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Fractions always yield decimals that terminate or eventually repeat periodically—that can be proven—and since this decimal does neither, it can’t be equal to the ratio of any whole numbers. It’s irrational.
Those of us who teach math should try to turn this bug into a feature. We should be up front about the fact that word problems force us to make simplifying assumptions. That’s a valuable skill—it’s called mathematical modeling. Scientists do it all the time when they apply math to the real world. But they, unlike the authors of most word problems, are usually careful to state their assumptions explicitly.
The upshot is that x² and 10x are now moving gracefully as a couple, rather than stepping on each other’s toes, by being paired within the single expression (x + 5)². That’s what will soon enable us to solve for x.
Imagine using a photocopier to reduce an image of a circle by, say, 50 percent. Then all distances in the picture—including the circumference and the diameter—would shrink in proportion by 50 percent. So when you divide the new circumference by the new diameter, that 50 percent change would cancel out, leaving the ratio between them unaltered. That ratio is pi.
Their confusion is understandable. It’s caused by our reliance on graphs to express quantitative relationships. By plotting y versus x to visualize how one variable affects another, all scientists translate their problems into the common language of mathematics. The rate of change that really concerns them—a viral growth rate, a jet’s velocity, or whatever—then gets converted into something much more abstract but easier to picture: a slope on a graph.
Newton knew nothing about chaotic dynamics, but according to his friend Edmund Halley, he complained that the three-body problem had “made his head ache, and kept him awake so often, that he would think of it no more.” I’m with you there, Sir Isaac.
The electric field’s undulations re-created the magnetic field, which in turn re-created the electric field, and so on, with each pulling the other forward, something neither could do on its own.
That’s the bleak message of the droopy curve. The primes are a dying breed. They never die out completely—we’ve known since Euclid they go on forever—but they fade into near oblivion.
