The amount of light an instrument can gather depends
on its area. Let's compare the area of an eye to the
area of one element of binoculars (since there are 2 eyes
and 2 elements).
Area(eye) pi * R(eye)^2 R(eye)^2
-------------- = ---------------- = ------------
Area(binoc) pi * R(binoc)^2 R(binoc)^2
(6/2 mm)^2
= ------------- = 0.0294
(35/2 mm)^2
So the eyes only gather 0.0294 times as much light as a
pair of binoculars; in other words, the binoculars gather
about 34 times as much light. So, a person may see a star
through binoculars which is about 34 times fainter than
a star he can see with his eyes alone.
Convert that ratio to a magnitude difference.
m(eye) - m(binoc) = - 2.5*log10 ( 0.0294 )
= - 2.5 * (-1.53)
= 3.83 mag
Thus, through binoculars, a person might see a star of magnitude
6.0 + 3.83 = 9.83