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’
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.