On Monday, November 6, 2017 at 12:21:29 AM UTC-7, bassam king karzeddin wrote: > 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?
The real number line is a nice, simple thing.
A line consisting only of the rational numbers is more complicated. And because squares have diagonals, and circles have both a circumference and a radius, we would have to add other numbers to that.
One can use functions like the log and trig functions on rational numbers to construct other numbers that are no longer rational.
But that can also be done with other less well-known functions, like the Bessel functions or the Gamma function. These were new functions that can be defined using things like integrals from the older, simpler functions, but they aren't just combinations of them by arithmetic operations.
Mathematicians can come up with new functions, making more numbers constructible. So those numbers must have existed all along.
Thus, a construct which refers simply to "all the numbers", which doesn't give the appearance of implying that if you take a stick and cut it with a knife, the exact length of the result has to be constructible with mathematical operations (and why would anyone want to think that?) avoids the situation where new numbers pop out of an impenetrable fog...