It is not an exercise worth ignoring: I had also once thought that pi should be replaced.
Slightly different motivation, though, as it seemed to me that pi/2 was really more interesting, and anyway it is really much easier to write multiplication than fractions when you run into a discrepancy. Here the roots of the sine are at even multiples and the roots of the cosine are at odd multiples, and e^(i pi/2) = i is more interesting than the previous incarnations of the Euler formula.
The thing is that the important parts of these formulae are the way they are derived and the structure they represent, not their appearance on a sheet of paper. If you can understand what relates the zeta function at even numbers to the ratio of a circle's radius and circumference, it really does not matter if you choose to represent this relation with a drawing of an elephant.
You got a good case with pi/2, it's truly useful, as it represents the angle between 3d axes, the boundaries of the tangent and inverse trigonometric functions, the monotonic regions of cosine and sine etc. It will even make spherical coordinates easier to visualize mentally. (It will also make a two-digit number, 16, a common resident in lots of formulas). I vouch for it, and i call for a move to name it, confusingly, pi-bar
Slightly different motivation, though, as it seemed to me that pi/2 was really more interesting, and anyway it is really much easier to write multiplication than fractions when you run into a discrepancy. Here the roots of the sine are at even multiples and the roots of the cosine are at odd multiples, and e^(i pi/2) = i is more interesting than the previous incarnations of the Euler formula.
The thing is that the important parts of these formulae are the way they are derived and the structure they represent, not their appearance on a sheet of paper. If you can understand what relates the zeta function at even numbers to the ratio of a circle's radius and circumference, it really does not matter if you choose to represent this relation with a drawing of an elephant.