Differential entropy under affine rescaling ($h_e(aX+b)=h_e(X)+\ln |a|$)

For real $a\ne 0$ and $b$, the change of variables formula gives $f_{aX+b}(ax+b)=f_X(x)/|a|$. ThereforeThe absolute value also covers a decreasing transformation. If a numerical length changes from metres to centimetres, $a=100$ and the differential entropy changes by $\ln 100\approx 4.60517$ nats, although the physical uncertainty is the same. The reference coordinate scale must therefore accompany an absolute differential entropy value.