I read recently that two supercomputer manufacturers were in a contest to determine who could calculate pi to the most digits. My simple question, simple for you at least, is, what data do they input to begin these calculations? Every schoolchild knows that pi is the ratio of a circle's circumference to its diameter. Obviously mathematicians do not draw a circle and then measure out the circumference with increasingly tiny rulers. But what do they do instead? --Maxwell Stephens, Washington, D.C.
Dreaming up "algorithms" (techie talk for "methods") to compute pi has occupied the world's great minds for more than two millennia. Clearly these aren't guys you'd want to go on a long fishing trip with. The ancient Greeks used a simple method: You draw polygons (e.g., hexagons) around a circle with a diameter of one--one inside, one out. Calculate the perimeter of the polygons (which is pretty straightforward), take an average, and you get a rough idea of pi. Use polygons with more sides and your approximation of pi gets closer and closer. The mathematician Archimedes got as far as 96 sides, calculating that pi was between 3.1408 and 3.1428.
Today mathematicians use far more sophisticated algorithms involving converging infinite series. A converging infinite series is a mathematical sequence that approaches (but never actually reaches) a target number called a limit. For example, the limit of the series 1+1/2+1/4+1/8+ . . . is 2.
Long ago it was realized that certain infinite series converge on fractions or reciprocals of pi. For example, in 1671 mathematician Gottfried Leibniz discovered that the series 1 - 1/3 + 1/5 - 1/7 + . . . converges on pi/4. This may seem strange--I mean, what do fractions have to do with the circumference of a circle, right?--but take my word for it, it happens. The discovery of ever more "efficient" infinite series--that is, that converge on pi faster for each term you add--coupled with the development of bigger and better computers has made it possible to calculate pi to thousands, millions, and now billions of decimal places. Cecil, knowing his readers' love of higher mathematics, would be pleased to reprint one of these magic formulas in full, but there isn't room and besides, I gave up Greek subscripts for Lent.
Why compute one billion digits? God knows. As one learned treatise notes, "thirty-nine places of pi suffice for computing the circumference of a circle girdling the known universe with an error no greater than the radius of a hydrogen atom." One pi-wars participant rationalizes by saying once you get beyond a billion digits subtle patterns may begin to emerge in the numbers, but give me a break. The real reason, many feel, is "because it's there." So immature. Thank God the rest of us have put such foolishness behind us.
Where did the word dollar come from, anyway? --Ellie Rosen, Santa Barbara, California
Finally, a semicute word-origin story that turns out to be true. In 1516 a local potentate opened a silver mine and later a mint at a locale in Bohemia called Joachimsthal (literally, "dale of Joachim"). The mint began churning out coins known as Joachimsthalers, soon shortened to thalers and called dalers by the Dutch. The English corrupted this to dollar and apparently began applying the word to any large silver foreign coin. In North America, for instance, English settlers referred to the Spanish piece of eight, then in wide circulation, as the Spanish dollar.
For several years after independence Americans used whatever oddball coinage they could get their hands on, Spanish dollars included. But by and by they began thinking it was time to establish their own currency. Thomas Jefferson strongly opposed using the English system and urged that the basic monetary unit be called the dollar, a term people were already familiar with. The Continental Congress said what the hell and declared the dollar the U.S. monetary unit in 1785, although no U.S. dollars were actually minted until 1794. By 1837 Washington Irving was making snide references to "the almighty dollar," and we've been slaves to the buck ever since.
Art accompanying story in printed newspaper (not available in this archive): illustration/Slug Signorino.