This may be a dumb question, but ...

You know how a hotel will place those little chocolates on your pillow, without charge? Or perhaps provide free use of an exercise room or a computer terminal?

Are those services complementary or complimentary?

Complimentary is the word you need. It means (in this sense, at least) "free".

Complementary means "completing".

From today's comics: go here and, if the site changes daily, go to the June 11 version.

[PS: arnie, how could i just put the image here? I tried the </> function in the edit menu, but didn't get the image; only the small red x.]

Oh, that's great, Shu!

The two senses of complimentary seem entirely unrelated to me. If the word means both 'free' and 'flattering', how on God's little acre did one meaning evolve from the other -- or how did each evolve from a common ancestor?

And how does complementary fit into this family tree? It's so similar in form to complimentary one can hardly think they are unrelated -- that the similarity is merely coincidence -- and yet its seems to have still an entirely different meaning, unrelated to either sense of complimentary.

For once the OED doesn't give us a nice graded series of senses. The words appeared in English with essentially the modern meanings. The two spellings are variations of the one word.

A complimentary gift is one given as a compliment. Formerly this would have been something a little more substantial, I would imagine, than a plastic tub of jam in a hotel, the only modern use.

I found something that may be relevant, in Chapter IX of Emma by Jane Austen.

'Oh!' said he directly, 'there is nothing in the way of fruit half so good, and these are the finest-looking home-baked apples I ever saw in my life.' That, you know, was so very. . . . And I am sure, by his manner, it was no compliment. Indeed they are very delightful apples.

But clearly it was a 'compliment' in the sense we would use the word, so Austen must be using it in another sense. She seems to mean "a polite formality, not to be taken seriously". A later passage, in Chapter XIV, is less clear but to my mind uses the same meaning, with the additional feeling of "a tedious polite formality".

She could not enter the house again, could not be in the same room to which she had with such vain artifice retreated three months ago, to lace up her boot, without recollecting. A thousand vexatious thoughts would recur. Compliments, charades, and horrible blunders ...

This, found in Tom Jones by Henry Fielding, accords with what wordnerd said about the 'compliment' being used to mean a mere formal politeness, not necessarily a praise.

Miss Bridget rings her bell, and Mr Allworthy [her brother] is summoned to breakfast, where I must attend, and, if you please, shall be glad of your company. The usual compliments having past between Mr Allworthy and Miss Bridget, and the tea being poured out, he summoned Mrs Wilkins, and told his sister he had a present for her. [Book I ch. iv]