calc/adele works on finite points.
Everyone knows the rationals $\mathbb{Q} = \varinjlim \mathbb{Z}/s$, the residues $\widehat{\mathbb{Z}} = \varprojlim n \backslash \mathbb{Z} = \prod_p \mathbb{Z}_p$, the reals $\mathbb{R} = \mathbb{Q}_\infty$ and the adeles $\mathbb{A_Q} = \left(\widehat{\mathbb{Z}} \times \mathbb{R}\right) \otimes \mathbb{Q}$.
$n \backslash r$ roughly means $r \bmod n$. But precisely speaking, it is an element of $\prod_{p \mid n} \mathbb{Z}_p$ with limited accuracies.
$n \backslash r/s$ means $r/s$ in $\prod_{p \mid n} \mathbb{Z}_p \otimes \mathbb{Q}$ which approximates $\mathbb{A_Q}$.
For example, $2^8 3^3 7 \backslash 7 / 2^{12} 3 \in (\mathbb{Z}_2 \times \mathbb{Z}_3 \times \mathbb{Z}_7) \otimes \mathbb{Q}$ is a representation of an element of $\mathbb{A}_\mathbb{Q}$ with following accuracies:
calc/arch works at infinity point.
It recognizes theory of relativity and quantum mechanics via $4\pi G = c = h/2\pi i = 1$.
It adopts logarithmic notation.
For example, $1.144730.750000\mathrm{X}$ means $\exp(1.144730 + 2 \pi i \times 0.750000)$.
\[ \log x = \int_1^x \frac{ds}{s} \]