By how many times a 3rd magnitude star is brighter than 5th magnitude star?
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The magnitude system started with Hipparcus and Ptolemy when they divided the stars into six magnitudes. About 20 of the brightest stars that they could observe from their location were assigned to the first magnitude. The next set of bright stars were assigned to second magnitude and so forth. Sixth magnitude stars were assigned to stars that were barely visible to the unaided eye under favorable conditions.
It was empirically determined that the ratio of first magnitude to sixth magnitude was 100 to 1. A logarithmic scale of 2.512 between magnitude levels is implemented.
For example, a first magnitude star is 100 brighter than a sixth magnitude star or the sixth magnitude star is 1/100 or 0.01 dimmer that a first magnitude star.
Hence the 3rd magnitude star is brighter by 6.25times or (2.5)2 times than the 5th magnitude star.