# Definition:Irrational Number

## Contents

## 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

- Square Root of 2 is Irrational: that such numbers exist was known to the ancient Greeks.

## 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

- 1937: Eric Temple Bell:
*Men of Mathematics*... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies - 1960: Walter Ledermann:
*Complex Numbers*... (previous) ... (next): $\S 1.1$. Number Systems - 1964: Murray R. Spiegel:
*Theory and Problems of Complex Variables*... (previous) ... (next): $1$: Complex Numbers: The Real Number System: $4$ - 1970: Lynn Arthur Steen and J. Arthur Seebach, Jr.:
*Counterexamples in Topology*... (previous) ... (next): $\text{II}: \ 31$ - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1.2$: The set of real numbers - 1977: K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*... (previous) ... (next): $\S 1.13$: Irrational Numbers - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): Glossary - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): Glossary - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction - 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic Of Shape