TITLE: Is the arc length always irrational between two rational points?
QUESTION [20 upvotes]: Recently I was wondering: Why does pi have an irrational value as it is simply the ratio of diameter to circumference of a circle? As the value of diameter is rational then the irrationality must come from the circumference.
Then I used calculus to calculate the arc length of various functions with curved graphs (between two rational points) and found the arc length two be irrational again.
Do all curved paths have irrational lengths?
My logic is that while calculating the arc length (calculus) we assume that the arc is composed of infinitely small line segments and we are never close the real value and unlike the area under a curve, there do not exist an upper and lower limit which converges to the same value.
If yes, are these the reasons irrational values exist in the first place?

REPLY [5 votes]: No.
Take any smooth curve between two rational points and deform it to change its length by a finite amount. During deformation, you will cross infinitely many rational lengths.
A simple example is a polynomial having two rational roots, times a variable factor.

Now consider the curve of parametric equations
$$\begin{cases}x=\dfrac{t^3}3-t,\\y=t^2\end{cases}$$
(a modified Tschirnhausen cubic).
We have
$$s=\int_a^b\sqrt{(t^2-1)^2+4t^2}\,dt=\int_a^b(t^2+1)\,dt=\frac{b^3-a^3}3+b-a,$$
so that the length between two rational $t$ (giving rational endpoints) is always rational.