Final answer:
A 1mg star is much brighter than a 6mg star; in fact, it is 100 times brighter because each magnitude step corresponds to a 2.5 times change in brightness, and a 5-magnitude difference correlates with a brightness ratio of 100:1.
Step-by-step explanation:
The question is asking which star is brighter between a star with a magnitude of 1mg (which represents a first-magnitude star) and a star with a magnitude of 6mg (a sixth-magnitude star), and by how much the brightness differs. Based on Hipparchus's system of stellar magnitudes, we understand that a change of 5 magnitudes (from magnitude 1 to magnitude 6) corresponds to a brightness difference of 100 times. So, the 1mg star is significantly brighter than the 6mg star, and in fact, it is 100 times brighter.
Specifically, this means that each step up in magnitude actually indicates a lower brightness, with the brightest stars having the lowest magnitude numbers. The magnitude scale is logarithmic with each magnitude difference of 1 corresponding to roughly 2.5 times change in brightness. Therefore, the brightness difference is derived using this ratio. Consequently, a 1mg star is much brighter than a star of a higher magnitude number.