Compliment - complement

A compliment, or a complimentary remark, is something you pay someone when you want to be nice: “What a nice dress!”, “That’s very clever”, “You played very well” etc. Sometimes complimentary means ‘free’: e.g. complimentary tickets.

A complement, or a complementary thing, is something that completes something: e.g. a ship’s complement is her crew; her assignment has a complementary Guide to experimental techniques; yin is complementary to yang.