Imagine if we used a base $\pi$ number system, what would it look like? Wouldn't it make certain problems more intuitive (eg: area and volume calculations simpler in some way)?
This may seem like a stupid question but I do not remember this concept ever being explored in my Engineering degree. Surely there is some application to the real world here.
I am interested in answers that demonstrate which problems would become more elegant to represent and compute. I am also interested in any visualizations that leverage the meaning of that scale. Never-mind a logarithmic scale, what would a $\pi$arithmic scale be and what would simple areas on it mean?
From the comments, I realise that the normal representation of numbers is flawed (or difficult to use) for this idea, so maybe it's worth modifying it slightly.
eg:
let:
$[1] = 1.\pi^0 = 1$
$[2][1] = 2.\pi^1 + 1.\pi^0 = 2\pi + 1$
$[2.3][1] = 2.3 . \pi^1 + 1.\pi^0 = 2.3\pi + 1$
$[1][2][3] = 1.\pi^2 + 2.\pi^1 + 3.\pi^0 = \pi^2+2\pi+3 $
Best Answer
No problem will became easier to solve by choosing a different base, because they depend on the numbers, not on their representation. The solution of some numerical problems might look nicer, for some definition of nice, but that's as far as you'd go.