# Number system and $\pi$

Ok, we all use the decimal system with numbers from 0 to 9. And we have $\pi$ with an infinite number of decimals. We also have a boolean system or hexadecimal. Is there any decimal system where $\pi$ has an ending number of numbers?

• No, only with number system with $\pi$ as a base. – m0nhawk Jun 4 '15 at 13:59
• No, because if we did then $\pi$ would be a rational number, which it is not. – Simon S Jun 4 '15 at 14:00

## 3 Answers

No -- since $\pi$ is irrational it does not have a terminating representation in any positional system whose base is an integer.

If a number has a finite expansion, in a rational base, using rational digits, then the number is rational. This is because the sum and product of rational numbers is rational.

Note: Even some rational numbers have non-terminating expansions in base 10. For example, $1/3$.

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.