The core meanings
"Compliment" (with an I) means an expression of praise or admiration. "Complement" (with an E) means something that completes, enhances, or goes well with something else. "Nice presentation" is a compliment. A wine that pairs perfectly with a meal is a complement to the dish.
Both words function as nouns and verbs, which doubles the opportunities for confusion. You can compliment someone on their work, and a good side dish complements the main course.
The memory trick
The most reliable mnemonic: "I like to receive compliments", compliment has an I, and it is about personal praise. "Complement completes", complement has an E, just like "complete." If the meaning involves completing or enhancing, use the E spelling. If the meaning involves praise, use the I spelling.
This letter-based trick works for the adjective forms too. "Complimentary" (with I) means either "expressing praise" or "given free of charge", a complimentary breakfast, a complimentary review. "Complementary" (with E) means "serving to complete or enhance", complementary colors, complementary skills.
- She paid him a compliment on his presentation. (praise, I)
- The red sauce complements the pasta perfectly. (enhances, E)
- The hotel offers complimentary breakfast. (free of charge, I)
- Their skills are complementary: she handles strategy, he handles execution. (completing each other, E)
Tricky cases in professional writing
Business writing is where this confusion appears most often. "Our services are complementary to yours" (they work well together) means something very different from "Our services are complimentary" (they are free). A proposal that promises "complimentary analysis" is offering it for free. A proposal that promises "complementary analysis" is offering analysis that rounds out other work.
Technical fields also rely on "complement." In mathematics, the complement of a set contains everything the original set lacks. In biology, complementary DNA strands fit together. In design, complementary colors sit opposite each other on the color wheel. In all of these, the E spelling signals completion.