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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?

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    $\begingroup$ " now this number is irrational we cannot say its 100% accurate since we don't know it's full decimal expansion" Yes we can say it is 100% accurate to by $\sqrt{2}$. Absolutely. "now if I take my ruler and measure the length of hypotenuse I get a value which doesn't seem to be never-ending" Look closer. The number you measure does not fall precisely on any division mark. If you make your division marks finer and finer by dividing the previous marks in 10 it will never land on any mark. But there's no reason your ruler can't have a $\sqrt {2}$ mark. Many mechanical rulers do. $\endgroup$ – fleablood Mar 21 '17 at 6:21
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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.

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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.

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    $\begingroup$ I have to absolutely disagree. If real numbers are more accurate than the real word (and the are) you can't say real numbers are "approximate" or a "guess". It'd be much more accurate to say reality is an approximation of the real numbers. Also in reality it's just as accurate and true to say the sides will never measure exactly 1. In fact, it'd be just as accurate to say the hypotenuse is exactly root 2 but the sides of 1 can never be measured. But in imperfect reality nothing is precise. But it's reality not the real numbers, that are the "guess". $\endgroup$ – fleablood Mar 21 '17 at 6:35
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    $\begingroup$ @fleablood I suspect we disagree over a philosophical rather than a mathematical issue. $\endgroup$ – John Wayland Bales Mar 21 '17 at 6:37
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    $\begingroup$ Not really. My point is you can't say that reals are an "approximation" of reality if the reals are more precise. You can and accurately say nothing can be measured to to precision. But if you say that it has to include the rationals and integers as well as the irrationals. If you like, if we can imagine in the real world machines that can measure and substances that can be engineered down to the Plank level then the hypotenuse would never line up with any marking for any decimal precision larger then the Plank level. I'll give you that. ... unless space is curved. That could happen... $\endgroup$ – fleablood Mar 21 '17 at 6:48

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