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Sunday, January 29, 2006

Static or dynamic systems

How do we think about systems of equations (or inequalities)? I know, most of us don’t think about them at all. But teachers and students of algebra certainly do. Whether we call them systems of equations or simultaneous equations, we traditionally think about them in a static way, as the word “simultaneous” suggests. For example,Wikipedia says this:
simultaneous equations are a set of equations where variables are shared. A solution consists of values for the variables which satisfy all of the equations simultaneously.
In other words, all the equations are given at the same time — i.e., simultaneously — and we find a common solution (if we are looking for a solution). If you prefer the more concise definition given by Mathematica, try this one:
Simultaneous Equations

A finite set of equations in the same unknowns of which the common solutions have to be determined.
Far be it from me to say that Wikipedia has better definitions than Mathematica, but in this case it certainly does, since the Wikipedia entry distinguishes between the use of an equation to state a relationship and the use of an equation to determine the value of an unknown. Mathematica ignores the former (and conceptually more important) definition. Far too many students — and even teachers — think of algebra as the study in which variables represent unknowns that have to be solved for, rather than the study of the relations among variables.

But that’s not really the point of this post. Whichever way you characterize equations, you also have to confront what simultaneity means. I always used to believe that it means what it says: we have a static view of a system, whether it’s equations or inequalities; all are true at once, like a snapshot.

But now, after conversations with my department head, I am thinking of a system in a dynamic way: each equation or inequality represents a constraint, and the constraints are imposed in time, not all at once. We have a video, not a snapshot. Consider, for examples, a system of three equations presented in an article by Joseph Ordinans in the February 2006 issue of the Mathematics Teacher:



Ordinans takes the standard emphasis on the geometric description of equations and presents it in a 3-D context: three planes that intersect in any of eight ways match the algebraic conclusions to be drawn from them (consistent or inconsistent, dependent or independent). This is the traditional, static view.

A dynamic view would take the equations one at a time:
  1. The first equation provides a constraint. The variables can have (infinitely) many tuples as their values, such as (10, 3, 10) or (-5, 1, 1). But there are also (infinitely) many tuples that do not satisfy the first equation, such as (1, 1, 1). So the constraint partitions the set of all tuples of numbers into two subsets, those that satisfy the first equation and those that don’t. In other words, at this point the first equation represents one constraint. If we take a snapshot, or freeze-frame our video, we can capture the set of solutions so far.

  2. Now the second equation provides another constraint. We can further subdivide our solution set to the first equation by finding the subset that also satisfies the second equation. Our video has moved on to its second frame.

  3. Finally, the third equation provides a third constraint: frame #3. In this particular case, only the empty set remains. After imposing the third constraint, we find that no tuples satisfy all three.
One thing that I like about the sequential, dynamic approach is that it translates well when we have a system of inequalities, or even a system containing both equations and inequalities. The human brain is not great at considering multiple issues in parallel; most of us tend to think sequentially, rather than simultaneously.

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