Irrational numbers

The opposite of a rational number is an irrational number. All real numbers which are not rational are irrational.

By definition, all irrational numbers are also real numbers.

An irrational number is a real number which can’t be written as a simple fraction of two integers.

Notation

The set of irrational numbers is usually represented by the symbol \mathbb{I}:

\mathbb{I} = \{ x \in \mathbb{R} \mid x \notin \mathbb{Q} \}

Examples of irrational numbers

• \sqrt{2} (cannot be expressed as a fraction of two integers)
• \pi (cannot be expressed as a fraction of two integers)
• e (cannot be expressed as a fraction of two integers)
• \frac{7\pi}{4} (cannot be expressed as a fraction of two integers)

Non-examples of irrational numbers

• \frac{1}{2} (rational)
• -3 (rational)
• 0 (rational)
• 4.75 (rational)
• 7.66666\ldots (rational)
• \sqrt{9} (rational, can be simplified to 3)
• \frac{4\pi}{2\pi} (rational, can be simplified to 2)
• i (not a real number)

Decimal representation

The decimal equivalent to an irrational numbers have non-terminating and non-repeating decimal.

This means that the digits after the decimal point go on forever without ending, and there is no repeating pattern in the digits (e.g. how \pi is ‘completely random’).