A Craps Tutorial

Other Topics Section    --    Bracket Function Properties

	Theorem   4
		If x is any real number then
			x is an integer → x   +   -x   =   0 x is not an integer → x   +   -x   =   -1

Theorem 4 is obviously true if x is an integer.
So, let's assume that x is not an integer.

Then there must exist a real number   α   such that
x   =   x   +   α      and   0   ≤   α   <   1.

Then   -x   =   - x   +   ( -α )      where   -1   <   -α   ≤   0.

By theorem 1,   -x   =   - x   +   -α   =   - x   +   ( -1 )

Adding   x   to the last equation completes the proof.