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The naive height function of a rational number $x=\cfrac{m}{n}$ (in lowest terms) is defined as

$$H(x) = H\left(\frac{m}{n}\right) = \max\{|m|, |n|\} $$

However, $0$ can be denoted by $\frac{0}{1}, \frac{0}{2}, ...$

Then what's the value of $H(0)$? Is it 1, or undefined?

Sumanta
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    $0/2$ is not in lowest terms. – David C. Ullrich Oct 01 '20 at 14:26
  • @DavidC.Ullrich Nice, that one went right by me. Still, +1 for the OP, re good question. – user2661923 Oct 01 '20 at 14:31
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    I am surprised by the (once again anonymous) downvotes. Personally, I was fooled until I saw David Ullrich's comment. Anyway, the OP is asking about a definition, or interpretation, rather than asking for help solving a problem. Therefore, I don't think that the OP is required to show work here. In fact, even without the downvotes, I would still have upvoted, re the OP's thoughtfulness in raising the question. – user2661923 Oct 01 '20 at 14:33
  • @DavidC.Ullrich Thank you for your answer. Would you post it as an answer so that I can mark it as solved? –  Oct 03 '20 at 09:36

1 Answers1

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It seems silly to post this as an "answer", but the OP asked me to do so.

The apparent ambiguity is resolved by noting that $0/2$ is not in lowest terms. In fact $\text{gcd}(0,2)=2$, while $\text{gcd}(0,1)=1$; so the height of $0$ is $1$.

quid
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David C. Ullrich
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