This is the layman's explanation:
A complimentary color is formed when two colors are mixed to form another color, and when the third color is added in an equal amount will create a neutral color (black, gray or white).
An example, pigments are three basic colors which mixed, create all other shades of color. The three basics are red, blue, and yellow. For the purpose of this example equal amounts of each are to be used. When you mix red and blue you create purple; the complimentary color for purple would be yellow, the third basic color. If you want to know what a complimentary color for any color is, say green. it's necessary to know that green is a combination of blue and yellow to know that the complimentary color would be red.
Using pigments, the combination of all three colors in equal amounts creates black.
Using light, the combination of all three colors in equal amounts creates white.
The three basic colors of light are magenta (a combination of blue and green), cyan (a combination of blue and green), and yellow (a combination of red and green). Using light is more confusing. See the site below for more information.
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Complementary colors are opposite each other on the color wheel. When you mix two complementary colors you get brown. If you are painting a shadow you use the complement of the color the shadow falls on.
The complementary color is the opposite of a color. In the case of green, it's magenta.