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Exploring Pi: Is Every Possible Number Hidden Within?

Pi's decimal expansion goes on without ever ending or repeating, but that's simply not the same as saying that pi must contain every possible sequence of digits. Check out the main channel ‪@polymathematic‬ ! This is, admittedly, a tough one to wrap your head around. Because of the nature of infinity, it can be just as hard to imagine pi containing every possible string of digits, and in that way encoding any information you want somewhere within its decimal expansion, as it is to imagine that it does NOT contain every possible string of digits. And, to be sure, it is totally possible that pi DOES contain every possible string of digits—not just once, but infinitely many times! Most mathematicians believe that's true, and they've even given that a name: normality. But no one has been able to prove that pi is normal, and just because it's an irrational number, that's not enough to guarantee it. Consider the weird number I construct in this video: the digits 1 through 9 repeating over and over again in that order, but with a different number of zeros between them every time. This is an irrational number. Because the zeros are changing, it would be impossible to represent this number with a ratio between two integers. Its decimal expansion also contains every possible digit an infinite number of times. And yet, there are tons and tons of sequences that will never be in this number. For example, 10 will never be in this number. Now, you might object that pi is just a typical number occuring all on its own out there in the world, and so it won't obey the weird rules this constructed number will. Maybe! I think that's probably true. But so far, no one can prove that it's true, and even more importantly, there's nothing about the nature of irrationality or any particular use of digits that requires it to be true. #PiMystery #InfiniteNumbers #IrrationalNumbers #NumberTheory Follow Tim Ricchuiti: TikTok:   / polymathematic   Mathstodon: https://mathstodon.xyz/@polymathematic Instagram:   / polymathematicnet   Reddit:   / polymath-matic   Facebook:   / polymathematic   Watch more Math Videos: Math Minis:    • Math Mini   Math Minutes:    • Math Minutes   Number Sense:    • Number Sense (UIL / PSIA)   MATHCOUNTS:    • MATHCOUNTS