The Rounding Problem
Created | Updated Jan 28, 2002
This is a mathematical problem I have found while working on determining the error of a result. I put it to the community of h2g2 to solve. It may have a trivial solution or error in which case I admit my stupidity and plead tiredness it being one in the morning here. If, however, the solution is mathematically interesting then I feel I may have achieved something by placing it here.
Let a, b be two points on the real number line such that their distance, c, is equal to b - a and c > 0.
Define a quantity gamma (henceforth in this version represented by 'g' for reasons of brevity. When seeing g, read gamma) for the system described by a and b such that d = c/g.
Let there be a point x such that a < x < b. For a given quantity of g, let x be equivalent to a when x < (a + d) and let x be equivalent to b when x > (a + d)
In this system, (a + d) is the balance point for an error constant g. In the conventional error system, g = 2 so that a + d is exactly halfway between a and b.
Consider the case where x is always equivalent to a where a < x < b. In this situation, d still equals c/g where a + d = b or d = b - a. This defines c = g(b - a) where x is always equivalent to a. We know that c = b - a and therefore g = 1 for this case.
Consider the case where x is always equivalent to b where a < x < b. What is g for this case? Does there exist a precise value of g to satisfy the above condition? In what set does g lie?
Let a, b be two points on the real number line such that their distance, c, is equal to b - a and c > 0.
Define a quantity gamma (henceforth in this version represented by 'g' for reasons of brevity. When seeing g, read gamma) for the system described by a and b such that d = c/g.
Let there be a point x such that a < x < b. For a given quantity of g, let x be equivalent to a when x < (a + d) and let x be equivalent to b when x > (a + d)
In this system, (a + d) is the balance point for an error constant g. In the conventional error system, g = 2 so that a + d is exactly halfway between a and b.
Consider the case where x is always equivalent to a where a < x < b. In this situation, d still equals c/g where a + d = b or d = b - a. This defines c = g(b - a) where x is always equivalent to a. We know that c = b - a and therefore g = 1 for this case.
Consider the case where x is always equivalent to b where a < x < b. What is g for this case? Does there exist a precise value of g to satisfy the above condition? In what set does g lie?
