Origins
The word 'complement' is about completion through filling — the act of adding what is missing to make something whole. Its Latin ancestor 'complēmentum' meant literally 'that which fills up,' and every modern use of the word, from mathematics to immunology to grammar, preserves this core idea.
Latin 'complēmentum' is a noun formed with the instrumental suffix '-mentum' (that which does something) from 'complēre' (to fill up, to complete). 'Complēre' itself combines the intensive prefix 'com-' (together, thoroughly) with 'plēre' (to fill), from PIE *pleh₁- (to fill). The same PIE root produced, through the Germanic branch, Old English 'full' and 'fyllan' (Modern English 'full' and 'fill'). So 'complement' and 'full' are cognates: different routes to the same ancient concept of filling.
The word entered English in the late fourteenth century and has since developed specialized meanings in numerous fields, all unified by the concept of completion.
Early History
In mathematics, the complement of a set A (within a universal set) is everything not in A — the elements needed to complete the universal set. The complement of an angle is the amount needed to reach 90 degrees. In both cases, the complement is what is missing: the piece required to make a whole.
In grammar, a complement is a word or phrase that completes the meaning of a verb or predicate. In 'She became a doctor,' the phrase 'a doctor' is a subject complement — without it, the sentence is incomplete. The grammatical term captures the word's etymology precisely: the complement fills the gap in the sentence's meaning.
In immunology, the complement system is a group of proteins in blood plasma that 'complement' (complete) the ability of antibodies to destroy pathogens. Discovered in the 1890s by Jules Bordet, the complement system was named because it completed the immune response that antibodies alone could not finish.
Figurative Development
In color theory, complementary colors are pairs that, when combined, produce white light (in additive mixing) or a neutral gray-black (in subtractive mixing). Red and cyan, blue and yellow, green and magenta — each pair completes the full spectrum when combined. The terminology reflects the filling metaphor: each color in the pair provides what the other lacks.
The distinction between 'complement' and 'compliment' is one of English's most frequent spelling confusions, and the history explains why. Both words derive from Latin 'complēmentum.' In the sixteenth and seventeenth centuries, the two spellings were interchangeable. But the word underwent a semantic split, influenced by Spanish 'cumplimiento' and Italian 'complimento,' which had developed the meaning 'a fulfillment of courtesy' — an act of social obligation that expressed respect or praise. English adopted this courtesy sense with the spelling 'compliment,' while retaining the original completion sense with the spelling 'complement.' By the eighteenth century, the two were treated as distinct words.
The phrase 'a full complement' — meaning the complete number of people or items required — is a particularly vivid example of the word's filling metaphor. A ship's complement is the number of crew needed to operate it fully. A regiment's complement is its authorized strength. In both cases, the complement is the number that fills the requirement to capacity.
Proto-Indo-European Roots
The broader Latin 'plēre' family in English is extensive. 'Complete' (filled up entirely), 'deplete' (un-filled), 'replete' (filled again, stuffed), 'supplement' (an under-filling, an addition from below), 'implement' (a filling-in, a tool or means of fulfillment), 'plenty' (fullness), 'plenary' (full, complete), 'plus' (more), and 'surplus' (over-full) all descend from the same PIE root *pleh₁-.
From PIE *pleh₁- through Latin 'complēmentum' to modern English, 'complement' preserves the intuitive idea that completion is a form of filling — that what is incomplete has a gap, and the complement is whatever fills that gap to make the whole.