On Wednesday, November 8, 2017 at 12:25:59 PM UTC+3, Quadibloc wrote: > 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... > > John Savard
Then it must be an easy task for you to design exactly a cube in our physical reality with any distance length of its side such that the cube represents a sum or a difference of two other cubes with positive distinct integer sides? can you? wonder!