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2:07pm on Tuesday, 14th May, 2013:

Piece of Pi


I've been told many times that the digits of pi contain, somewhere within them, every finite sequence of digits. This means that anything that can be encrypted as such a sequence — image files, DNA, digital music — appears somewhere in pi. For example, according to Pi-Search, my favourite number (142,857) occurs at position 335,037 after the decimal point; my phone number is up in the 2,000,000s.

The rationale for this is that pi is infinite and non-repeating, therefore it must be the case that all strings of integers appear in it somewhere. Personally, I don't buy this as a proof: pi with all the 9s removed looks as if it would satisfy the same criterion but it clearly does not contain every finite sequence of numbers. It may be there's an actual proof that pi does contain every sequence, but I haven't come across it if there is.

Let's assume pi does have this property, though. It wouldn't only be pi that had it: there would be an infinite number of real numbers that had it. Pi/10 also has it, for example. The number e is also reputed to have it, using the same argument employed to show pi has it.

Hmm, so that means if you were to choose any finite sequence of numbers it would appear in both e and pi.

The first 6 digits in e are 2.71828 . These appear in pi 33,789 places after the decimal point. This means that if you were to multiply pi by 10^33789 then take the remainder after dividing by 10, the result would be the same as e to five decimal places. Put another way, if you were to shift the digits in pi left 33,789 times and discard anything in the 10s column and beyond, you'd get e correct to five decimal places.

This shift-left works for any finite sequence of integers, assuming that both e and pi do indeed both contain every such sequence. You can shift pi left a finite number of times and it will be congruent with e for as many decimal places as you specify. If you want a longer match, you just shift it left more until you get one. The first 7 digits in e are 2.718281; they occur in pi at position 1,526,800 after the decimal point. The first 8 are 2.7182818 and occur at position 73,154,827. No matter how much of e you want to find in pi, it'll be there: just look further. It's only not going to be there if you want all of e, or perhaps if you're allowed to shift left an infinite number of times.

Yes, since you ask, I am supposed to be composing some slides for a talk I'm giving next month that I've been putting off writing.

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