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?
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 $1$.