The exact value of an irrational number As we know we can not get an exact value of the irrational number like $\sqrt{2}$ but as we know also we can represent the irrational number on the number so we can measure the length of the line segment whose length is $\sqrt{2}$  , this means that we got an exact value of it 
So what is the problem with this ? 
 A: I'm going to assume that an exact value is a number, $n$ which is equal to, not an approximation of, an irrational number (Example: $n=\sqrt{2}$). Obviously, this isn't a very rigorous definition, but it gets the point across.
Let's take your example of a line segment which is perfectly equal to the $\sqrt{2}$. The hypotenuse of a 45-45-90 triangle is equal to $\sqrt{2}$ when the legs are each equal to 1, so just visualize that in your mind. Okay, we have our segment, now all we have to do is measure it...right? Well, it's not that easy. Every single measurement is within a certain degree of accuracy -- it can't be perfect. We might get a very good approximation of an irrational number, such as 1.4, but we won't get the exact value. Now, even if we had some impossible ruler that could perfectly measure any line segment, $\sqrt{2}$ (and all irrational numbers) are non-terminating and non-repeating, meaning that we would need an infinitely long screen to display all of the digits (and an infinitely long amount of time if the numbers appear sequentially on our hypothetical ruler). Therefore, it is impossible to get the exact value of a rational number even if we have an impossibly perfect measuring tool.
