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?