The regulator of a number field is essentially the covolume of the unit group embedded into the vector space $\{(x_1, \ldots, x_{r+s}): \sum_i x_i=0\}$ under the log embedding: $$x \mapsto (\log |\sigma_1(x)|, \ldots \log |\sigma_r(x)|, \log |\sigma_{r+1}(x)|^2, \ldots , \log |\sigma_{r+s}(x)|^2).$$

But you need to take some care in terms of what the right Euclidean structure is for defining volumes in $\{(x_1, \ldots, x_n): \sum_i x_i = 0\}$. What you're supposed to do is to use the Euclidean measure induced by any coordinate projection $\{(x_1, \ldots, x_{r+s}): \sum_i x_i = 0\} \rightarrow \mathbb{R}^{r+s-1}$. Equivalently, you could use the subspace Euclidean measure, but then normalize by dividing by $\sqrt{r+s}$.

My question is why is this the best normalization?

So far the only answer I have is that for the quadratic real case this makes the regulator exactly the log of the fundamental unit, which seems a very sensible convention. I guess it also makes the analytic class number formula slightly cleaner, but it's not obvious to me why this normalization exactly comes into the class number calculation.

My apologies if this question is too elementary.