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The universal real constant pi, the ratio of the circumference of any circle and its diameter, has no exact numerical representation in a finite number of digits in any number/radix system.
It should be noted that the fraction part, unlike the integer part, is not limited to a finite number of digits.
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In this question the term computable sequence is defined to mean that the numbers ak are real and there is a Turing machine that, for any pair of positive integers A, B, will print out in order the first A digits of all ak, for k ranging from 1 to B, in a finite number of steps.
"We have a finite number of troops.
"There are a finite number of jobs.. . .
There are a finite number of junior doctors.
A finite number of rule-driven, repetitive steps.
The world has a finite number of Tyrannosaurs rexes.
Are they finite (ie terminating after a finite number of steps)?
Our own universe has a finite number of jobs, a finite number of educational places and a requirement by the inhabitants to have shelter and food.
Brouwer, Nash demonstrated that such an equilibrium exists in any game with a finite number of players and a finite number of moves to choose from.
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