Is it possible to write any rational number using base pi

Question

Is it possible to write any rational number (say 1 or 2 or .15) using a number system that was base pi instead of a number system that used a rational number as its base?

Answer 1

We can write rational numbers in irrational bases; easily shown by $100$ base $\sqrt 2$ is 2. There are in fact problems and papers that I have seen that use base phi, $\frac{\sqrt 5 + 1}{2}$.

However, because pi is transcendental, it is not the solution of any polynomial with rational coefficients. 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 $2\cdot \pi^0 = 2$. I was talking about everything else.

Answer 2

EDIT : The issue : The general setup for a given base $b$ is to write a given Real number $a$ as $a_0 b^0 + a_1b^1+....+a_k b^k$ with $a_k$ an integer less than the floor of $b$.

EDIT 2: STILL needs some work, which I am doing right now. Will be back soon to rewrite, or, if necessary, delete. Comments are welcome.

In our case, for $\pi$ we want to represent ( I assume) a Real number $a$ as $ \pi^0 a_0+ \pi^1 a_1+....+ \pi^n a_n ; a_j < \pi$

at each stage, you choose the largest number $a_k$ , so that $$a_0 \pi^0+a_1\pi^1+...+a_k \pi^k \leq a $$.

This is a monotone, non-decreasing sequence of Real numbers bounded above by $a$, so it will converge to its $lub=a $.

I think we need as a basis a Real number $x; |x|>1$.

Answer 3

This isn't an answer. But it's too big to be a comment.

There is a problem with non-integer bases. Let $B$ be an integer and let $b=B-1$. Then $$(0.bbbbbb\dots)_B = 1_B$$ just as we would expect.

In base $\pi$, however, $\pi - 1 < 3$. So it shouldn't come as a surprise that $$(0.333333\dots)_B = \dfrac{3}{\pi-1} \approx 1.40083_{10} > 1_\pi$$

So ordering numbers base $\pi$ is more complicated than ordering numbers in an integer base. It also seems that some numbers can be expressed in more than one, non-trivial, way.