PRH | Educ | 6.4 • A Gentle First Course in Adelic–Quantum Arithmetic

This booklet offers a gentle, linear-algebra friendly path toward the adelic-quantum (AQ) viewpoint on arithmetic. The slogan is that addition behaves like a wave (shifts), while multiplication behaves like a scale change (dilations and prime steps). These two actions generate the adelic $a x+b$ symmetry and satisfy the commutation rule

$$
\left[H_{\text {idele }}, H_{\text {adele }}\right]=i H_{\text {adele }},
$$

which is the adelic avatar of the classical $[D, P]=i P$ on $L^2(\mathbb{R})$. We build up the picture from first principles - numbers, primes and composites; vectors and inner products; functions as vectors; Fourier transform; a gentle look at $p$-adic numbers; then adeles/ideles and their basic actions - with many short exercises (odd-numbered answers included).