The Calculus Diaries - Jennifer Ouellette [16]
Or so historians and mathematicians believed. That is why the rediscovery of the Archimedes palimpsest in the 1990s is so significant. Heiberg had been unable to transcribe the relevant pages in 1908 because they were so badly damaged. Modern analytic methods uncovered that long-hidden text. This time around, the task of transcribing the fully restored text fell to Reviel Netz, a professor of mathematical history at Stanford University. Netz’s transcription hints that Archimedes had a far more sophisticated understanding of the infinite than historians have generally credited him with. In particular, the Greek mathematician flirted with the notion of actual infinity while calculating the volume of a fingernail-shaped figure.
This is not the same as fashioning a rigorous mathematical proof to deal with infinity, however. The man who gets the credit for resolving the problem of infinitesimals is an eighteenth-century mathematician named Jean le Rond d’Alembert. D’Alembert’s life story is fodder for an Oscar-worthy biopic. He was a foundling who took the name of the church in Paris on whose steps he was found: Saint Jean Baptiste le Rond. He was raised by a glazier but later discovered his birth parents were a general and a noble-woman.
D’Alembert’s insight into Zeno’s problem of motion seems obvious in retrospect: that arrow shot from a bow is on a journey, and it makes that journey in an infinite number of smaller steps, but its travel does not continue indefinitely—it has a destination, namely, your calculus teacher’s heart. That ultimate destination is the limit, even though the arrow makes an infinite number of subjourneys before arriving. All those subjourneys, added together, mean that the arrow hits its mark—and that summing process is integral calculus.
Let’s go back to Catherine and Ed to illustrate. Ed moves closer and closer to Catherine, halving the distance between them with each step. We can add those numbers together: 1 + ½ + ¼ + ⅛ + , and so on, and notice that the sum of this “infinite series” gets closer and closer to exactly 2. We can check this by “taking the limit”: if 2 is the limit, then 2-1 = 1, 1-½ = ½, ½-¼ = ¼, and so on into infinity. With each iteration, the result gets closer and closer to 0. Ed still crosses a distance of 2 feet, but he does it in an infinite number of steps.
This gives rise to one of the most common stumbling blocks for the beginning calculus student: the notion that 0.9999 . . . is actually equivalent to 1. My mind, too, balked at accepting this mathematical fact when my spouse, Sean, patiently explained it to me late one night as I was struggling to understand the limit. Intuitively, we think of a fraction as being a finite sum. We have all eaten one-ninth of a pie, after all. But we also learn about irrational numbers—like π, or the so-called golden ratio, ø—where the string of numbers in the decimal expansion really does go on forever.
That was my mistake: assuming that 0.999 . . . is like an irrational number, and therefore represents a sequence of numbers that get closer and closer to 1 but never reaches it exactly. Such is not the case; it is a rational number, in which the same decimal number endlessly repeats, and thus can have a finite sum. Eventually the limit of the sequence equals 1, and that means 0.999 . . . has a fixed value, rather than being an infinite progression. It might seem paradoxical, but two very different mathematical expressions can nonetheless represent the same number. Ergo, to a calculus teacher, 0.999999 . . . is just another way of writing 1.10
One could argue that calculus itself was invented via tiny infinitesimal bits of accrued knowledge that, taken together, added up to a revolutionary new whole. But like the function, calculus is far more than the sum of its parts, making it possible to understand the world around us in dynamic, rather than static terms. “We live in a world of ceaseless growth and decay, with things in fretful motion on