There is often confusion over the words 'complement' and 'compliment'.
Compliment
A compliment (with an i) is an expression of praise.
Examples:
 My compliments
to the chef for such a wonderful starter. 
(my praise to the chefs)
When I said
your eyes looked misty, I meant that as a compliment.
(I meant that as praise)
Tell the cook of this restaurant with my compliments that these are the very worst sandwiches in the whole world, and that, when I ask for a watercress sandwich, I do not mean a loaf with a field in the middle of it.
(Oscar Wilde)
(compliments can also mean good wishes, regards or respect)
ComplementA complement (with
an e) is something that enhances something else or goes well with it. (For
example, cranberry sauce is a complement for turkey.) It is not common,
but complement can also mean composition or make-up. (When used, it is often
seen in the term 'full complement', meaning 'the whole number'.)
The cashew
nuts were an excellent complement for the soup.
(cashew nuts went well with the soup)
The
drums were a perfect complement to their dancing style.
(drums enhanced their dancing style)
The Verbs 'To Compliment' and 'To Complement'
The examples above all relate to nouns.
However, both words exist as verbs too.
I would like
to compliment the pilot on such a smooth landing.
(pass praise to the pilot)
The jade and
silver cufflinks complement the green tie.
(go well with the green tie)
He has worked
hard for many years to break my record. I can only complement him for such
terrific
dedication. 
(should be 'compliment' - i.e., offer praise to him)
Select the correct version:
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THE LETTERS CAN HELP YOU REMEMBER
 The word "compliment"
has an "i" in it - just like "praise".
The word "complement"
has an "e" in it - just like "enhancement".
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COMPLEMENT ALSO MEANS COMPOSITION OR MAKE-UP
Do you have
a full complement of men for your mission.
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Easily Confused Words
