Hey while going through this answer by Charles Slade to this question on quora :
He states that
"the length of any line segment you can construct [with compass and straightedge] is, in fact, algebraic. Meaning, the length is a root of some polynomial with rational coefficients."
SO how can I go about proving it?
EDiT:put on hold as unclear what you're asking by mrf, LutzL, Sawarnik, Joey Zou, Shailesh 2 days ago Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking
TO Mods/Whomever-Concerned : I'm a High-school student and I don't have graduate mathematical training.The question indirectly asked on quora about squaring a circle,I also had same question but I was given same excuse like OP:Galois theory,Algebra,etc... I stumbled across this answer posted on quora and came across the given :
"the length of any line segment you can construct [with compass and straightedge] is, in fact, algebraic. Meaning, the length is a root of some polynomial with rational coefficients."
I attempted to prove it myself but I can't prove it.Tried to google it but couldn't do it
So I posted it here to see if anybody out here can help me prove or intuitively understand the above thing using only high-school level math?
^To be more precise :a standard math course equivalent to that taught in Soviet/Russian schools
The linked question is of course non-sensical, the quoted "was only off by 0.06 of a square inch in area" tells you exactly that the allegedly algebraic number that was constructed is not $2\pi$, the circumference of the unit circle.
An approximation is not an exact value. The algebraic numbers are dense, as are the rational numbers, in the real numbers. The properties of an approximation do not translate automatically into properties of the exact value.
After guessing what you really mean, i.e., that you referred to the first answer in the linked site, not the question, I first invite you to read again the full answer on quora. One can not provide a better answer in the frame provided here.
As was said in the comments, by arctic tern esp., your original question does not describe what you probably mean. The fact that any length you can construct with compass and straightegde is algebraic (in the coordinates of the given points) is a key fact of analytical geometry and contained in the theory of constructible numbers.