Wikipedia:Reference desk/Archives/Mathematics/2006 November 11

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November 11

Hamlet written into pi

I was thinking about the Infinite monkey theorem and about the number pi. If pi is infinite does that mean that it has every combination of numbers possible? If it does that means that it has it could have the entire script of Hamlet written in binary. So somewhere along the infinite decimals of pi there are a bunch of 1's and 0's that are in order to spell out Hamlet completely. Ed Dehm 18:16, 11 November 2006 (UTC)

I seriously doubt that. I see where you are coming from, but pi has numbers from 2-9 as well. It is very likely that a number is going to break the binary chain somewhere along the line. It's statistically possible, but HIGHLY improbable. PullToOpen ^talk 18:24, 11 November 2006 (UTC)

Actually, from what I remember reading elsewhere the infinite monkey theorem also suffers from the "statistically possible, but highly improbable" problem. On a related note, read the end of Contact for a more interesting (IMHO) related example (at least from a fictional mathematical perspective). Virog_{It's not}^{my fault!} 18:53, 11 November 2006 (UTC)

The fact that pi is (an infinite decimal/irrational/transcedental) does not, by itself, imply that it contains every finite sequence of numbers. For example, Liouville's constant has all those properties, but its decimal expansion contains only 0s and 1s, so you won't find the sequence "8675309" anywhere in its decimal expansion. Pi, of course, contains all 10 digits, so this objection doesn't apply to pi. As far as I know, the answer to your question isn't known. It is conjectured but not proved that pi is normal, meaning that its digits are distributed randomly. Our article goes on to say that no one knows if all 10 decimal digits appear infinitely many times in pi. This means that the answer to your question is not known. To see this, consider all sequences of numbers with a 2 in them somewhere. We can see pretty easily that most of them cannot overlap, so if all such sequences appear in pi, we would have infinitely many occurrences of the digit 2. This just means that no one has proved that you're right -- as Trovatore correctly states below, it is very widely believed that pi is normal; we just haven't found a proof yet.Tesseran 20:14, 11 November 2006 (UTC)

The answer is, yes, almost certainly there is somewhere in the decimal representation of π where Hamlet is written out in binary. Infinitely many places, in fact. Currently, however, no one knows how to prove it.

The Hamlet thing would follow if we knew that π were a normal number. Which it is, beyond any reasonable doubt. But currently no one knows how to prove it. --Trovatore 19:04, 11 November 2006 (UTC)

I had heard the following about normal#s:they are algorithmicaly random & have the same density of each digit & are measure 1 in[0,1]. What you are saying sounds like that a minor corollary to being normal is having, for example,in base ten, a sequence of 999!^999!^999!^999 consecutive 8s in the decimal expansion infinitely often. Did I understand you correctly?Rich 05:36, 13 November 2006 (UTC)

A normal number has a disjunctive sequence for its expansion in the base for which it is normal. So yes to my large number of 8s above. I may have been thinking vaguely of simply normal in my near disbelief above.Rich 09:21, 13 November 2006 (UTC)

http://www.xkcd.com/pi.jpg —Keenan Pepper 21:03, 11 November 2006 (UTC)

• Thanks, that's funny!Rich 05:54, 13 November 2006 (UTC)

Ok, this question has kept me up too many nights!! I've read the infinite monkey article but this still bothers me. Firstly, the fact is Hamlet is NOT a random set of letters. It follows MANY determined rules, 'z' don't appear as often as 's', there are only five vowels but almost all words have at least one of them, etc... These rules are anything BUT random. Secondly, 'random distribution' becomes 'less random' the larger your sample set right? If you choose ten thousand numbers between 0 and 9 perfectly randomly you will have roughly a thousand of each right? Doesn't that mean an infinite set of random letters will have 1/26th of each letter? Hamlet is far from 1/26th of each letter. Ok, I know small chunks of that infinity (say billions of samples) don't have to follow that rule strictly but isn't there any limiting statistical rule? For example, couldn't you argue if the set contains 'hamlet' that it could theoretically contain only the letter 'a'? However small the chance? I think that's not possible, even if the set doesn't explicitly exclude it. In short, I personally don't think Hamlet is 'random' enough to be included in an infinite set of random letters but I haven't worked out a way to prove it and I haven't seen proof to convince me otherwise. Vespine 05:44, 13 November 2006 (UTC)

We'd better help you get some sleep. Consider a simpler problem, flipping an unbiased coin. If I ask, "Can I get the sequence 'heads-tails-heads'?", the answer is yes. In fact, we probably will not have to flip many times before it happens. But what about a sequence of 86 heads? The answer is still yes, it is sure to occur eventually. However, we probably will have to flip a great many times before we see it. In fact, any given finite sequence of heads and tails will eventually happen, with probability 1. Likewise, the text of Hamlet is a finite sequence of letters, so is guaranteed to occur eventually in a sufficiently random (and uncorrelated) sequence of letters. Good night! --KSmrq^T 09:22, 13 November 2006 (UTC)

Sorry, that still hasn't helped me. The issue that I have is that this assumes that unlike with coins the alphabet has 26 letters and not all combinations of them occur with the same frequency. So if you break up hamlet into 26 letter blocks, ignoring punctuation the 1st 26 setters of speech are:

• Whos there nay answer me stand a

Broken down that's 4a e, 3s n, 2w h t r, and 1 o y m d. Your 'random' assumption is that on a much larger scale, the chance of the above 26 letters occurring is the same as any 26 letters in a row, which I think is wrong. The chance of each letter occurring (almost) once in any sample of 26 letters is MUCH higher. Obviously there is the word chance again, so your going to say, as long as there is a chance, not matter how small, at infinity the probability becomes 1. But isn't there a point where probability rules a chance out? Zeno's paradox style or something? No wonder people go crazy thinking about stuff too much ;) Vespine 00:26, 14 November 2006 (UTC)

No, you are wrong about what I assume. Look again at my example of 86 heads in a row. That sequence is very unlikely (except in Tom Stoppard's delightful play Rosencrantz & Guildenstern Are Dead), yet occurs with probability 1. Perhaps Zeno's paradoxes are slightly relevant, in that they also stem from confusion; but they shrink numbers down to infinitesimals, whereas here we only have tiny. --KSmrq^T 21:00, 14 November 2006 (UTC)

That first line of Hamlet converted to binary is "01010111 01101000 01101111 01110011 00100000 01110100 01101000 01100101 01110010 01100101 00100000 01101110 01100001 01111001 00100000 01100001 01101110 01110011 01110111 01100101 01110010 00100000 01101101 01100101 00100000 01110011 01110100 01100001 01101110 01100100 00100000 01100001 " so basically we are looking for that in pi. Pi has all digits 0-9 and they are in pretty random order so looking for that sequence seems pretty hopeless. In fact I spent a few minutes on http://3.141592653589793238462643383279502884197169399375105820974944592.com/index1.html and found no sequence of 1's and 0's that is longer than 6 digits. That is not even enough to get one letter into Hamlet. But there are not a million digits in pi, there are an infinite number which means the probability of finding that sequence is almost certainly 1.Ed Dehm 07:08, 14 November 2006 (UTC)

But couldn't you convert pi to binary first to increase the probability? --WikiSlasher 11:50, 14 November 2006 (UTC)

Seeing the line of hamlet converted to binary has really helped me grasp more of the concept, thanks… Essentially, I think what I'm getting at is that an infinite random set contains EVERY SINGLE POSSIBLE sequence/combination of the random elements. Thinking about the problem in terms of "letters" with different frequencies and stuff was really messing with my head. ;) Vespine 21:39, 14 November 2006 (UTC)

Where is the mistake?

• 1/-1 = -1/1
• rt(1/-1) = rt(-1/1)
• rt(1)/rt(-1) = rt(-1)/rt(1)
• 1/i = i/1

but 1/i should be -i!! Where is the mistake? -JianLi 00:00, 12 November 2006 (UTC)

It occurs moving from the second line to the third: roots or exponents may not distribute across multiplied negative numbers. Consider $1={\sqrt {1}}={\sqrt {-1\times -1}}=^{?}{\sqrt {-1}}{\sqrt {-1}}=i^{2}=-1$. This equation has the same problem. --TeaDrinker 00:05, 12 November 2006 (UTC)

(I assume the notation rt(−1) means √−1.) The problem is that 1 has two square roots, +1 and −1; and −1 also has two square roots, +i and −i. As an analogy, if Andy says "I'm dating one of the Olsen twins" and Bob says "I'm dating one of the Olsen twins", does that mean Andy and Bob are dating the same girl? --KSmrq^T 09:06, 13 November 2006 (UTC)

it is because when you divide a positive number by a negative one, you don't switch the numbers around, you just switch the negative for continuality

$x\div -y\not ={\frac {-y}{x}}$

$x\div -y={\frac {-x}{y}}$ Englishnerd 15:05, 13 November 2006 (UTC)

No it isn't. That mistake is not in the original question. The answer is as Teadrinker and KSmrq said - the 'proof' includes a non-unique step and then chooses an inconsistent pair of from the non-unique values. --ColinFine 00:12, 14 November 2006 (UTC)

As TeaDrinker explained above, the rule ${\sqrt {ab}}={\sqrt {a}}{\sqrt {b}}$ holds for real numbers but not for imaginary numbers. --MathMan64 01:46, 17 November 2006 (UTC)