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If we make a square with an area of $2 m²$, its side is square root of $2$ then. Wouldn't that mean that the square root of 2 has a concrete length (and therefore point in the real numbers axis). We could measure the side and draw the conclusion square root is $x$.

Wouldn't the same apply to $\pi$, when you get a circle with well defined area. The result of $\pi r²$ is a concrete number, not something that need to be approximately calculated.

I just don't get how some well-known thing can have a non-concrete area.

Shevliaskovic
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Carlos Perez
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    The word "concrete" has no meaning. $\sqrt{2}$ is a real number, it is simply irrational. Just because it doesn't have a terminating or repeating decimal representation doesn't mean it is less real, or cannot represent a real length of something. – Emily May 05 '14 at 13:04
  • Why is this listed under the "infinitesimals" tag? $\pi$ is not infinitesimal – Mikhail Katz May 06 '14 at 12:44

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I am not sure what you mean by a "concrete" number. I think there are just numbers, real numbers, rational numbers, etc.

Yes, $\sqrt{2}$m has a "concrete length", it is just $\sqrt{2}$ in the set of real numbers, which is by definition on the real axis. Likewise, $\pi r^2$ is a real number as well - it may not be a rational number, depending on $r$, even though you can always approximate it with rational numbers by virtue of the denseness of the rationals in the reals - but if you want to be precise, it is just $\pi r^2$, a uniquely defined real number, rational or otherwise. I don't think there is anything not concrete about these numbers.

tvk
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