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The purpose of the question is to understand why the number $\sqrt[3]{2}$, that was proven rigorously by ancient Greek is an impossible number (even at infinity), by their three famous impossibility problems,

this one was simply the impossibility of doubling the cube,

But, much later and oddly after many centuries, particularly (in the middle ages), mathematicians fond $\sqrt[3]{2}$ also at infinity, and they widely accepted it as a real number, which would imply the invalidity of the most famous Ancient Greek rigorous proof!

Of course it so clear to even a school student that $$a^3 = 2b^3$$ in positive constructible numbers is impossible equation, thus $\sqrt[3]{2}$ is also impossible number as a result, this was the basis of Greek famous rigorous proof

I had chosen this number $\sqrt[3]{2}$ in particular since it was the oldest famous number that encountered the Greek with a tremendous challenge that is still valid!

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    $\begingroup$ Wikipedia gives a clear answer on this: Pierre Wantzel in 1837. $\endgroup$ – Alexandre Eremenko Jan 13 '16 at 21:50
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    $\begingroup$ Trivial search on Wikipedia gives the answer (and the proof too). $\endgroup$ – Alexandre Eremenko Jan 13 '16 at 21:51
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    $\begingroup$ You appear, based on past questions, to be obsessed by the idea that a number not being constructible by straightedge and compass might mean it does not exist. This viewpoint is not mainstream mathematics. Suggesting that $\sqrt[3]{2}$ does not exist in the real numbers is not an attitude shared by 99.44% of mathematicians (probably more). By the intermediate value theorem, $\sqrt[3]{2}$ exists in the real numbers. $\endgroup$ – KCd Jan 14 '16 at 4:28
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    $\begingroup$ It is not a matter of attitude shared by the vast majority of mathematicians, it is rather the fact that is very bitter & misleading all other branches of science "especially physics", the truth must rule the science regardless of our opinions and sweet games, the proof of nonexistence of all non constructible numbers, & the infinite decimal expansion for any constructible numbers is too silly to state even to a non mathematicians, but how would you know if your margin is too narrow to contain answers without even a mathematical reason, fortunately, the margin of internet is too vast .. $\endgroup$ – Bassam Karzeddin Jan 17 '16 at 15:51
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    $\begingroup$ I think this is the most important question in the history of mathematics that some find too difficult to follow or may know the consequences for this most innocent question where they don't want to encounter, every body almost knows that the ancient Greek that cube root of two was proved rigorously "impossible number" to exist even at whichever infinity you wish, despite this and many centuries later in the middle ages, they accept this number as a real number by means of approximations, neglecting the fact that constructible numbers are so dense to the limit that they never allows such numb. $\endgroup$ – Bassam Karzeddin Jan 23 '16 at 15:17
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Gauss considered the algebraic background behind straightedge and compass constructions, and from his work it is clear that e.g. $\sqrt[3]{2}$ is impossible to construct, as it can't be expressed via a finite number of sums, products, and square roots starting with $1$.

That it can't be constructed with some random selection of tools doesn't mean it doesn't exist.

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    $\begingroup$ I looked into those proofs mentioned by Alexandre, I'm afraid they aren't rigorous but intuitive conclusions, I wish there is a rigorous one somewhere else, note also the impossibility of constructing $\sqrt[3]{2}$, was prior to Wantzel, (wondering Why), also ancient Greece were not considering $\sqrt[3]{2}$ as a number. the proof of doubling the cube was originally by Greek, the intermediate value theorem is not rigorous in this regard since again it is a conclusion "it must between two points, because there is no finite steps for this conclusion. I say this is more important to physics. $\endgroup$ – Bassam Karzeddin Jan 14 '16 at 7:07
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    $\begingroup$ @bassamkarzeddin, for the ancient Greeks, "number" was a natural number (really not even that, 1 was "unity", not a number). They didn't conceive even of rationals as numbers. But they clearly considered the length of the diagonal of the square, the side of the cube of twice the volume, and the circumference of a circle as bona fide lengths of line segments. $\endgroup$ – vonbrand Jan 15 '16 at 10:02
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    $\begingroup$ If you assume that $\sqrt[3]{2}$ really doesn't exist on the real line number (with rigorous proof), what would you conclude then!? $\endgroup$ – Bassam Karzeddin Jan 19 '16 at 11:42
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    $\begingroup$ @bassamkarzeddin, that the "real number line" which includes all real numbers but is missing some of them is a mighty peculiar beast... In the proof of said hole, I'd go chasing for the mistake. $\endgroup$ – vonbrand Jan 19 '16 at 17:38
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    $\begingroup$ certainly you would find a hole, It would save the world from all those fectious numbers that never existed on the real line number, but, you must be very careful here, note that constructible numbers are so dense to the limit that they don't accept any infinite number to be within the real line number, even for their own string expansion, to arrive at this fair truth, don't accept any kind of approximation given by (infinite, limit, convergence, famous cuts,...etc), mathematics facts require exactness even at whichever infinity you like, nothing else more is required, Good luck... $\endgroup$ – Bassam Karzeddin Jan 23 '16 at 12:53

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