Measuring the length of hypotenuse of a right triangle. We all know about Pythagoras and we can use his THEOREM to figure out the hypotenuse of a right triangle.Lets just say the sides of my triangle are both 1 cm and now I use Pythagoras THEOREM and get my hypotenuse as $\sqrt  2$, now this number is irrational we cannot say its 100% accurate since we don't know it's full decimal expansion, now if I take my ruler and measure the length of hypotenuse I get a value which doesn't seem to be never-ending so is that value measured by the ruler still an approximation of that irrational number that I got using Pythagoras THEOREM?
 A: If your ruler is subdivided into milimeters, the resulting measure will not lie precisely on any mark.  It will lie a little over one tenth the way past 1.4 cm.  If your ruler has .1 milimeter markers of even down to the nanometer markings it will never lie precisely on any marking.
But, your ruler doesn't need to be marked only in metric units and factors of 10.  The manufacturer of the ruler could very well include markings for $\sqrt{2}$ on it.  (By doing the exact construction you described and imposing the distance onto the ruler.)  
Of course he can't include every possible measure because then the ruler would have only markings and you couldn't tell them apart.
A: Real numbers are an approximate, idealized version of measurement. Mathematics abstracts from reality omitting certain factual aspects such as the limits on the precision of measurements. This makes mathematics less real than reality, not more.
In a Platonic world of the imagination, the length of the hypotenuse is exactly $\sqrt{2}$. In the real world there are no irrational values and the hypotenuse will never be measured to be exactly $\sqrt{2}$, that is merely the best guess as to what the measurement will reveal.
