April 11, 2012
My neighborhood bar used to be a coffee shop named the Arabica at Shaker Square.
Twice each day I’d walk across the square from my apartment to enjoy a cup of coffee with a diverse group of friends.
One of them was a priest. He was one of those “one of the guys” priests. You never felt awkward around him. You never felt there were topics you couldn’t bring up when he was there.
Every once in a while the priest would talk about good and evil. He believed deeply in the strength of the force of good. I remember being surprised at how strongly he believed in evil. A priest. He believed that evil was all around us and yet he believed in the power of good over evil.
In mathematics there is a lot more evil than good. We just spend most of our time focusing on the good.
For example, consider the graph of the function f(x) = 3x.
The graph is a straight line with slope three that passes through the origin. If you know what these words mean, the graph is continuous and smooth everywhere.
Let’s define a new function g that has the same value as f everywhere but x = 0. When x = 0, choose any value for g(0) except 0.
Now g is not continuous or smooth at 0. This is true for any value you can choose for g(0) except 0. In other words, of the infinite values available for you to choose for g(0), only one of them makes g continuous. This is true about every other x value.
Much more evil than good — we just spend our time on the good.
As another example, there are a lot of rational numbers. You could, however, count them.
Actually, you couldn’t. But you could organize them in a way where there is a first, a second, a third, … you could build a list of all of the rational numbers so that all of them appear on the list at least once. We say that the rational numbers are countable.
Real numbers can’t be counted. There is a famous proof by Cantor that shows that any time you think you’ve listed all of the real numbers, I can find one that is not on your list. So there are more Real numbers than rational numbers.
We’ll look at proofs on rational and irrational numbers in our next couple of posts.
Here’s another high school flashback. We punish students by making them factor quadratics for several weeks during Algebra and Algebra II. We give them ax^2 + bx + c and ask them to factor it as (Ax + B) (Cx + D).
The sad thing is that there are relatively few choices of a, b, and c that we can factor by hand. So we teach students the few ones we know how to do handle and then we only ask them those. For most choices of a, b, and c — even where a, b, and c are positive integers — we’re stuck using the quadratic equation to find the roots of the quadratic y = ax^2 + bx + c.
More evil than good.
We spend most of the student’s time on good because, when we go to higher ordered equations, we don’t have formulas like the quadratic equation. By the way, the reason we don’t have these formulas is not that no one has figured them out yet. We don’t have these formulas because someone has proved that they can’t exist.
I first heard this explained by Joan Countryman twenty-some years ago when she held up a calculator and said “this calculator could get a B+ in my Algebra II class. Shouldn’t we change what we’re teaching?”
Imagine we live in a world where only rational numbers exist. Tomorrow we’ll see that we quickly are led to a world where they aren’t enough.