Monday, September 21, 2015
Rational Numbers and Randomized Digits
Consider randomizing decimal digits in an infinite string (say, by using a standard d10 from a roleplaying game, shown above). How likely does it seem that at any point you'll start rolling repeated 0's, and nothing but 0's, until the end of time? It's obviously diminishingly unlikely, so effectively impossible that you'll roll a terminating decimal. Alternatively, how probable does it seem that you'll roll some particular block of digits, and then repeat them in exactly the same order, and keep doing so without fail an infinite number of times? Again, it seems effectively impossible.
So this intuitively shows that if you pick any real number "at random" (in this case, generating random decimal digits one at a time), it's effectively certain that you'll produce an irrational number. The proportion of rational numbers can be seen to be practically negligible compared to the preponderance of irrationals.