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At some level you really shouldn't care why 1 isn't prime. It's not-prime because we defined it that way. 1 has some mathematical properties in common with the numbers we happen to call "prime numbers" and some mathematical properties which are different; in another world we could have called it prime and replaced "the prime numbers" with "the prime numbers greater than 1" in assorted theorems and EVERYTHING ABOUT MATH WOULD BE THE SAME.

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Well, yes and no. If you're just talking about integer primes then sure. If you consider gaussian integers though (a + b*i where a and b are integers) then you would have to say "the gaussian prime numbers that aren't 1, -1, i, or -i.

The definition of primes that excludes units makes sense in more places than one that includes units (especially once you start talking about prime ideals instead of prime numbers)

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