Rationals and irrational numbers as coordinates

In the pair (r,theta) where theta is an angle ...of course the
irrational pi is used to describe the theta coordinate.

If one didn't have an irrational number to describe theta.....then the
curve fragment (s=r* theta) would not be a curved line but would be a
series of straight lines approximating a curve. So it seems that when
we go to two dimensions.....the irrationals have to be included in the
domain.

We do this and we get a basic curved surface acting as a coordinate.
Think about how much easier some math operations would be with such a
fundamental change. One major problem with imaginary numbers is that
it is a rectilinear coordinate system.

 

Do you mean as in 0 <= theta < 2*pi ? In what sense is pi used to
describe the theta coordinate? How is the irrationality of pi
relevant? One might measure angles differently and write
0 <= theta < 360.

In any number of dimensions one uses real numbers (rational and
irrational) as coordinates: R, R^2, R^3, ... This isn't something that
just happens in two dimensions with polar coordinates.

What is "a rectilinear coordinate system"? If it's that "one major
problem", why is it a problem? If it's the imaginary numbers, then they
aren't a rectilinear coordinate system. Each imaginary number is a real
number multiplied by i. i, if you like, is the ordered pair
(0, 1), so the imaginary number iy is just (0, y) but neither it, nor
the set of all of them, is a rectilinear coordinate system.