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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!

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  • $\begingroup$ $1/2+1/4+1/8+1/16+...$. This series is infinite but it converges to the value of 1. $\endgroup$
    – user3001408
    May 25, 2014 at 0:16
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    $\begingroup$ 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 $\frac{\sqrt{5}-1}{2}$. $\endgroup$
    – André Nicolas
    May 25, 2014 at 0:26
  • $\begingroup$ 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. $\endgroup$
    – Neil W
    May 25, 2014 at 0:26
  • $\begingroup$ I guess some rationnal numbers could also be included in the problem! $\endgroup$
    – user108343
    May 25, 2014 at 0:28
  • $\begingroup$ Ok, thanks André Nicolas, I'll go read it ! $\endgroup$
    – user108343
    May 25, 2014 at 0:29

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We all know a measure that can be with some approximation,i think you are saying about this

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.

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