B. Complex-Number Reference

Complex numbers in one page. Form: z = a+bi, i²=−1; conjugate z* = a−bi; modulus |z| = √(zz*) = √(a²+b²); argument arg(z) = atan2(b,a). Polar: z = |z|e^{iθ} — the amplitude's phase θ is what interference machinery manipulates; |z|² is what measurement sees. Euler: e^{iθ} = cos θ + i sin θ; the global phase e^{iθ}|ψ⟩ is physically identical to |ψ⟩; relative phases between components are physical. Arithmetic: multiply → moduli multiply, arguments add ((|z₁|e^{iθ₁})(|z₂|e^{iθ₂}) = |z₁z₂|e^{i(θ₁+θ₂)}) — the entire mechanism of phase kickback and QFT. Born rule: probability of outcome k = |αₖ|² — modulus squared, killing the phase; phases matter only through interference before measurement. numpy: np.angle(z), np.abs(z)**2, complex128 everywhere; silent dtype bugs are 90% of first-week quantum-coding errors.