>Why do we really need those real non-constructible numbers, if it is impossible to express them exactly except only by constructible numbers or as meaningless notation in mind only?
First of, the ability to express anything is irrelevant. Your squareroots are just as unexpressable as cuberoots for example.
Secondly, the reason is simple. Rational numbers (or any finite extensionso f htem and even the extension of 2^n'th roots) don't possess all the properties that are needed to make a good foundation for analysis. the supremum property is of outmost importans in a lot of things, including calculus.
>And this intuitive and so simple question not necessarily directed to those described as professional mathematicians since we know exactly how are those captured by so many meaningless definitions that make it impossible for them to think outside their so tinny box or far behind their long noses
Actually they have thought on all of this long before you were born so don't put yourself on a pedestal. They are smarter than you, more knowledgable than you and know the value of these things.
>People hiding their true identities by using many fictional and masked names
You mean like YOUR OWN FUCKING NAME!? Hypocrite!
>Neither does anyone need them nor can anyone use them. At least not in mathematics. The only occupation for such non-existing creatures is matheology.
We can use a lot of them and it is not about the individual numbers. The fact is, mathematics cares very little about the individual numbers or elements of anything. It is the general structure of things that is important. The fact that reals have the supremum property is what is valuable about them.