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 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.

Also see

• Definition:Rounding

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Introduction
• 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: Taming irrationals