As far as I can tell, the logic goes something like this:
- pi is an irrational number
- since it is irrational, it's decimal expansion never repeats
- since it never repeats and it goes on forever, every possible sequence must eventually come up somewhere.
That third step might sound reasonable, but it is wrong. Let me illustrate with a quick example, I will provide an irrational number in which every sequence does not come up:
.10110111011110111110...If we continue this in the obvious way, we get an irrational number in which many sequences of numbers never show up, 00 is an example. So what do we have here? It is a sequence that never repeats, that goes off to infinity, and yet many finite sequences never show up inside. This may seem counter intuitive, but the example is before you.
Let's return to the statement made in Person of Interest. The statement remains that every number shows up within the digits of pi. I have provided an example of an irrational number that does not have this property, which I would argue at least puts a bit of doubt on this statement, but it doesn't prove it wrong. It demonstrates that the above logic is faulty, it can't be true simply because pi is irrational, but perhaps there is something else about pi that makes this true. I couldn't think of a way to prove this either true or false, time to consult the internet! It turns out that this is actually an unsolved problem. Nobody knows whether or not what Finch said is true or not (which I why I put mistake in quotation marks in the first paragraph). That is how hard these problems are to solve.
One last thought here, if the decimal expansion to pi has every possible finite sequence contained within it, that means that somewhere we will find a billion zeroes in a row. Put in those terms, this seems obviously false...and yet, it could be true. No one knows. Infinities are weird!
I was thinking of another angle to the billion zeroes thing this morning. It will take me a couple of steps to get there.
First, think of the 10 available digits, we would probably expect that each one would show up at some point. No single digit seems to have a greater chance to show up than any other. There's no reason to think that a 5 has a better chance of turning up than a 7 right?
Now, the digits that we have are somewhat arbitrary, we have 0,1,2,3,4,5,6,7,8,9 because we count in base 10, and there's nothing really special about 10 except that is the number of fingers that we have. So what if we counted in a different base? We would need a bunch of new symbols, but fundamentally nothing would really change. What if we instead counted in base 100? Every 2 digits of our normal decimal expansion of pi would be a single digit in the other base. I don't want to come up with 90 new symbols, but I could write the decimal expansion of pi in base 100 like this:
In base 100 this is 6 digits. Given this view of the decimal expansion, I see no reason to think (00) would have any less chance to come up as any other number. By thinking in base 100=10^2, I can argue that there is no particular reason to think 2 zeroes won't show up at some point.
Now, if I want to think about getting a 1,000,000,000 zeroes instead of 2, why can't I use the same idea and write the digits of pi in base 10^(1,000,000,000)? I would argue that there is no reason to think the zero of this base has any less chance of showing up compared to any other number in that base.
I want to stress that this is absolutely not an argument that we will definitely get a billion zeroes somewhere. It's more an argument that it is much less impossible sounding than it might initially seem. A little change of perspective can really screw with your intuition.