Notes on English usage - compliment or complement?
These two words are often confused with each other, but they have quite different meanings.
If I compliment someone, I say something nice about them - "I like your new hairstyle", for example. A complimentary remark or statement is therefore one that shows approval.
However, a complement is something that accompanies something else, maybe offered as a gift. For example, if my restaurant meal comes with a complementary glass of wine I would assume that it is free of charge because it was not on the original bill of fare but something extra that the restaurant supplies to its customers.
Confusion sometimes arises when a situation can involve both meanings. For example, in the case given above the glass of wine could arrive with a note that reads "with the compliments of the manager". In other words, the "complementary" (i.e. extra) glass is offered because the manager wishes to "compliment" his customers for having the good sense to eat in his restaurant!
How to remember which is which? A compliment (with an I) is nice to get, and nice has an I as well. A complement (with an E) completes something, and complete has Es.