The first irrational numbers

As early as 1800–1600 BCE, the Babylonians knew how to calculate the length of a square's diagonal from its side; so did Indian mathematicians in the 6^th century BCE. The real question for Greek geometers was whether the ratio of a square's diagonal to its side—$\sqrt{2},\backslash sqrt\{2\},$ in modern notation—was rational, that is, a quotient of two integers?

√2, the rebel! ----------------

It was probably the Pythagoreans, for whom "all is number," who proved that the number in question stubbornly resisted being written as a fraction: it is indeed irrational.

This discovery, dating from 530 BCE, is attributed to Pythagoras's disciple Hippasus of Metapontum, who, according to the philosopher Proclus (412–485), "was the first to bring the study of the irrational out of obscurity."

While Aristotle proved by contradiction that $\sqrt{2}\backslash sqrt\{2\}$ is not rational, the Pythagoreans preferred geometric arguments, for example subtracting the side of a square from its diagonal. If the diagonal of a square with side 29 were equal to 41—in other words, if $\sqrt{2}\backslash sqrt\{2\}$ were equal to ${\displaystyle \frac{41}{29}}\simeq 1,413793\dots ,\backslash dfrac\{41\}\{29\}\; \backslash simeq\; 1,413793\; \backslash dots,$ then, since the colored triangles have the same shape, we would also have $\sqrt{2}={\displaystyle \frac{17}{12}}\simeq 1,416666\dots \backslash sqrt\{2\}\; =\; \backslash dfrac\{17\}\{12\}\; \backslash simeq\; 1,416666\; \backslash dots$. What a contradiction!