you ask an arbitrary mathematician, the chance is actually not so slight that they'll choose a nonrational number, most likely one of the obvious $\sqrt{2},e,\pi,2\pi,\ldots$. –  leftaroundabout Jun 7 '12 at 17:55
We can run an experiment, it's not too hard. –  Asaf Karagila Jun 7 '12 at 18:04
What do you mean by 'highly unlikely'? –  user1708 Jun 7 '12 at 20:47
The last paragraph is, of course, a comment on mathematicians, not mathematics, but it's highly pertinent. Short form: the human brain is a terrible uniform random number generator. It doesn't visualise distribution well. Which is also why it's so hard to understand that the probability of hitting a rational being 0 means that you'll never hit one with a finite number of darts - your brain can find rationals easily, so it has no intuitive grasp of what it means that they are sparse. –  Tynam Jun 8 '12 at 12:49
(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! :-) –  Asaf Karagila Jul 3 '12 at 21:41

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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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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If we're already hypothesizing that we can somehow work with a physical number line that actually contains all real numbers and also hypothesizing that we have darts sharp enough to hit a single number on this line, why not hypothesize that we can verify that we did or did not hit a rational number instantaneously? –  Qiaochu Yuan Jun 7 '12 at 20:57
If you're using a physical number line then you would always hit a rational, because physical space is quantized - there would always be an integral number of atoms (or planck lengths perhaps) on both sides of the point you hit. –  evil otto Jun 7 '12 at 22:30
@QiaochuYuan I was assuming that the only hypotheses being made were that the dart was sharp enough to hit a single point, that the number line contained all real numbers, and that neither of those hypotheses implied that it would be possible to verify if the dart hit a rational in a finite amount of time. I was trying to offer insight as to why one might think a dart could never hit a rational, because the intuitive thing to do after the dart hits is to check to see where it hit. –  Thomas Jun 8 '12 at 0:43
@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. –  Qiaochu Yuan Jun 8 '12 at 1:37

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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I don't think they are exagerating--the accepted answer seems correct. Another way to think about it, how do you write the number "0" exactly? "0" is just an abrevation/truncation. It's 0.00000... but you have to keep on writing. Any time you write anything but a zero, you can stop (because n != 0) but you'd have to write infinitly long for it to actually be zero, and nobody can write infinitly long, so you can't actually even truly write "0", how could you hit it with a dart? –  Bill K Jun 8 '12 at 0:30
@BillK If your argument held, one could say the same for any real number and then it would be impossible to get any real number by picking a random real number, but my dart must hit somewhere, right? It is certainly possible to pick any one of the elements in $\mathbb{R}$ (and not $i$ or $x^2 + 3x$ for example), but each one has probability zero. This is a counterintuitive idea, but if each real number had probability $\epsilon \gt 0$ (all numbers should have the same probability of being chosen), then the probability of picking any number, summing all probabilities, would not be $1$. –  talmid Jun 8 '12 at 2:10
@BillK And yes, the author is exaggerating. "the infinity of the rationals is nothing more than a zero" is a nice example because it specifically says $\aleph_0 = 0$, if taken literally. So I understand the author is speaking figuratively. One could say, without exaggerating: "the infinity of the rationals is nothing more than that of the natural numbers". –  talmid Jun 8 '12 at 2:12
@BillK "0" is not an abbreviation/truncation for 0.00000..., the two are equivalent but this is for an entirely different reason and has nothing at all to do with the reason you can't hit 0 with a dart. –  process91 Jun 8 '12 at 3:38
@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.) –  Kundor Apr 14 '13 at 0:22

protected by Qiaochu Yuan Jun 7 '12 at 20:54

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