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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  • $\begingroup$ See this duplicate. $\endgroup$ – Dietrich Burde Mar 21 '18 at 19:48
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    $\begingroup$ Possible duplicate of What would a base $\pi$ number system look like? $\endgroup$ – Blue Mar 21 '18 at 19:49
  • $\begingroup$ "In this new basis can $pi$ have an exact amount?" The answer is yes because from your definition $\pi=1_{\pi}$ $\endgroup$ – KKZiomek Mar 21 '18 at 19:49
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    $\begingroup$ @KKZiomek $\pi$ has an "exact amount" also in basis $10$. $\endgroup$ – Dietrich Burde Mar 21 '18 at 19:49
  • $\begingroup$ Also worth noting that a similar "system" is used and is useful. There's a thing called the diameter tape, basically you measure something circular with it, and it gives you automatocally the diameter of that object $\endgroup$ – KKZiomek Mar 21 '18 at 19:51

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".


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