A Craps Tutorial

Other Topics Section    --    Bracket Function Properties

	Theorem   6B
		Let x be any real number.
		if x lies exactly halfway between two integers then   - 0.5 - x   is the smaller of those two integers. Otherwise,   - 0.5 - x   is the closest integer to x.

Let N be the closest integer to x unless x is halfway between two integers,
in which case let N be the smaller integer.

Then there exists a real number ψ such that   N = x + ψ    and
-0.5 ≤ ψ < +0.5     statement 1

In the figure below k is the closest integer to x, and ψ ( not shown in the figure )
would be some negative number whose magnitude is ( the distance between k and x ) less than 0.5.

Since x = N - ψ,   we have
-x = ψ - N   and
0.5 -x = 0.5 + ψ - N
So,   0.5 - x = -N + 0.5 + ψ         statement 2

But statement 1 implies [ simply add 0.5 throughout ] that   0 ≤ 0.5 + ψ < 1
From this it is now obvious that   0.5 + ψ = 0
So, statement 2 gives us 0.5 - x = -N + 0
Therefore   - 0.5 - x = N

Last Update: 21 August 2006