Tuesday, August 14, 2007

Powers of x?

I'm ramping up into a new school year, which means training my students on some basic classroom protocol. One of my little canons is to have “real paper” (i.e., a notebook or folder) as opposed to “nice garbage” (i.e., random sheets of paper from class to class). Another canon is, when taking notes, to write the date on each new day's section. This is because one of my little mottoes is, “Everything is history.” Math equations have a history, medical disorders have a history, novels have both a creative and narrative history—everything has a history. Well, as I was indicating the date to write down, August 14th, I said in my mind, “My brother's birthday is August 28th, which is 14 * 2, or 14 + 14. So if I wanted a snappy way to tell someone my brother's birthday, I could say, 'Double today's date.'” This transformation led me, reflexively, to envision the idea, being so numerical, as an equation: 14ˆ2 = 28— which you'll notice is totally incorrect, as 14ˆ2 = 196. All the same, my psychological tendency (and, I found out, others' as well) to imagine 14 * 2 as 14ˆ2 got me thinking about exponents.

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

does work.

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!


CrimsonCatholic said...

I hate to kill anyone's enthusiasm about math, but your use of exponential notation is wrong. Recall the definition of exponentiation:
x * x = x^2
x(sub 1) * x(sub 2) ... * x(sub i) = x^i

You can establish rules for dealing with exponents, like:
x^m*x^n = x^(m+n)

One of the necessary results is that fractional exponents are roots of the base. For example, the square root is x^0.5, so that x^0.5*x^0.5 = x^1 = x. Hence, 14^(1/7) is the seventh root of 14 (i.e., the number which, when multiplied by itself 7 times, equals 14), and that is 1.458, not 2.

However, so you won't feel too bad, your intuition that there is a link between the exponent's square root and the anti-log is correct. Because SQRT is simply x^0.5 and 0.5 = 2^(-1), there are lots of log relationships that related log 2 with SQRT(2). But I've gotta beat you up again, because 1.1414/10 is .11414, not .1414!

the Cogitator said...

Hey Jonathan,

Fear not, you shan't kill my enthusiasm for math. The chance of seeing it truly obliterated was, first, in seventh grade when I got my first true F (in pre-algebra) and was not only mortified by such defeat but also outraged at such an intellectual affront; second, over the course of the next decade and a half as I hobbled my way through maths, feeling elation only ninth-grade in geometry and then, dimly, like a candle flickering to fumes, in my first year of undergrad taking my first true calculus course. Once I saw math approaching philosophy -- goal-directed mental problem-solving -- and looked past the numbers, then I started to get a hold of it. The fact is, however, I simply had such a poor foundation in, well, the foundations (algebra, trig, pre-calc, etc.) that I am having to unwind the ball of yarn and learn much again with a much clearer mind and set of intentions. I've had lots of time to feel bad about math; there's nowhere to go but up, I say.

In any case, as for my goof with the square root, I realized it after I packed up and was driving to my next teaching gig and only got to clean it out of the post here at home, which is when I saw your comment. I'm actually glad to find out my entire method of exponentiation is wrong, since otherwise, I'd continue in a dry rut. (More good news is that, considering the intricacies of log and anti-log you unveil here, which my students would have to exploit to find the law I set a bounty for, I have almost no risk of shelling out a class prize!)

"Fractional exponents are roots of the base." This orphic gem I must place in my memory palace and ponder as I study. I've got great gobs of ignorance to overcome -- a project to which one of my blogs is dedicated -- across the board: physics, calculus, Latin, Greek, history, theology, biology... oh, and let's not forget that little matter of wisdom and virtue. Many thanks!