Is Pi in Pi? And is 1/3 in Pi? And other numbers?

Question

Note: According to my high school math teacher, people have seriously gone insane because of musing over stuff to do with infinity, so unless you are fine with going insane, or are sure not to ponder too much, stop reading now :P

Relatively basic mathemetics is that irrational numbers, such as Pi, have an infinite number of decimals that do not repeat forever. Numbers with an infinite number of decimals that do repeat, however, are considered rational(such as 10 / 3).

Another relatively basic piece of mathemetics is that any irrational number contains any possible combination of numbers, so given that you keep calculating for long enough, your birth date, your credit card number, all of your passwords in binary, and basically every possible series of numbers is somewhere in Pi.

But think about this: Does that mean that the same goes for e, the square root of 2, the square root of 3, and all irrational numbers? And if every possible combination of numbers is in Pi, then is Pi in Pi? And what about 10 / 3, which is rational? And if Pi is in Pi, would that mean that it repeats, so it is no longer irrational - but 10 / 3 would also be in Pi, along with Pi itself, then it would not repeat, meaning that it is irrational AND rational at the same time?

Now, since you have read this, make sure you do everything you have ever wanted to do in your life before you go insane. :P

Answer 1

Something is wrong with your basic notion of 'containing' if you think this 'basic fact' is true. I am not sure we know that much about the decimal expansions of transcendental numbers, but this idea is clearly an exaggeration based on some of the odder things we do know.

It seems obvious that no infinite rational representation appears literally inside the infinite representation of a transcendental number. If 1/3 were in the decimal expansion of pi, in the sense I think you mean, there couldn't be anything after it, since the repetition never ends. Then pi would be rational, as it would end with a repeating pattern.

It is tempting to think it is embedded in there less directly, maybe by skipping digits, or something. But there is a standard 'trick' for constructing irrational numbers that contain only certain sets of digits, say 1 and 0, that is often used as a first-pass test at proofs about the irrationals. The number following the pattern .01001000100001... with each run of zeroes increasing over the last, is irrational. It clearly doesn't contain '2' much less 'every sequence of numbers'.

Answer 2

That Pi does this has not been proven and is merely a hypothesis. See here.

For sure not every irrational number has this property. A counter example is the number 0.011000111100000111111... which doesn't contain the sequence '37', see also here.

Lastly, the claim about this property only considers finite sequences. That doesn't mean Pi doesn't contain any finite sequences though.

Of course, naturally Pi is in Pi. However, it's not possible that both the number 0.0110001111... from before and 1/3 are in Pi: they can't contain each other, and they can't be put after each other since they're both infinite. That doesn't mean that Pi is the only infinite sequence contained in Pi: also the sequence 'Pi without the first 3' is in Pi, as is the sequence 'Pi without the 3.1', etc. - and since Pi doesn't repeat itself, there are infinitely many different infinite sequences in Pi (whatever you want to do with that though).

To read up more on this, you may find the concept Cardinality interesting. It's about different kinds of infinities. For example, the set of positive Natural numbers {1,2,3,...} and the set of whole numbers {...,-2,-1,0,1,2,...} have the same cardinalty, meaning they contain equally many numbers in some sense. However, the set of real numbers (all numbers p/q where p and q are elements of the set of whole numbers) has a different cardinality - see here.

Answer 3

This is really more of a math than a philosophy question (although there are certainly valid philosophical questions about pi). In this case, the claim about pi is not that it contains all sequences of numbers, but that it contains all finite sequences of numbers. With that in mind, it no longer follows that it can or must contain itself, e or 10/3, although it does follow that it must exactly resemble those numbers for arbitrarily long stretches of time.

If you find concepts such as these of philosophical interest, you might also be interested in Jorge Luis Borges' famous short story "The Library of Babel" and in the iconic fractal, the Mandelbrot Set, which contains infinitely many non-identical copies of itself.