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!
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.