Definition:Irrational Number

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Definition

An irrational number is a real number which is not rational.

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


The set of irrational numbers can therefore be expressed as $\R \setminus \Q$, where:

$\R$ is the set of real numbers
$\Q$ is the set of rational numbers
$\setminus$ denotes set difference.


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:

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

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.


Geometrical Definition

In the words of Euclid:

With these hypotheses, it is proved that there exist straight lines infinite in multitude which are commensurable and incommensurable respectively, some in length only, and others in square also, with an assigned straight line. Let then the assigned straight line be called rational, and those straight lines which are commensurable with it, whether in length and in square or square only, rational, but those which are incommensurable with it irrational.

(The Elements: Book $\text{X}$: Definition $3$)

And let the square on the assigned straight line be called rational and those areas which are commensurable with it rational, but those which are incommensurable with it irrational, and the straight lines which produce them irrational, that is, in case the areas are squares, the sides themselves, but in case they are any other rectilineal figures, the straight lines on which are described squares equal to them.

(The Elements: Book $\text{X}$: Definition $4$)


Also see


Historical Note

It was the ancient Greeks who discovered that the irrational numbers were indispensable in geometry.

They were famously discovered by the Pythagoreans.

Some sources suggest that this discovery is one of their more important contributions towards civilisation.


Linguistic Note

The name irrational number is strictly speaking derived from the fact that such a number cannot be defined as the ratio of two integers.

It is arguable as to whether the name also has connotations of being nonsensical or beyond reason.

The ancient Greeks, whose term for an irrational number was alogon, which evoked a feeling of undesirably chaotic and unstructured, or, perhaps more literally: illogical.


Sources