Granularity

There is a factor that can cause error in various mathematical calculations that I believe should be given a name. To correctly set up a calculation of something, it is first necessary to fully understand what we are dealing with. I find that units are a significant factor in causing error in calculations if not handled properly.

I am not referring to defined units such as meters, pounds, degrees and so on. Sometimes calculations are inadvertantly set up as if it was a smooth, homogenous medium that was being dealt with when, in fact, it is not. One example is the difference between water and sand. Water is such a smooth and homogenous medium, at least down to the level of the molecules. Sand, however, is not. Sand is composed of grains, or discreet units, hence the name "granularity".

Sand may flow like water but failure to consider the size of the grains of sand and deal with it as if it were the smooth medium that water is may invite error in calculations. Granularity involves the difference between a medium that is smooth and homogenous, like water, and one that may resemble it in appearance or behavior (behaviour) but which is, in fact, composed of discreet units, such as sand.

What I will term "granular errors" occur when we assume that a medium that is grainy is in fact smooth or when we, during calculations, mistake an entity that is finite for one that is either infinite or infinitesimal. I have determined that there are basically two types of granular errors: those of the infinite and those of the infinitesimal. With either of these conditions, units become meaningless. But since we are actually dealing with the finite, this throws off our calculations.

An ideal example of granularity is the compounding of interest. Most people know that we will not come up with the same answer if we compound the interest by day as if we use continuous compounding. This is because the daily compounding revolves around a certain unit (or grain), the day, whereas continuous compounding does not.

Let me give you may favorite (favourite) example of a granular error. Suppose you have a drawer full of gloves, well-mixed and with an equal number of right- and left-hand gloves. Now suppose you reach into the drawer without looking and pull out two gloves. What are the odds that you will have a matching pair, one left and one right?

The first reaction of many people is to say that there is a fifty percent chance of having a matching pair. But this is not correct and the reason is granularity. Suppose there are ten gloves in the drawer, five right and five left. When you pull out your first glove, that will leave five gloves that would form a matching pair with it and four that would not.

Therefore, your odds are pulling out a matching pair of two gloves when you pull out the second glove are not 50/50 but 5/9, five out of nine. Due to granularity, the odds become better than even. The odds would only be 50/50 if there were an infinite number of gloves in the drawer. As long as there is any finite number of gloves, the odds must be above fifty percent. This is an example of granularity, treating something as infinite, when in fact it is finite due to the fact that the pile of gloves are composed of discreet units, the gloves.

Another type of granular error is considering something in the calculation as infinitesimal when it is not. An example of this is the neutron "bullet" that is fired toward a uranium or plutonium nucleus to begin a nuclear chain reaction. In calculating the probable distance that the neutron must travel before striking a nucleus we cannot consider the nucleus as a dimensionless point if we want the best answer. The neutron has certain dimensions of it's own that increase it's chances of striking a nucleus in a given distance over what those chances would be if the neutron was a dimensionless point.

One reason that I believe Planck's Constant shows up all over physics formulas is that space is not a smooth, homogenous medium as it appears to us. It is, in fact, composed of certain particles, as in the theory of Loop Quantum Gravity and to get accurate answers, the size of these particles or grains of space must be taken into account.

To sum up granularity, when we are doing calculations involving either the infinite or the infinitesimal, units tend to become largely meaningless. But if we are dealing with the finite that we perceive as the infinite or the infinitesimal than we open ourselves to error in such calculations.