Over at Cosmic Variance, they had a recent post on big surprises that one could have in physics. In the comments someone suggested that we should be looking for messages from *the creator* in the digits of pi. I’m sure this was said in jest, but I’ve seen enough similar attempts forwarded into my e-mail box to know that this goes on. Of course, this is just one more version of what we in physics pejoratively describe as numerology: some random collection of facts about some really important *mystical number, *that has no physical mechanism to describe a physical situation *. *Apart from pi, another very popular set of such numbers are 42 and 137. The plan of this approach is that everything there is to know about the world is encoded in these numbers, if you only have the correct algorithm to decode it. One of the popular algorithms is to look for the digits of pi or some other irrational number and to try to see patterns in them.

Let us begin with periodic digit sequences. If you see a number with periodic sequences of digits that repeat themselves endlessly, then that number is a fraction. This is easy to prove. Take the number and multiply it by a suitable power of 10, so that the digits on the tail can be matched. We obtain this way that if we subtract the original number from the new bigger number

where χ is the number we are given, and m, N, s are all integers. Conversely, if you are given some fraction, you will get a repeated sequence of digits in the tail. This is because for any integer k that is relatively prime to 10, there exists another integer m such that

This equation means that k divides the number on the right, and one can massage the expression to the representation so that the first equation I wrote down is true. The other thing that is easy to show is that m is less than k. However, I will leave that for another day.

What is the point I’m trying to make? That if you want a long message to be in a number, it better be that the number is rational and that it is a fraction with a rather large denominator (for most people that is unnatural). The other possibility is to pick a number where there are no repetitions like above: an irrational number.

So what do the irrational numbers typically look like? They would look as if their digits are random. As a matter of fact, they should be typically truly random.

Here is the punchline: a random sequence of digits has every possible sequence of digits of any length showing in it an infinite number of times. So let us assume that you have an algorithm for turning digits into letters. We do this with computers all of the time, so you can try it here as well. This result states that any message that you want is written in the digits of a normal number an infinite number of times. Every single message.

This includes the messages `God does exist’, `God does not exist’, ‘You idiot’, ‘Die’, ‘URstoopid’, ‘Pi is the source of all knowledge’ and ‘You have just wasted your whole life in this, and for consolation you have dug up this message’.

The one that you will see the most is the fourth one, because it has the smallest number of letters. This is followed by 3 and 5, from which you should already be getting the message …

You could say pi is not just any number, it is a more special number than any other number, so those misleading messages are not there.

I would still call your bluff. Research has shown that pi is pretty normal.

Here are some further websites to aid your search:

Pi searcher, which describes itself in it’s own words as

`The Pi Searcher has proven both exceptionally useless and occasionally useful to math & early science classes’

And then there is research on how random the digits of pi truly are. Here is the website of the researcher: David Bailey, where you can read a lot of fun stuff about pi.

I think the biggest surprise would be to find that certain messages are completely missing: then pi would not be normal. In the meantime, for all of you who believe that pi has the answer for every question: it seems that it does, and moreover it has all the wrong answers too. So DON’T WASTE YOUR TIME ON NUMEROLOGY.

on February 10, 2009 at 9:47 amAstronomy Link ListAdded to the Astronomy Link List

on February 10, 2009 at 5:28 pmUncle AlPi is obviously a natural compilation of all important constants – including itself – albeit with a certain amount of managerial overhead. Starting

Ndigits after the decimal:271828182845 from 1,016,065,419,627 -th

314159265358 from 1,142,905,318,634 -th

PiFastVer. 4.3 or later by Xavier Gourdon will crunch pi, e, sqrt(2)… much faster than you can download the digits. All it asks for ridiculous speed is GB of RAM.on February 11, 2009 at 1:32 amRandom pieces of pie « Peculiar Velocity[…] David at Shores of the Dirac Sea explains, they’re all encoded in the digits of pi: This result states that any message that you want is written in the digits of a normal number an […]

on February 11, 2009 at 8:22 amsaintnekoDepending on how you’re looking at it, there’s an infinite amount of numbers possible on either side of a decimal point – if you look hard enough you’ll find anything, but that’s pareidolia, not hidden truths.

on February 11, 2009 at 8:23 amsaintnekoOops. I meant apophenia, of which pareidolia is a subset (but a visually or sonically oriented one).

on February 11, 2009 at 9:23 amPlatoDavid,

I guess you do not watch the show “Numbers”very often do you?:) Your not very Liminocentric orientated either. That’s no body fault but your own. If you do not want to have a topological relationship with the world, that’s your choice.

Anyway Pi day, Match 14 is an international event. Can that many people be wrong just cause John Baez saids so?:)

Sure, Pi the movie can have you drilling holes in your head too.

But more seriously, who knows what the descending order of things are to take?:)

Best,

on February 11, 2009 at 9:45 amPlatoMore useless knowledge:)

I mean sure, some people like to think constants are not really constants so they explore the world under that premise.

Oh and the Ulam spiral probably means nothing to you either:) If your a 3D visualizer it can become really quite interesting. Mersenne Prime. The basis of the atomic bomb?

Best,

on February 12, 2009 at 10:23 amOliverA particularly interesting feature of pi is that, despite its digits being random it is, in some sense, not a random number. In algorithmic complexity theory, the randomness or otherwise of number is defined along the following lines: try to write a computer program to generate said number. If the program is appreciably shorter than the number, then the number is not random (its information content can be compressed). For example, the number 10101010… can be generated by a loop containing the command `print 10′. Obviously, this number is not random.

I’m guessing that this can be related to the fact that, whilst pi’s digits are random, no useful messages are encoded by them.

As an entertaining aside, there is a nice application of Godel’s incompleteness theorem in this subject. As shown by Gregory Chiatin, it is not possible to prove that a given digit string is random (with the above definition of randomness). Suppose that one has tried one’s best to write the shortest possible program to generate a putative random number, and this program is no shorter than the number itself (suggesting randomness). Chiatin showed that there is no way of proving that a shorter program cannot be written. So although it is known that the vast majority of digit strings are random, it is impossible to prove whether a given string is random!

on February 13, 2009 at 4:48 pmHelen CheneyPure academicians are unlikely to realize that 42 stands for a swear. I refrain from being explicit.

HC

on February 20, 2009 at 3:53 amUte MedleyFor a demonstration of a numerological pattern in number Pi, clusters of numbers especially number 7, up-to-date explanation of the metaphysical attributes of numbers in context with current scientific facts, and the demonstration of living authentic lives see my book “Invisible Cloak – Know Thyself” (www.sumuniversal.com).