Past exam of the physics A course of the University of Cambridge / 2015 / Paper 1 / Section A / 4 / Solution

For a fixed magnitude zero point, photometric magnitude uncertainty follows from propagation of uncertaintyThis is the absolute standard uncertainty of the apparent magnitude. The requested fractional uncertainty is insteadThis is the requested fractional standard uncertainty as a function of $m$; there is no single numerical fractional error independent of the magnitude. At $m=0$ it is undefined. Moreover, its value changes when the magnitude zero point changes; the absolute magnitude uncertainty is the useful invariant of such a zero-point shift. Treating the stated percentage as a finite perturbation rather than a standard uncertainty gives changes $-0.1035$ mag for $f\mapsto 1.1f$ and $+0.1144$ mag for $f\mapsto 0.9f$, consistent with the linear estimate.