Why does convention not recognize $2\pi$ as the fundamental quantity? It looks as though my question might turn out to be a duplicate. If so, it does not need an answer, after all, thanks.
ORIGINAL QUESTION
Why does convention recognize $\pi\approx 3.14$, rather than $2\pi\approx 6.28$, as the fundamental quantity?
You have some contour integrals that come out to $\sqrt{\pi}$ or $1/\sqrt{\pi}$, so that's kind of neat; but I don't know that that arises more often than, say, $1/\sqrt{2\pi}$, as in the Fourier transform. Anyway, all that special-function action is so advanced that it misses what might seem to some to be the main point: $2\pi$ is a circle. How much more fundamental can you get than that?
But mathematicians are smart people (and I am just a building-construction engineer), so I do not doubt that a good reason exists. I just do not know what the reason is. Hence the question.
Why wasn't some other symbol defined, $\kappa\approx 6.28$?
 A: The short answer to your question is that there is another symbol for $2\pi$, namely $\tau$. Some people promote that notation, but it hasn't caught on in the mainstream of mathematics, at least yet (and I don't think it will).  
The longer answer to your question comes from history. The reason that $3.1415...$ got its own symbol $\pi$ before $6.28...$ did is that $\pi=3.14...$ is the ratio of the circumference of a circle to the diameter of the circle. Finding that ratio was a question that Babylonian, Chinese and Greek mathematicians/geometers in antiquity would quickly encounter while studying geometric shapes. Measuring/approximating that constant became an interesting problem for them. Mathematics had to develop much further before the value $2\pi$ would begin to show up often enough to conceivably merit its own symbol. The value $2\pi$ started showing up frequently only after the idea of the radian measure was invented in the 18th century, because then it became natural to think of a full $360^\circ$ rotation as a rotation of $2\pi$ radians. Then as  complex analysis, fourier analysis, etc developed, it would show up more often. 
I have no definite answer as to why people have not started using $\tau$ or some other symbol instead of $2\pi$. I suspect that simply there is no real need for it; $2\pi$ is easy enough to write as it is. 
A: Surely you can find this discussed in many place on the web.  The diameter of a circle is much easier to measure than the radius.  The ratio of the circumference to the diameter was taken as the fundamental constant describing the geometry of the circle.  This is from nearly 2000 years ago in Greek geometry.  (And similar time-frame for other places, perhaps India or China??)  So it is far earlier than contour integrals and Fourier transforms.
See, for examlple, LINK
