After interviews with despairing professors, a few examples of recent student papers appeared on the screen. Here was our favorite, a vocabulary quiz answer, wrong in a couple of quite delightful ways but also unintentionally right.
définition: l'étude de fossiles
How the geologist on the sofa next to me laughed. . . .
The other surprise nugget that appeared this week isn't amusing but was unexpected in the middle of Jonathan Franzen's The Corrections. I had tried reading this a few years ago, but found the will to live as well as the will to read seeping away by about page 70. Relentless downward spirals ( cf Bonfire of the Vanities) affect me this way. This time I persevered, however, and was rewarded by quite an amazing novel and — fanfare of trumpets — a math proof I had forgotten.
Having had it drummed into our heads so often, we all remember how to square a + b algebraically. What I didn't remember was the geometric representation of this, so elegantly simple. Once again, I'll copy and paste an online version.
The natural way to understand the concept of squaring is through looking at the area of a square—which is calculated by squaring. So below is a picture of a square whose sides are each a + b long. To make that more clear, those sides are broken up into their separate a and b parts.
Asking about (a + b)2, then, is just like asking about the area of that whole square. But the whole square is broken up into smaller squares and rectangles, and we know enough information to calculate each of those smaller parts separately. The areas of the two smaller squares are calculated below.
Notice that the areas of the two smaller squares together come nowhere close to totaling the area of the large square. In algebra terms, we'd have to say that (a + b)2 must simply be greater than a2 + b2. Of course that means they can't be equal, which is exactly what we've been trying to understand! This picture actually tells us even more, though. It tells us how much greater. Each of the blue rectangles has a length of a and a width of b, so they each have an area of a times b. And there's two of them. Which means precisely that (a + b)2 = a2 + 2ab + b2, just as we saw in the algebra.