# Definition:Irrational Number/Approximation

## 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

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.2$: Numbers, Powers, and Logarithms: $(2)$ - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic Of Shape