All numbers can be expressed with different number bases.
The beginning of pi’s decimal expansion expressed with the base 10 is familiar to most of us:
Another number base that occurs from time to time when you want to express Pi is 27. That means that you need 27 different symbols. Usually you pick the 26 letters of the alphabet (a to z) representing the values 1 to 26 and you use a blank (space) to represent zero.
Expressed with the base 27 the “septemvigesimal” expansion of Pi begins like this:
c.cvezcvbmlyzxmswprpiijzhweemupdrxou jhcfmobyhsijlpjsca zgxlhqunzwkhdfphtst…
For the following thought experiment to work we need to divide the never ending string of letters of the septemvigesimal expansion of Pi into appropriate chunks. We’re going to need some pretty big chunks. The larger the chunks the better the chances for us to find whatever special combinations of letters we seek. If we were looking for a 5 letter word like “hello” maybe a chunk size of say 10 million (107) digits would suffice, but in this case we need even bigger numbers.
I suggest we divide the letters of the base 27 expansion of Pi into chunks comprising of 3745452 letters.
For pedagogical reasons I suggest we give this number a name and symbol. How about the “Bonds Integer” and the greek letter β.
It’s a somewhat unruly number. Written out it consists of 355671 digits.
Here are the first 50 and the last 50 digits.
β = 98531088776004089478532579092311073356917231337969…
[355571 digits omitted]
Expressed with scientific notation it can be approximated as 9.8531 * 10355670.
Even though it’s size is massive you still need an infinite number of them to cover “the whole” of the septemvigesimal expansion of Pi.
I suggest we divide Pi into parts using the following framework:
and so on…
Imagine that we examine the ”decimal” expansion in one of these parts and all of a sudden we encounter the following strange sequence of letters…
…uqwporutqjfhlsdkfhlskj gfeigfurhe jihpiuqytpiquy topwieugh dpfjtytykghepiyepituhe ojkhsfd kheprye tpiu hgljsfd ghwelitu yepitu weogjkfhdlgjk shdlghiesguysep iryuighbaorjxhjgqwktj the house stood on a slight rise just on the edge of the village it stood on its own and looked over a broad spread of west country farmland not a remarkable house by any means it was about thirty years old squattish squarish made of brick and had four windows set in the front of a size and proportion which more or less exactly failed to please the eye the only person for whom the house was in any way special was arthur dent and that was only because it happened to be the one he lived in he had lived in it for about three years ever since he had moved out of london because it made him nervous and irritable he was about thirty as well dark haired and never quite at ease with himself the thing that used to worry him most was the fact that people always used to ask him what he was looking so worried about he worked in local radio which he always used to tell his friends was a lot more interesting than they probably thought it was too most of his friends worked in advertising it hadnt properly registered with…
It feels like an infinitely long string of nonsense and then all of a sudden we discover words with meaning: ”the house stood on a slight rise just…” In fact, the words feel familiar. We have read them before. Suddenly we realize that it’s the beginning of Douglas Adams bestselling science fiction trilogy Hitchhikers Guide to the Galaxy. You go and get the book from the shelf and follow the expansion of Pi further. You realize that it corresponds word for word with the book. It’s an exact reproduction of the book. All the way until the last page that ends with the words: ”take in a quick bite at the restaurant at the end of the universe”. After this it goes back to the seemingly endless string of gibberish again.
Now you may say that this is just a dream scenario. The chance of finding the complete Hitchhikers Guide to the Galaxy in the septemvigesimal expansion of Pi is something that doesn’t belong to the arena of reason and mathematics. The probabilities involved belong to philosophy perhaps or poetry.
”Au contraire” – I say. You only have to pick a portion of Pi reasonably big – then the probability of finding it can be rather substantial. Let us examine the problem from a strict mathematical point of view and let us use the division in “bonds-parts” as suggested earlier where each part encompasses 9.8531 * 10355670 digits.
So what’s the real probability of finding the complete Hitchhikers Guide to the Galaxy, word for word, in an arbitrarily chosen bonds-part of the base 27 digit expansion of Pi?
First we need to know how many letters there are in the book. That’s easy. The answer is 248484 letters (including spaces as word dividers).
Let’s start with a completely different question: How small would be the probability of finding the entire book at any given start digit.
The answer is which is extremely unlikely because we get, so to speak, only one try.
If we, on the other hand, scans a bigger portion of Pi we get more chances. The probability of finding the Hitchhikers Guide in at least one place in a portion consisting of x possible places is calculated like this:
Let’s go back to the original question. How big is the probability of finding the Hitchhikers Guide somewhere within an arbitrarily chosen bonds-part of Pi? The first thing you need to calculate is the number of possible places there are within such a part. Using some basic maths you can calculate that there are approximately at least 27248484 places for the book to be in within such a part. The probability of finding the complete book within a bonds-part can then be calculated like this…
it has been shown that this approaches the limit
The answer is approximately 0.632 or close to 63%. The probability is therefore pretty substantial of finding the complete Hitchhikers Guide within an arbitrarily chosen bonds-part of Pi.
I’m aware, of course, of the practical problems. That scanning through ridiculously large portions of the septemvigesimal expansion of Pi, requires a ridiculous amount of time… but why should trifling practicalities like that let us down?
With this small paper I’ve been trying to investigate the existence of a specific integer – “the Bonds Integer” defined as:
The number of base 27 ”digits” in the septemvigesimal expansion of Pi constituting a chunk that makes the probability of finding the complete Hitch Hikers Guide to the Galaxy exactly (or rather as close as possible to)
That is my special number – β.
I’ve wanted to show that the seemingly outrageous problem of finding the Hitchhikers Guide in Pi can be formulated in a way that makes sense using limit values (“gränsvärden” in Swedish). Infinity on both sides of a fraction bar can under certain conditions be analyzed as outweighing each other to produce an easy to comprehend managable value.
The reason why I picked Douglas Adams’ classical trilogy as the object of this thought experiment is of course a consequence of my opinion that no one has more brilliantly described how outrageous improbabilities can be the subject of a meaningful discourse. =)
May 19, 2010