82.10 Estermann and Pythagoras

102.21 An addendum to Estermann's proof of the irrationality of

“…we obtain a proof that the square root of an integer is either an integer or an irrational.". Flanders [2] carried out the extension that Estermann remarked upon, and Siu [3] gave a geometric interpretation of Estermann's argument, making a plausible historical case that it could well have occurred to the Pythagoreans. Apostol [4] gave a pretty geometric proof of the irrationality of the square root of 2, which aligns perfectly with Estermann's proof.…”

Using A4-sized paper to illustrate that is irrational

“…Alternatively (*) may be embedded into the proof by contradiction via the well-ordering principle: this is Estermann's charming proof [1] with follow-ups in [2,3,4]. Finally, (*) may be illustrated geometrically as in Apostol's proof [5] based on Figure 1(i): see also [6]. A variation starting from two superimposed triangles is shown in Figure 1 Our last two proofs make use of the fact that A3, A4, A5 rectangles are all similar and that folding each one in half generates its successor.…”

95.42 Irrational square roots of natural numbers — a geometrical approach

“…This can be written as [2] discussed this method in 1998 and showed that it was equivalent to the algebraic statement that if V2 =~then also V2 =~b_-t with a smaller denominator. Then in 2000 Apostol [3] gave the same method, but with a simpler construction, and pointed out that it also applies to any square roots of the form {ii'f+f or~where n > I is an integer.…”