What would a base $\pi$ number system look like?

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:

$[1] = 1.\pi^0 = 1$

$[2][1] = 2.\pi^1 + 1.\pi^0 = 2\pi + 1$

$[2.3][1] = 2.3 . \pi^1 + 1.\pi^0 = 2.3\pi + 1$

$[1][2][3] = 1.\pi^2 + 2.\pi^1 + 3.\pi^0 = \pi^2+2\pi+3$

• How do you write the first few integers in this representation? – Simon S Jun 10 '15 at 17:46
• Should it not be $\pi=10$ and $\pi^2=100$ ? – Peter Jun 10 '15 at 17:51
• In this representation, no rational number $x$ can be represented exactly because some coefficient in $x=a_n\pi^n+...+a_1\pi+a_0$ must be transcendental. – Peter Jun 10 '15 at 17:53
• Moreover, the number $\pi^2$ has no intuitive meaning. It is related though to the sum $$\sum_{j=1}^{\infty} \frac{1}{j^2}=\frac{\pi^2}{6}$$ – Peter Jun 10 '15 at 17:58
• Some related information en.wikipedia.org/wiki/Non-integer_representation – pjs36 Jun 10 '15 at 18:01