September 26, 2017, by Alan Krome, former LCA teacher
This is a salute to Bill Harvey, my algebra teacher at Norfolk Academy.
I salute him because he was able to take a word-nerd and imbue him with awe at the mysterious and echoing beauty of mathematical pattern. I already knew by the time I was in his tenth-grade class that I wanted to be a poet, so most of my demigods were wordsmiths. I was good at math, but it didn’t engage me any more than history or geography, two other courses that seemed to me encrusted with facts as dull as dried mud cracking in the sun. (The fact that I later learned to love these disciplines is another story for another day . . .)
But Mr. Harvey took a few minutes one day to point out that any time you took a square, such as 10 x 10 and instead multiplied its square root + 1 times the square root - 1, you would always get one less than the square: 10 x 10 = 100, 11 x 9 = 99; 7 x 7 = 49, 8 x 6 = 48, etc. I don’t recall why he pointed it out—possibly as a short-cut for finding the product of two numbers separated by 2? What I do remember is wondering how far this pattern went; surely when you got into larger numbers, the difference must get larger. So I tried 80 x 80 and got 6400; 81 x 79? Whoa, it was 6399! The fact that this held true despite the size of the numbers defied my understanding of multiplication. [So I continued noodling with the numbers. Before long I tried adding and subtracting 2 from the square roots. 12 x 8 = 96, which was 3 less than the 99, 4 less than the 100, and 9 x 5 3 less than the 48, 4 less than the 49. Subtracting 3 gave me 91 and 40, each 5 less than its predecessor or 9 less than the original 100 or 49 . . .]
Highly excited, I took these findings to Mr. Harvey, who—may blessings shower upon his memory—said, “Wow, Alan, that’s a fascinating discovery. It certainly means something—I want to you to figure out what.” [So after several hours of chasing this elusive mathematical mystery down into a simpler form, I discovered that if (x + 1)(x - 1) = x² - 1 and if (x + 2)(x – 2) = x² - 1 - 3 = x² - 4 and if (x + 3)(x – 3) = x² - 1 -3 - 5 = x² - 9, then possibly (x + y)(x – y) = x² - y². I tried it by adding and subtracting 4, then 5—Eureka: I’d found a pattern powerful enough to predict things I hadn’t tested yet!]
Then felt I like some watcher of the skies when some new planet sweeps into his ken . . .
It was the first time I was aware that numbers are magical because they can give us some slight insight into all the rhythms that pulse beneath the surface of the universe, keeping it in balance. I became aware of it because Mr. Harvey planted a seed that happened to take root in my imagination. But much more importantly, because he didn’t short-circuit the power of that seedling’s growth by explaining it to me: he made me explore it for myself. For that insight and the awe it still can evoke within me, I owe Bill Harvey more than I can ever explain, much less repay.