Euclidean geometry and irrational numbers.

I was wondering, given a square that is $1 \times 1$, how can we know that the diagonal is an irrational length geometrically??? We could use the Pythagorean Theorem to see that the diagonal of a square is $\sqrt{2}$... But how can a finite length have an infinite sequence of numbers.... I think there's two main questions then :

How to determine geometrically that a length is irrational and How can a finite length be irrational ?

Thank you!

• $1/2+1/4+1/8+1/16+...$. This series is infinite but it converges to the value of 1. May 25, 2014 at 0:16
• There are undoubtedly discussions of geometric incommensurability proofs on MSE. The following Cut the Knot article is very good. Too bad I don't feel like drawing pictures, there is a very nice one for $\frac{\sqrt{5}-1}{2}$. May 25, 2014 at 0:26
• Do you have any problem with a rational number such as $1/3 = 0.33333...$ being a finite length? It also has an infinite sequence of numbers. May 25, 2014 at 0:26
• I guess some rationnal numbers could also be included in the problem!
– user108343
May 25, 2014 at 0:28
• Ok, thanks André Nicolas, I'll go read it !
– user108343
May 25, 2014 at 0:29

We all know a measure that can be with some approximation,

In the diagram we will find why square root of 2 is irrational, you can easily see that the image represents a sequence of square. now look at the arms of any two consecutive squares, the arms are not parallel, but they are having a fixed ratio! so that can not be written as the form of (p/q). Actually you should notice we can't measure the perimeter of any circle or circular loop. Because our pencil compass is unable to take the loop in a form of straight line. with regards supriyo saha.