The parameter a for the closed curve #r = f(theta;a)# is rational. For example, a is rational in #r=a(1+cos theta)#. How do you prove that its length, area enclosed and the volume of the solid of revolution, about an axis, are all transcendental?
The trigonometric functions are all
convenience that we convert the transcendental
It is impossible to graduate, in mathematical exactitude,
the x-axis for y = sin x, for the amplitude y = 1., on the same scale.
Also, this is the problem in evaluating orbital characteristics of
None of them, like eccentricity, period, speeds, ..., could be
expressed in mathematical exactitude, in the form of finite-sd digital
strings. All we use, like 1 year = 365.256363004... days, are finite-sd
It is important that, as arc of a unit circle,
A ticklish issue here is the marking of the edge between irrational
and transcendental numbers, to identify indubitably which is which.. .