3 ms·

Would we assert that Pi contains Pi?

• lutusp 13y ago
  Pi can only be expected to contain a subset of itself. Specifically, if we choose a finite digit sequence of length N, for any value of N there are an infinite number of copies of that sequence within Pi. Let's try this with an experiment. Let's say we want to find the first four digits of Pi (3141) within Pi. For the first 100,000 digits of Pi, we find the substring "3141" at these index locations (skipping the first match): 3497 26613 42126 45903 53277 68919 88009 91355 For the substring "31415", one more digit in length, within the first 100,000 digits of Pi we find only one example, at index 88009. This makes sense, since the binomial theorem provides a probability of 0.632 for one or more successes within 100,000 normal digits and a substring length of 5. Binomial theorem calculator: http://arachnoid.com/binomial_probability http://arachnoid.com/binomial_probability Based on this example, with an arbitrarily large set of Pi's digits, it's reasonable to say that any reasonable digit sequence can be found within it. And for Pi's infinite sequence, any subset of Pi -- or any other numerical sequence of any length -- can be located within it. One more example. For the substring "33333", within the first 100,000 digits of Pi we find occurrences at indices 28468 and 89086. I include this to suggest that any digit sequence, for that matter any sorts of numbers, for example representing letters, can be found. The old saw about finding the works of Shakespeare within Pi is obviously true, if difficult.