Theorem. $\sqrt{2}$ is irrational.

This is an ancient theorem, about 2400 years old, and its modern proof is identical to the one appearing in Euclid's elements. A simple number theoretic proof, where you get the chance to use the abductio ad absurdum (or εἰς ἄτοπον ἀπαγωγή).

Note. As Victor Protsak noted, the number-theoretical proof is not the first one. The first one is believed to geometrical, using anthyphaeresis (ἀνθυφαίρεσις), i.e., proving geometricallly that the euclidean algorithm of dividing $1+\sqrt{2}$ by $1$ is periodic: \begin{align} 1+\sqrt{2}&=2\cdot 1 +v_1, \\ 1&=2\cdot v_1+v_2, \\ v_1&=2\cdot v_2+v_3, \\ \text{etc} \end{align} and thus $1+\sqrt{2}$ and $1$ are inconsummerable (ἀσὐμμετρα). It is noteworthy that, although the number theoretical proof appaears Euclid's Elements, which were written c. 300 BC, the fact that there is a proof that the square roots of positive integers less than 19 is mentioned in Theaetetus of Plato, writeen c. 380 BC. Anthyphaeresis works for every $n$, but it can get extremely complicated, as $n$ gets larger. In fact, for $n=19$, in order to establish periodicity of Euclidean algorithm, 6 steps are required, and huge geometrical figures to observe it! A few years ago I supervised a Master's thesis on this proof, and I think it makes an extremely interesting lecture.

Theorem. $\sqrt{2}$ is irrational.

This is an ancient theorem, about 2400 years old, and its modern proof is identical to the one appearing in Euclid's elements. A simple number theoretic proof, where you get the chance to use the abductio ad absurdum (or εἰς ἄτοπον ἀπαγωγή).

Note. As Victor Protsak noted, the number-theoretical proof is not the first one. The first one is believed to geometrical, using anthyphaeresis (ἀνθυφαίρεσις), i.e., proving geometricallly that the euclidean algorithm of dividing $1+\sqrt{2}$ by $1$ is periodic: \begin{align} 1+\sqrt{2}&=2\cdot 1 +v_1, \\ 1&=2\cdot v_1+v_2, \\ v_1&=2\cdot v_2+v_3, \\ \text{etc} \end{align} and thus $1+\sqrt{2}$ and $1$ are inconsummerable (ἀσὐμμετρα). It is noteworthy that, although the number theoretical proof appears Euclid's Elements, which were written c. 300 BC, the fact that there is a proof that the square roots of positive integers less than 19 is mentioned in Theaetetus of Plato, written c. 380 BC. Anthyphaeresis works for every $n$, but it can get extremely complicated, as $n$ gets larger. In fact, for $n=19$, in order to establish periodicity of Euclidean algorithm, 6 steps are required, and huge geometrical figures to observe it! A few years ago I supervised a Master's thesis on this proof, and I think it makes an extremely interesting lecture.

Theorem. $\sqrt{2}$ is irrational.

This is an ancient theorem, about 2400 years old, and its modern proof is identical to the one appearing in Euclid's elements. A simple number theoretic proof, where you get the chance to use the abductio ad absurdum (or εἰς ἄτοπον ἀπαγωγή).

Theorem. $\sqrt{2}$ is irrational.

This is an ancient theorem, about 2400 years old, and its modern proof is identical to the one appearing in Euclid's elements. A simple number theoretic proof, where you get the chance to use the abductio ad absurdum (or εἰς ἄτοπον ἀπαγωγή).

Note. As Victor Protsak noted, the number-theoretical proof is not the first one. The first one is believed to geometrical, using anthyphaeresis (ἀνθυφαίρεσις), i.e., proving geometricallly that the euclidean algorithm of dividing $1+\sqrt{2}$ by $1$ is periodic: \begin{align} 1+\sqrt{2}&=2\cdot 1 +v_1, \\ 1&=2\cdot v_1+v_2, \\ v_1&=2\cdot v_2+v_3, \\ \text{etc} \end{align} and thus $1+\sqrt{2}$ and $1$ are inconsummerable (ἀσὐμμετρα). It is noteworthy that, although the number theoretical proof appaears Euclid's Elements, which were written c. 300 BC, the fact that there is a proof that the square roots of positive integers less than 19 is mentioned in Theaetetus of Plato, writeen c. 380 BC. Anthyphaeresis works for every $n$, but it can get extremely complicated, as $n$ gets larger. In fact, for $n=19$, in order to establish periodicity of Euclidean algorithm, 6 steps are required, and huge geometrical figures to observe it! A few years ago I supervised a Master's thesis on this proof, and I think it makes an extremely interesting lecture.