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 $\sqrt 2$?
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3The construction ensures it – Soumyadwip Chanda Sep 16 '20 at 02:16
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2There are irrational constructible numbers – J. W. Tanner Sep 16 '20 at 02:16
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1There is no such thing as the last digit of $\sqrt{2}$. – Thorgott Sep 16 '20 at 03:03
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1We also don't know the "last digit" of 1/7, does that mean we can't divide a line segment into 7 equal pieces? – Sep 16 '20 at 04:48
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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 $\sqrt 2$.
Carl Schildkraut
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@Mathfun The length is $\sqrt{2}$ in an ideal scenario, i.e. disregarding human error. Like what Carl Schildkraut said, this assumes that all lines are of zero thickness and we can thus invoke proven mathematical theorems, such as the Pythagorean Theorem. – Sage Stark Sep 16 '20 at 02:27
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@Mathfun Unsure if this is what you are looking for. Extending Carl Schildkraut's answer, once the length of $\sqrt{2}$ is identified, you can then draw a circle centered at the origin with that length as its radius. The place where the circle intersects the positive side of the $x$-axis will represent the location of $\sqrt{2}$ on the number line. – user2661923 Sep 16 '20 at 02:29
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1@user2661923 I know the construction,but it amused me that even we are not sure the exact quantity of $\sqrt 2$ we can still pointing it on number line. But as we see in the answers that it is an ideal scenario which is impossible in the real world. – Shreya Chauhan Sep 16 '20 at 02:36
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1@Mathfun You don't need to talk about the "last digit of $\sqrt{2}$", which doesn't even make sense. Suppose the coordinate of the constructed point differed from $\sqrt{2}$. Then there would be some least decimal place in which it differed. The circle through the origin with the constructed length as radius would then miss the off-axis vertex of the right triangle in the construction by at least a certain amount, depending on that least decimal place. This would be a contradiction. – Will Orrick Sep 16 '20 at 02:52