Whilst venturing out along the darkened paths of the internet at midnight, I stumbled upon the following image (and not for the first time!). What caught my interest was not the somewhat flawed perception of π, but the attempt of explaining why that view is a little wrong. Here, have a read:
“Mathbusters to the rescue! This really cool “fact” is actually not necessarily true. Just because there are infinitely many sequences of digits in the decimal expansion of π doesn’t mean that all sequences are in it. In a similar sense, just because there are infinitely many rational numbers pq, and between every two real numbers is a rational number, doesn’t mean that all of those infinite numbers are rational numbers. Infinity’s not very intuitive!
Mathematicians don’t even know if every digit occurs infinitely often in the sequence of π, though it does certainly appear to us so far that the digits are randomly distributed. This randomness, not irrationality, is actually the key to understanding whether or not π contains all possible strings of information.
What’s more, if this randomness characteristic of infinite nonrepeating decimals were actually true, it wouldn’t be the case that just π contains all of this fascinating information to humans. Rather, every irrational number would have this characteristic — and therefore, exactly the same information somewhere hidden in its infinite decimal expansion.
That’s obviously weird enough to think about — but to make things worse, there are a heck of a lot more irrational numbers than rational numbers, so I guess if every infinite nonrepeating decimal had this characteristic, statistically* speaking, every number would contain all the information strings of the universe — including this post, Tumblr’s logo in every possible image format, the exact chemical makeup of every star every born and infinitely many more that weren’t — and, of course by which popular mathematical result this image is probably inspired, all of Shakespeare’s plays.
What’s even weirder still is that since our universe is finite in age and extent, all the information contained in our universe is statistically zero compared to the possible information content of a random, infinite decimal. So while that information is significant to us, it would be quite literally trivial to a being that could conceivably comprehend all of π (or rather, any random infinite decimal) at once.
Anyway, moral of the story is, while this result may seem awesome, it’s a little bit deceiving. But it’s always cool to see people looking at math in interesting ways!”