What is the difference between complement and compliment?

A compliment is praise. A complement is something that completes or goes well with something else. They sound identical and differ by one letter, which is why the mistake is so common.

She paid him a compliment. — she said something nice. The wine was a good complement to the meal. — it completed it.

The spelling hook: compliment contains an i, and so does the word like — a compliment is something you like hearing. Complement shares its e with complete, which is exactly what it means.

Both work as nouns and verbs, which is part of the trouble. He complimented her cooking (praised it) versus the sauce complemented the fish (completed it).

The adjectives diverge usefully. Complimentary has two senses: expressing praise, and free of charge — a complimentary breakfast is one you are not billed for, the sense arising from something given as a courtesy. Complementary means combining well, as in complementary colours or complementary skills.

That pair causes real confusion in business writing, where complimentary services and complementary services mean completely different things: one is free, the other pairs well with what you already offer.

Complement also has a precise sense meaning a full quantity — a ship's complement is its full crew, and in geometry two angles are complementary when they sum to 90 degrees.

Spellcheck will not catch either error, since both spellings are valid words.

Complement has a specific grammatical sense too: a subject complement is the part of a sentence that completes the verb, as tired does in she seemed tired.

And in letter writing, the sign-off — Yours sincerely — is called the complimentary close, with an i: it is a courtesy, not a completion.