Level of Resolution
March 3rd, 2008 by SplineGuy
In both my Algebra course and Differential equations course, I have been stressing something called the "level of resolution" that goes in to setting up a mathematical model. When determining the components of the models, some assumptions must be made in order to translate the real world phenomena into a mathematical context. Changing the assumptions will drastically change the model’s ability to predict what happens in the real world and how well the model will mimic the actual phenomena. We must identify the variables which are responsible for changing the system. In general, we may choose not to incorporate all these variables into the model at first.
The simplest example I give in class is when we cover the simple model of a mass being dropped with the sole force acting on the object being gravity as described by classical Newtonian mechanics. For example, we assume no effect due to air resistance. When covering differential equations in Calculus or on the first or second day of the Differential Equations course that I teach, we change the level of resolution by introducing a force due to air resistance. We consider a model where we assume the force due to resistance is proportional to the velocity of the object. Then we consider the case where it is proportional to the square of the velocity. We don’t get to the point that we consider other factors that affect air resistance such as exposed surface area.
I was reminded of this point in reading about the MPAA mathematical mistake on 360’s blog entry a couple of days ago. Apparently, a commissioned study released in 2007 by the Motion Picture Association of America (MPAA) reported that 44% of the industry’s domestic losses could be attributed to piracy by college students. And guess what? They were wrong. I guess credit goes to them for admitting it but nevertheless, they had made certain assumption that all downloaded films are ones that would have been purchased. The revised figure assumes that only a proportion of the pirated movies would have been purchased otherwise.
I’ll quote the blog, 360’s, extra credit question here and ask that anyone interested can distract themselves for a few moments:
If the change from 44% to 15% was entirely due to assuming that college students would only buy x% of the movies they download, can you find x?
My Very Important Principle: While it is important to solve the model correctly, it is equally important to solve the correct model.
HT: 360
