= Magnitude difference
{title2=$\Delta m$}
For two positive measurements $F_1,F_2$ in the same <photometric passband>, using the same <magnitude zero point>, units, and weighting convention, their <apparent magnitudes> obey
$$
\Delta m=m_2-m_1=-2.5\log_{10}\left(\frac{F_2}{F_1}\right),\qquad \frac{F_2}{F_1}=10^{-0.4\Delta m}.
$$
The <logarithm> takes a dimensionless ratio. An <energy>-weighted measurement must be compared with another <energy>-weighted measurement, and a <photon>-counting measurement with another consistently calibrated <photon>-counting measurement. Changing <photometric passband> or <magnitude zero point> requires the corresponding calibration, as in a <colour index>. https://arxiv.org/pdf/1510.06262[IAU 2015 Resolution B2: Energy-flux magnitude convention]{ref}. https://xmm-tools.cosmos.esa.int/external/xmm_calibration/calib/documentation/CALHB/node597.html[ESA XMM-Newton: Filter-specific count-rate calibration]{ref}.
For example, $F_2=100F_1$ gives $\Delta m=-5$: source 2 is brighter. Conversely, $\Delta m=+5$ gives $F_2/F_1=0.01$, and $\Delta m=+1$ gives $F_2/F_1\simeq0.398107$. Even fractional steps work: $\Delta m=+0.1$ gives $F_2/F_1\simeq0.912011$, an approximately $8.80\%$ decrease. These are illustrative ratios, independent of any particular <star> catalogue.
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