A Craps Tutorial

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

The idea for this proof is from page 79 of
An Introduction to the Theory of Numbers by Ivan Niven and Herbert Zuckeerman
© 1960 John Wiley & Sons, Inc.

Let N be the closest integer to x unless x is halfway between two integers,
in which case let N be the larger 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 + 0.5 = N - ψ + 0.5 = N + 0.5 - ψ      statement 2

But statement 1 implies [ simply multiply thru by -1 ] that   0.5 > -ψ ≥ -0.5
Adding 0.5 to this inequality gives us   1 > 0.5 - ψ ≥ 0
which we can write backwards as   0 ≤ 0.5 - ψ < 1
From this it is now obvious that   0.5 - ψ = 0
So, statement 2 gives us x + 0.5 = N + 0

Last Update: 21 August 2006