# When and who was the first mathematicians to prove rigorously that $\sqrt[3]{2}$ was impossible number? [closed]

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!

• Wikipedia gives a clear answer on this: Pierre Wantzel in 1837. Jan 13 '16 at 21:50
• Trivial search on Wikipedia gives the answer (and the proof too). Jan 13 '16 at 21:51
• 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.
– KCd
Jan 14 '16 at 4:28

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 $1$.
• 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. Jan 14 '16 at 7:07
• If you assume that $\sqrt[3]{2}$ really doesn't exist on the real line number (with rigorous proof), what would you conclude then!? Jan 19 '16 at 11:42