### Irrational numbers

The number \(\sqrt{2}\) arises as the hypotenuse of a right-angled triangle of side length 1.

The Greeks realized that this number was not rational, which means that it could not be written as a fraction \(\dfrac{a}{b}, b \neq 0.\)

There are many proofs of this.

This caused great upset to the Greek mathematicians, since it introduced a new sort of number which they called an irrational number. An older word for this is incommensurable, which meant that it could not be measured as a ratio of two whole numbers. This discovery caused a dramatic rethink into the nature of number. The validity of many of their geometric proofs, which assumed that all lengths could be measured as ratios of whole numbers, was also called into question.

Numbers such as \(\sqrt{2},\) \(\sqrt{7},\) \(\sqrt{29}\) are called surds.

You will have come across to the number \(\pi\). This number is irrational.