Pointing out $\sqrt 2$ on number line We know that using a compass and a ruler we can point out $\sqrt 2$ on the number line. But we don't know the last digit of $\sqrt 2$. So how can we be sure that the pointed number is $\sqrt 2$?
 A: This isn't a direct answer to your question, but it should give some intuition for how such a thing could work.
Say you draw a circle of radius $1$. You know how to find the circumference of a circle, so you know the circumference of a circle is $2\pi$. You can't measure it to be exactly $2\pi$ (maybe if you found a way to measure it with a ruler you'd measure it to be $6.3$ or $6.28$ or something), but you know that, because of mathematical properties of a circle, the circumference must be $2\pi$.
Now draw two segments of length $1$ perpendicular to each other that share an endpoint, and connect the two other endpoints with a segment. You've now made a right triangle with legs both of length $1$, so you know by the Pythagorean theorem that the hypotenuse is of length $\sqrt 2$. If you measure it, you may get a length of $1.4$ or $1.41$ or $1.42$, but mathematically, you can prove that (in an idealized version of your drawing where lines have zero thickness, etc.) the length is $\sqrt 2$.
