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 ...

In base pi, pi is represented as 10. The same way that 10 is represented as 10 in base 10. If you are asking if pi is normal in base pi then the answer is no because it only has the digits 1 and 0 in it.

Jul 24, 2013 · A number is rational if it can be written as the quotient of two integers. All of these concepts can be developed without any reference to what base you work in. And as such, rational is indepent of the base. Thus, ##\pi## is irrational in any base you work in. Regardless whether you can write it as ##10## in base ##\pi##.

See here for formal definition of representation in a non integral base. With such a definition $2$ in base $\pi$ is simply... $2$. This because $2<\pi$ hence it can be represented by itself. In any base $\beta$ most numbers (all but a countable set) have an infinite representation. Only sums of powers of $\beta$ have a finite representation.

Aug 6, 2017 · This means that no rational number can be written as a sum of powers of pi, which is equivalent to the fact that you can not write any rational number base pi. Edit: Of course, you can always write the numbers non negative integers less than or equal to the base in that base, like 2 base pi is still $2\cdot \pi^0 = 2$.

May 11, 2015 · One can use any number as a base. For example, if we were to use [itex]\pi[/itex] as a base, then any number would be written as [itex]a+ b\pi+ c\pi^2+ d\pi^3+ ...[/itex], writing only the coefficients, a, b c in reversed order, That is [itex]2+ 3\pi+ \pi^2[/itex], which is about 21.29438 in base 10, would be written as 132 in base [itex]\pi[/itex].

Jun 4, 2015 · The decimal system by definition has 10 digits, or at least this is my understanding. Do you mean perhaps a base (like binary, octal, hex, decimal) in which pi has a finite (an ending) number of numbers? If so, then e.g. base pi would do the job, though representing other numbers, like e, or ten (zehn, dix, ..) might be difficult in this system.

Feb 7, 2019 · I don't know if the numerical system with base $\pi$ has any practical use, but the golden ratio one has: you can for instance look at Beta Encoder, a system to convert from analog to digital. Share Cite

May 27, 2017 · In other bases, $\pi$ wouldn't start off like the familiar $3.14\dots$, and interesting coincidences like the Feynman point won't exist any more. On a deeper level, we don't know if the digits of $\pi$ appear "uniformly" in base 10, nor in any …

Feb 3, 2015 · I was just thinking about why there are 360 degrees in a circle, and why an hour is composed off 60 minutes, a minute of 60 seconds... That is, all because the Babylonians used a sexagesimal (base-60) positional numeral system. Now I was wondering about what would happen if we used a base-Pi numeral system.