Of late, I have been trying to get a much better handle on physics, calculus and chemistry, so equations and numbers are running through my mind much more frequently. I've got books by and on Einstein, Hawking, Feynman, Gödel, et al. all lined up and I'm even working my way through Hugh Neill's Teach Yourself © Calculus and Michael Kelley's Humongous Book of Calulus Problems. The other night, for example, before I fell asleep I found myself working out a simple fractions problem (viz., “What fraction of Fr. Keefe's Covenantal Theology is endnotes and, therefore, how much of the text have I actually read?”). Instead of just waving the question off with a lazy approximation, I went through the right steps as correctly and systematically as I could: reducing fractions, cross-multiplying, etc. Math has always been the bane of my education, but this does not negate the fact that I have a wary respect for it and even an enjoyment of the math I can grasp. My motivation for learning calculus is not simply autodidacticism (though it is that); it's vocational as well. Realizing I want to study the philosophy and history of science means I must understand physics at a more than elementary level, which, in turn, means I must understand calculus at least an elementary level.
In any case, with numbers on the brain, I pondered how I could express my birthday-equation exponentially. Given that 14ˆ2 = 196, how can I get from 14 to 28 exponentially? Formally,
14ˆx = 28 ∴ x = ?
At first I thought it just entailed raising the number to the desired product's (2's) square root, as if the squaring process would nullify the square root and double 14. But that process seemed not only too simplistic and magical, but also seemed to be mathematically inaccurate. It was beyond me to fathom how the square root of an exponent could yield a product. Incredibly, after only a few minutes of “thinkering” (props to Michael Ondaatje), I came up with x = 1/7. I grasped the fact that squaring 14 to get 28 requires reducing one of the 14's to 2, so that 14 * 14 becomes 14 * 2 = 28. To anyone competent in math, this must seem like a drooling parlor trick. But to me it was a real mental victory!
But something still bothered me. Is there a law, a rule, I can find to help me with this kind of problem using any number? In other words, formally,
pˆx = 2p ∴ x = ?
I ran this by my students, and have offered a class prize if they can figure out a law.
My own gambit is this:
pˆx = 2p iff x = 2/p
14ˆ(2/14) >> 14ˆ(1/7) = 14ˆ(.1414) ≅ 28
But the rule seems not to hold as a rule for all other numbers. Consider:
2ˆx = 4 iff x = 2/2 >> 2ˆ1 = 2, not 4.
On the other hand,
3ˆx = 6 iff x = 2/3
I seek a rule in this matter, especially because I'm trying to extend the transformation to triplets and quadruplets, as in 14ˆx = 3 * 14, x = ? Is it just the cube or fourth root?
My best formulation, after more thinkering in and out of class tonight, is this:
pˆx = n * p iff x = n/p (uhm, except, for some reason, for 2).
I really want some input on this! I'm so dumb. This is a pre-calculus quandary. Please contribute!