Magnification is the process of enlarging something only in appearance, not in physical size. Magnification is also a number describing by which factor an object was magnified. When this number is less than one it refers to a reduction in size, sometimes called minification.
Typically magnification is related to scaling up visuals or images to be able to see more detail, increasing resolution, using optics, printing techniques, or digital processing. In all cases, the magnification of the image does not change the perspective of the image.

Magnification as a number (optical magnification)

where f is the focal length and S is the distance from the lens to the object. Note that for real images, M is negative and the image is inverted. For virtual images, M is positive and the image is upright. Additionally, this can be written as the following, where di is the image distance and do is the object distance:

Note again that a negative magnification implies an inverted image.

where f_o is the focal length of the objective lens and f_e is the focal length of the eyepiece. The angular magnification is given by

If instead the lens is held very close to the eye, and the object is placed close to the lens, a larger angular magnification can be obtained, approaching

Here, f is the focal length of the lens in centimeters. The constant 25 cm is an estimate of the "near point" distance of the eye—the closest distance at which the eye can focus.

where M_o is the magnification of the objective and M_e the magnification of the eyepiece. The magnification of the objective depends on its focal length f_o and on the distance d between objective back focal plane and the focal plane of the eyepiece (called the tube length):

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The magnification of the eyepiece depends upon its focal length f_e and can be calculated by the same equation as that of a magnifying glass (above).
Note that both astronomical telescopes as well as simple microscopes produce an inverted image, thus the equation for the magnification of a telescope or microscope is often given with a minus sign.

Single lens: The linear magnification of a thin lens is
Telescope: The linear magnification is given by
Magnifying glass: The angular magnification of a magnifying glass depends on how the glass and the object are held, relative to the eye. If the lens is held such that its front focal point is on the object being viewed, the relaxed eye can view the image with angular magnification
Microscope: The angular magnification is given by