Decimals of Pi
Number Theory
Level
3
True or false: because pi is irrational, any finite sequence of digits MUST be found somewhere in its infinite decimal expansion.
False
True
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While it seems unlikely, a finite sequence of digits does not HAVE to be in the decimal expansion of an irrational number. Here is a counter-example: 1 . 1 0 1 0 0 1 0 0 0 1 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 ⋯ . This number is irrational, yet you will never find a 2 , 3 , 4 , 5 , 6 , 7 , 8 , or 9 in its decimal expansion.