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What do the Binoculars Numbers refer to?  
The first number refers to the magnification of the binoculars, the second number indicates the size of the objective lens, the aperture of the binoculars.  
Magnification  
The first number in a binoculars' designation indicates how much larger, or closer, an image will appear.  This magnification is often indicated with an "x", indicating the "power" of the binocular.  For example, an image will appear 8 times larger through 8 x 40 binoculars than when viewed with the naked eye. More magnification is not necessarily better.  As magnification increases, brightness and clarity may diminish, and the field of view is usually more restricted.  
Objective Lens/Aperture:  
The second number of a binoculars' designation refers to the diameter of the objective lens (the larger end of the binoculars). It is almost always directly related to the size and weight of the binoculars. Compact binoculars will have objective lenses ranging from 15mm to 25mm in diameter, while truly "full-size" binoculars' objective lenses range around 35mm to 42mm.  A larger objective lens will gather more light and, theoretically, provide you with greater detail and a clearer image.  The trade-off with larger objective lenses is that the binoculars themselves become rather large and can be unwieldy for some users.  
Different designs of binoculars:  
Porro prism binoculars:  
The classic look of binoculars, shaped like the letter M.  The Porro prism design can deliver a wide field of view with excellent image sharpness at very affordable prices.  Traditional Porro prism binoculars have off-set barrels that are large in size and can be bulky.  In compact models, the prisms are designed differently in order to reduce the overall size of the binoculars.  
Roof prism binoculars:  
A newer prism design, roof prism binoculars are generally smaller and more streamlined than Porro prism binoculars.  They have straight barrels and a compact optical design that is able to tolerate rough treatment better, too.  Roof prism models require precise tolerances when manufacturing, and their complex prism configurations often mean that they cost significantly more than Porro prism models of the same quality  
Exit Pupil  
The exit pupil of a binocular is the point at which all of the light rays that entered the objective lens and passed through the binocular exit through eyepiece to form a magnified, circular image.  If you hold the binocular away from your eyes and look through the ocular lens, you will be able to see the clear circular exit pupil.  
Eye Relief  
Eye relief in a binocular refers to the distance images are projected from the ocular lens to their focal point.  The eye relief of a binocular can vary from 5mm to as much as 23mm.  Eye relief is of importance to eyeglass wearers, who will need a minimum of 14-15mm of eye relief to be able to use binoculars effectively. A binoculars' eyepieces will have either rubber, fold-down eyecups or more modern retractable style eyecups for accommodating the user, whether they wear eyeglasses or not.  
Field of View  
The field of view is the size of the area you can see with a pair of binoculars.  This measurement is expressed as the width, measured in feet, of the area you would see at 1000 yards.  
Close Focus  
The close focus of a binoculars refers to the minimum distance which it will focus to.  The close focus of a particular model depends on its optical design and, to a small extent, on your own ey'es characteristics.  For many birdwatchers and butterfly watchers, it is desirable for binoculars to focus to at least 10 feet.  
Interpupillary distance (IPD)  
The distance from one eye pupil to the other, expressed in millimeters.  For many people, IPD is not a consideration, but for people with either narrow-set or wide-set eyes it may be important to know your personal interpupillary distance.  
Twilight factor  
A commonly used measurement of viewing efficiency and image detail in low lighting conditions.