A star's "magnitude" is a measure of its brightness, not its size. (Although most people would say that a brighter star appears to be bigger than a fainter one, they're all just points of light to the unaided eye.) The number of stars that you can see in the sky depends on the darkness of your observing site, the number of stars that are bright enough to be seen from that site, and the kind of observing equipment you use. It also depends on the shape of your horizon; a sea-level site surrounded by an sea-level horizon will always be able to see one-half the full-sky sphere, but a site near a mountain range has a portion of the half-sky dome cut off from view.
Stars can be classified by their "apparent" visual brightness, their brightness as seen from the Earth. The first formal classification seems to have been done by Hipparchus for a star catalog created about 150 BC. He called the brightest group of stars "first magnitude", the next brightest "second magnitude", and so on. Modern measurements of their brightness indicated that the stars in the first group were two to three times brighter than the stars in the second group.
In 1856, N. R. Pogson standardized the brightness scale. He set it so that the brightest stars would be assigned magnitudes near zero, and stars which differed by one magnitude would have a brightness ratio approximately the same as the ratio between Hipparchus' first and second magnitude groups. On this scale, stars which differ by five magnitudes have a brightness ratio of 100, and, therefore, stars which differ by one magnitude have a brightness ratio of 2.512 (= the fifth root of 100).
The scale is numerically inverted -- the brighter stars have lower magnitudes than the fainter ones. In fact, the brightest stars have negative magnitudes.
|Sirius||-1.44||Canis Major (Big Dog)|
|Canopus||-0.63||Carina (Ship's Keel)|
|Rigel Kent||-0.01||Centaurus (Centaur)|
|Procyon||0.40||Canis Minor (Little Dog)|
|Achernar||0.54||Eridanus (River Eridanus)|
|Fomalhaut||1.23||Piscis Austrinus (Southern Fish)|
|Acrux||1.28||Crux (Southern Cross)|
|Mimosa||1.31||Crux (Southern Cross)|
|Polaris||2.00||Ursa Minor (Little Bear)|
The source data for this table (and the others below) is the Sky2000 Catalog (I used version 3 of this catalog.) The goal of this catalog is to include all stars with a visual magnitude brighter than 9.0. (The reference document for the catalog states that "the visual magnitude [included in this catalog] is generally observed in the photoelectric V passband of Johnson and Morgan (1953)", and the "effective wavelength [is] 5,500Å.)
Table 3 (below) lists all of the primary stars in the catalog, by half-magnitude group. The table does not include the secondary stars of any double- or triple-star group, and, therefore, does not include their magnitude contribution to the primary star of the group.
|Brightest||Brightest magnitude in this row|
|Dimmest||Dimmest magnitude in this row|
|Count||Count of primary stars from the brightest magnitude (inclusive) to the dimmest magnitude (exclusive)|
|CumCnt||Cumulative sum of Count down to this row|
|Growth||CumCnt of this row divided by the CumCnt of the previous row, expressed as a percentage|
The table shows that we can see more and more stars as we include dimmer magnitudes; there are about 75% more stars for each half-magnitude group. The dropoff in cumulative count near magnitude 9.0 is an artifact of the catalog goal; the process which built the catalog simply did not include most of the stars beyond the catalog limit. I suspect that a deeper catalog, with a dimmer limit, would continue to show the same kind of growth until its own limit.
To get an estimate of the number of stars that you can see from your observing site, you need to know the site's "limiting magnitude", the dimmest stars that you can see. The simplest way to do this is to look at various parts of your sky, then use a catalog that shows the magnitudes of the stars that you can see. Of course, this will be a personal limiting magnitude. Then you can use the table above to show the total number of stars in the entire sky and divide by 2 to give the number of stars visible in your half-sky dome.
When you estimate your site's limiting magnitude, you should include many different areas of the sky. For example, when observing from my home, the limiting magnitude along the north horizon (over large homes whose owners use upward-pointing lights to illuminate their trees, and the sky above them) and the south horizon (over the bright lights of shopping malls and large cities beyond) is about 3.0 to 3.5. Near the zenith, I can see stars that are dimmer, down to magnitude 4.5.
Here's a site that can help you determine the darkness of your observing site: Visual Satellite Observers: Determing Brightness using the Magnitude System
John Bortle's Dark-Sky Scale gives another way to estimate the darkness of your observing site. Go to the Sky and Telescope web site and search for "The Bortle Dark-Sky Scale". On his scale, my home observing site, almost 36 miles north of the middle of Chicago, should be Class 6 or Class 7, and, therefore, my limiting magnitude should be about 5.0 to 5.5. Perhaps my eyes are just not good enough.