A Craps Tutorial

Other Topics Section    --    Bracket Function Properties

	Theorem   2
		If x is any real number and p is any positive integer then
			x / p     =       x     / p

To prove theorem 2 we will need to use a well-known theorem called the division algorithm. It is not really an algorithm but rather a statement about the division process. Its proof can be found in almost any book on elementary number theory.
It simply says that when you divide an integer b into an integer a you will get a unique quotient q and a non-negative remainder r whose magnitude is smaller than that of the divisor b.

	x   =     x     +   α        where the real number α satisfies   0 ≤ α < 1	statement 1
	By the division algorithm,   there exists an integer q   such that
	x   =   qp   +   r        where   0 ≤ r < p	statement 2
	In statement 2's equation,   divide through by p to get
	x   / p   =   q   +   r / p      0 ≤ r/p < 1	statement 3
	Applying theorem 1 to statement 3,   we get
	x   / p       =     q   +   r / p       =   q   +   r / p     =   q	statement 4
	Combining statements 1 and 2   gives    x   =   qp + r + α	statement 5
	In statement 5,   divide thru by p to get    x / p   =   q   +   ( r + α ) / p	statement 6
	Apply theorem 1 to stmt 6      x / p     =   q   +     ( r + α ) / p	statement 7
	0   ≤ α < 1   →    r   ≤   r + α   <   r + 1
	Since r and p are integers,   statement 2's inequality   can be rewritten as
	0   ≤   r   ≤   p - 1
	which implies     1   ≤   r + 1   ≤   p
	Add r throughout the inequality of stmt 1 and
	combine with the above inequality to get
	r   ≤   r + α   <   r + 1   ≤   p   →   r   ≤   r + α   <   p
	r / p   ≤   ( r + α ) / p   < 1   →     ( r + α ) / p       =   0
	From stmt 6,     x / p     =   q + 0   =   q
	Now combine this with statement 4
	x   / p       =   q   =     x / p