Definition:Irrational Number

A number which is not rational is irrational.

That is, an irrational number is one that can not be expressed in the form $\displaystyle \frac p q$ such that $p$ and $q$ are both integers.

The fact that such numbers exist was known to the ancient Greeks.

Approximation by Decimal Expansion
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:
 * $\displaystyle n + \sum_{j=1}^k \frac {d_j}{10^j} \le x < n + \sum_{j=1}^k \frac {d_j}{10^j} + \frac 1 {10^k}$

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.

However, it is usually more complicated than this - see Rounding.