Definition:Irrational Number/Approximation

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Definition

From its definition, it is not possible to express an irrational number precisely in terms of a fraction.

From Decimal Expansion of Irrational Number, it is not possible to express it precisely by a decimal expansion either.

However, it is possible to express it to an arbitrary level of precision.


Let $x$ be an irrational number whose decimal expansion is $\left[{n.d_1 d_2 d_3 \ldots}\right]_{10}$.

Then:

$\displaystyle n + \sum_{j \mathop = 1}^k \frac {d_j}{10^j} \le x < n + \sum_{j \mathop = 1}^k \frac {d_j}{10^j} + \frac 1 {10^k}$

for all $k \in \Z: k \ge 1$.

Then all one needs to do is state that $x$ is expressed as accurate to $k$ decimal places.


Also see


Sources