It seems to me that when mathematicians consider circles, they normally think of them in terms of the radius. A “unit circle” is a circle with a radius of 1, not a diameter of 1. Angles are measured in terms of the radius, not the diameter. And yet, pi is defined as the ratio of the circumference to the diameter, not the radius. This leads to the factor of 2 cropping up all the time when we use pi. E.g. there are 2.pi radians in a circle, not just pi radians. It does sometimes work the other way, with pi.r[sup]2[/sup] being the obvious example, but generally speaking there seem to be a lot of "2.pi"s knocking around in famous formulas and equations in maths and physics. Other times, there is one pi when maybe it would be clearer if it were only half of pi, like in Euler’s recently-discussed equation. If it were e[sup]i.pi/2[/sup] = -1, that might better convey the key idea of it being analagous to a rotation of half a circle.
So when we make contact with alien mathematicians, are they going to think we’re nuts for using 3.14159…, which to them will be half of their “pi”? They’d have “pi” radians in a circle, which would seem neater, and their textbooks will be blessedly free of “2.pi” all over the place.
(I realise it doesn’t really matter, it doesn’t change the mathematics of it. But just for aesthetics’ sake.)
The way I think of it is: the area of a circle is pi * r^2, so obviously the circumference is 2 * pi * r (the derivative).
Also, Think about how you draw a circle. You have a string, tacked into a point, and you stretch the string and draw with the pencil. The string length is the radius, not the diameter. It makes sense to me to think of that as the base of a circle.
Funny you should ask, because I was thinking of the wide variety of places pi turns up and wondering if I could write a computer program to seek them out and test whether pi or 2pi is more common.
Though, it would be a shame to spoil e^(pi*i) - 1 = 0, because every single token in the expression is either one of the five most essential numbers or one of the four most essential operators. I had to take liberties to write it here - exponentiation is more neatly represented with a superscripted expression, which would have saved using parentheses.
Not all arcs are complete circles, or even half circles. All arcs have radii but not all arcs have diameters. Pi is just as useful for little bitty pieces of circles as for ones that have an actual diameter.