Pi, a number beyond reason

The Pythagoreans, a mathematical school of the 6^th and 5^th centuries BCE founded by Pythagoras, initially believed that all numbers were rational. They soon realized that the golden ratio, which appeared in their symbol, the pentagram, was not. They then discovered that $\sqrt{2}\backslash sqrt\{2\}$ was not rational either. These two discoveries caused an enormous scandal: "everything is number" (meaning an integer or a fraction), the foundation of Pythagorean philosophy, was crumbling! But Pandora's box had been opened: there are, in fact, infinitely many irrational numbers. The Greeks may have wondered whether π was one of them. A priori, they could not answer the question, which was not settled until much later.

Although some numbers were already known to be irrational in antiquity, this does not mean that proving a particular number irrational is easy. It was not until 1978 that Roger Apéry proved the irrationality of $\zeta (3)={\sum }_{n\ge 1}{\displaystyle \frac{1}{n^3}}\backslash zeta\; (3)\; =\; \backslash sum\_\{n\; \backslash geq\; 1\}\; \backslash dfrac\{1\}\{n^3\}$ (see the article "Towards Transcendence").

An effective tool -----------------

Continued fractions are widely used to study irrational numbers. This involves expressing them as a sequence of quotients