Etymology
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complement (n.)

late 14c., "means of completing; that which completes; what is needed to complete or fill up," from Old French compliement "accomplishment, fulfillment" (14c., Modern French complément), from Latin complementum "that which fills up or completes," from complere "fill up," from com-, here probably as an intensive prefix (see com-), + plere "to fill" (from PIE root *pele- (1) "to fill").

From c. 1600 as "full quality or number, full amount;" musical sense of "simple interval that completes an octave from another simple interval" is from 1873. In 16c. also having senses which were taken up c. 1650-1725 by compliment.

complement (v.)

"make complete," 1640s, from complement (n.). Earlier in a now-obsolete sense of "exchange courtesies" (1610s), from complement (n.) in a 16c. sense "that which is added, not as necessary, but as ornamental," now going with compliment. Related: Complemented; complementing.

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Definitions of complement
1
complement (n.)
a word or phrase used to complete a grammatical construction;
complement (n.)
a complete number or quantity;
a full complement
complement (n.)
number needed to make up a whole force;
a full complement of workers
Synonyms: full complement
complement (n.)
something added to complete or embellish or make perfect;
a fine wine is a perfect complement to the dinner
Synonyms: accompaniment
complement (n.)
one of a series of enzymes in the blood serum that are part of the immune response;
complement (n.)
either of two parts that mutually complete each other;
2
complement (v.)
make complete or perfect; supply what is wanting or form the complement to;
I need some pepper to complement the sweet touch in the soup
From wordnet.princeton.edu