How to find out any digit of any irrational number?

Patatiniuh 2022-07-07 Answered
How to find out any digit of any irrational number?
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Answers (2)

persstemc1
Answered 2022-07-08 Author has 18 answers
Let α be an irrational number. As long as there exists an algorithm the can decide whether α > q or α < q for any given rational q, you can obtain arbitrarily good rational approximations for α. Especially, you can find upper and lower bounds good enough to uniquely determine any desired number of decimals.

For α = 2 , the decision algorithm is quit simple: If q = n m with n Z , m N , then α < q n > 0 n 2 > 2 m 2 .
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Kolten Conrad
Answered 2022-07-09 Author has 1 answers
In general, no.

Suppose that for every irrational number r there were an algorithm that takes a natural n as input and returns the n-th digit of r. The possible algorithms are countable, all the irrationals are not, hence it is not possible to have such algorithms for every irrational.

However, such algorithms do exist for the so-called computable number.
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