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I've always been penalized for using contractions in formal writing. It makes since. It's like going down to the level of slang. English, formally, needs to keep with a strict standard or we'll all...

The basic primis is any contraction; can't, I'm, you're, I'd (etc) is informal speach and writing. In any formal writing you would not use any form of contractions. I don't know about formal speach...

can't is the contraction of cannot. They are the same thing. Much like I'm vs I am.

And yes, you can use can not. It's quite common for people to spread the two words out, especially in the...