I have a question, is it possible to use $\pi$ as a single unit like we get $3,14$ as $1$, and use it to develope a new numerical basis based on $\pi$ in which number $2$ will be $\pi \cdot 2$ , and if so does it exist? in this new basis can $\pi$ has an exact amount?
closed as unclear what you're asking by Lord Shark the Unknown, TheSimpliFire, Dietrich Burde, Blue, futurebird Mar 21 '18 at 22:32
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Yes. Non-integer bases can be used to represent numbers. See https://en.wikipedia.org/wiki/Non-integer_representation#Base_%CF%80 But $\pi$ is $\pi$ (and is an exact number) no matter what, so I don’t understand the last part of your question. In base $\pi$, the representation of $\pi$ (assuming you use the usual single digits) will be $10$, but that doesn’t make $\pi$ equal to “ten.” It’s still a little bigger than three.
It kind of depends...
There's no problem in having base $\pi$ in a positional number system, but it's hard to find $\pi$ different symbols, so most (all?) numbers will have multiple representations.
And obviously $\pi$ will representable as $10$ in that system, if that's what you mean by "has an exact amount".