"Check digits" on credit card numbers are a way to make sure you've typed in the number correctly. If you get any of the digits wrong, the software can automatically determine this before it tries to charge your purchase to it. This makes it a test similar to the "make sure the credit card number has exactly 16 digits" test that can also be done on the client side.
Telephone numbers could benefit from this. If someone were to give you an invalid phone number, the address book on your PDA or other software wouldn't accept it, so you'll know right away that you need to ask for the number again. If you try to call an invalid number, your telephone will inform you without trying to put the call through, and so you won't run the risk of calling a wrong number.
Note that check digits won't tell you who the number belongs to (if anyone), only that you've dialed it in correctly.
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I can see this feature annoying the crap out ladies at bars the world over.
A good idea on the surface, but would take redoing of some infrastructure, mostly in where you enter phone numbers into (the actual telephone infrastructure need not be reworked for the check digit).
It is my opinion that telephone numbers are already too long. I am not in favor of doing anything that makes them more difficult to remember.
Maybe a better way to avoid dialing wrong numbers would be to give the subscriber the option to get a spoken word confirmation of the name of the party being dialed before the call goes through. (This would be a software-generated voice, of course.) The phone company's computers are fast enough now to provide this feedback instantly, so it would add only four seconds or so to a call. Speed-dialed numbers, and numbers entered followed by a pound sign, would not be voice-confirmed.