\begin{definition}[Definition:Irrational Number/Approximation]
From its definition, it is not possible to express an irrational number precisely in terms of a fraction.
From Decimal Expansion of Irrational Number neither Terminates nor Recurs, 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 $\sqbrk {n.d_1 d_2 d_3 \ldots}_{10}$.
Then:
:$\ds n + \sum_{j \mathop = 1}^k \frac {d_j} {10^j} \le x < n + \sum_{j \mathop = 1}^k \frac {d_j} {10^j} + \frac 1 {10^k}$
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'''.
\end{definition}