# How to find out any digit of any irrational number?

How to find out any digit of any irrational number?
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Let $\alpha$ be an irrational number. As long as there exists an algorithm the can decide whether $\alpha >q$ or $\alpha for any given rational $q$, you can obtain arbitrarily good rational approximations for $\alpha$. Especially, you can find upper and lower bounds good enough to uniquely determine any desired number of decimals.

For $\alpha =\sqrt{2}$, the decision algorithm is quit simple: If $q=\frac{n}{m}$ with$n\in \mathbb{Z},m\in \mathbb{N}$, then $\alpha 0\wedge {n}^{2}>2{m}^{2}$.
###### Not exactly what you’re looking for?
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.