Which of the following numbers is constructible?
1) $3.14141414\ldots$
2) $\sqrt{3}$
3) $5^\frac{1}{4}$
4) $2^\frac{1}{6}$
Also,
Given a segment of length $\pi$, is it possible to construct, with a straight edge and compass, a segment of length $1$?
I dont need full on proofs, just a little explanationif they could be constructed or not.
i have a feeling number 2 and 4 are constructible, just a educated guess because we could get $\sqrt{2}$ from a unit square and the diagonal is that and then we can just extend it I believe.
I just dont get it. Please help out
Best Answer
I'll only answer the last part, because everything else was done by anorton. Given a segment of length $1$ you can't construct a segment of length $\pi$ -- this is what we mean when we say $\pi$ is not constructable.
Starting with a segment of length $\pi$ is just a re-scaling of the constructability problem by a factor of $\pi$. The constructable numbers starting from $\pi$ are precisely the constructable numbers starting with $1$ multiplied by a factor of $\pi$. You can only construct $1$ from $\pi$ if you can construct $1/\pi$ from $1$. This, however, is impossible -- constructable numbers form a field, so if you could construct $1/\pi$ you could also construct $\pi$.