Irrationality of square root of n
- Author:
- Christian Mercat
The A4 format is defined by a ratio of width and length of √2. That is, a half A4, an A5 sheet, has the same shape, which is obtained by folding an A4 in half.
By including an A5 in an A4 in the same direction, in a corner, it appears a diagonal rectangle of the same shape and strictly smaller than the A5.
In the same way, the number √n is defined by the fact that a rectangle whose sides are in this ratio can be folded into n small rectangles, bands of the same shape.
If N is the integer part of √n, we can place this band N times along the diagonal. When √n is not integer, there remains a rectangle (orange) of the same shape but strictly smaller.
If the measurements of the sides of the strip are whole, those of the small orange rectangle are also (if it exists). This is a contradiction because they are strictly smaller and the process can be repeated. Therefore √n is whether integer (the orange rectangle vanishes) or irrational.