0

It is possible for an irrational base to represent the same number using two finite strings. For example, there is a number $\varphi>1$ such that $\varphi^2-\varphi-1=0$ and therefore $100_\varphi=11_\varphi$.

Is this also possible for a rational base?

(This is a follow-up to the comments on the question "Are exact representations of numbers in fractional bases unique?")

Community
  • 1
Chris Culter
  • 26,806

1 Answers1

0

No.

Suppose there is a rational $b>1$ such that $\sum_{k=0}^na_kb^k=0$ and the integer coefficients $a_k$ are not all zero. Without loss of generality, $a_0$ and $a_n$ are nonzero. And since this polynomial came from comparing two $b$-expansions, suppose that each $|a_k|<b$.

Let $b=p/q$ in lowest terms. By the rational root theorem, $p$ divides $a_0$, so $p\leq |a_0|<b=p/q$, which is absurd.

Chris Culter
  • 26,806
  • Interesting, will go through this carefully... –  Mar 28 '17 at 02:08
  • @selfawareuser Cool, please let me know if you have more questions! – Chris Culter Mar 28 '17 at 18:51
  • Isn't $a_0$ having to be nonzero a problem? Also, I'd probably see it if I stared for long enough but why is the final equation absurd? Rational root theorem –  Apr 30 '17 at 02:58
  • @selfawareuser If necessary, divide the polynomial by $b$ until the constant coefficient $a_0$ is nonzero. And $p<p/q$ for positive integers $p$ and $q$? – Chris Culter Apr 30 '17 at 05:06
  • Okay, I can see why it's absurd - $q$ would have to be less than one, which by definition it isn't. But how does that prove what we want? And I'm still not sure how we can force $a_0$ to be nonzero... –  May 02 '17 at 07:53
  • Suppose for example that $b^4-3b^3+7b^2=0$. Then also $b^2-3b+7=0$. – Chris Culter May 02 '17 at 23:00
  • Okay, I think the implication is that the number we are trying to represent is not zero, which is fine. But how does any of this prove that base $p/q$ yields unique representations? –  May 03 '17 at 03:02
  • It proves that base $p/q$ cannot represent a number using two different finite strings. – Chris Culter May 03 '17 at 04:19
  • I'm sure I'm missing something obvious, but why does it prove that? –  May 04 '17 at 04:08
  • Going back to the example in the question, do you understand why $100_\varphi=11_\varphi$ is saying the same thing as $\varphi^2-\varphi-1=0$? – Chris Culter May 04 '17 at 05:26
  • Yes, I understand that very well. –  May 05 '17 at 00:32
  • So you understand that any other example of a base representing a number using two different strings involves a similar polynomial? – Chris Culter May 05 '17 at 04:45
  • It is a direct generalization of the equivalence between $100_\varphi=11_\varphi$ and $\varphi^2-\varphi-1=0$. But I give up; feel free to ask a new question about how this works, and probably someone else can explain it better. – Chris Culter May 11 '17 at 04:46