One of the great challenges of swimming, for me, is remembering how many laps I’ve swum. Mostly I just repeat the number in my head as I swim: “This is lap twelve, this is lap twelve, this is lap twelve,” for example, which is a little boring. I have a few mnemonics. Lap number fifteen, for example, is the lap especially devoted to daydreaming about the snack that I will eat when I get home. To most numbers, however, I don’t have any particular association, and my mind strays.
Yesterday, on lap twenty-five, I strayed into thinking about why the fraction one quarter is written in decimals as 0.25. I had always assumed that the way fractions appear in decimal form is more or less arbitrary, but yesterday, maybe because I was swimming, and therefore thinking about my hands at the same time that I was thinking about decimals, it started to seem a little suspicious to me that one quarter, when written in decimals, should take the form of five times five—and especially suspicious given that humans ordinarily write their numbers in the decimal system (a ten-based system—that is, with ten numerals, 0 through 9) in part because they have five fingers on each hand, and two times five equals ten. Is one quarter a special fraction? Is it a coincidence that one quarter, in decimals, looks like five squared?
I couldn’t figure it out while swimming, but this morning, while walking the dog, I tried again.
Suppose you’re writing numerals in a b-based system. (In the usual, decimal system, b is 10, but b could just as well be 8 or 12 or 47.) And suppose that b is the product of the two smaller numbers, the number 2 and a number that we’ll call f, for fingers. (In the world we live in, b = 10 and f = 5.)
Then here’s another way of asking my question: what does the fraction 1/4 look like when expressed in “decimals” in a b-based system?
Suppose we’re going to need at least two “decimal” places to express the fraction 1/4 in the b-based system. (They shouldn’t really be called “decimal” places, of course, in a non-decimal system, and I’m guessing when I say that two is the number of “decimal” places that we’ll need to shift, but I’m pretty sure my guess is kosher.) In order to “see” the two numerals to the right of the “decimal” place, we need to move the “decimal” place over two notches—in other words, we need to multiply by whatever 100 is in the b-based system. In our usual decimal system, 100 is 100—more properly written, 10010 = 10010. In an 12-based system, 10012 is 14410. In general, 100b = b2.
To find the numerals to the right of the “decimal” point that express the fraction 1/4 in a b-based system, in other words, you need to multiply 1/4 by 100b, or 1/4 x b2.
But b = 2 x f. So the numerals to the right of the “decimal” point that express 1/4 in a b-based system also equal 1/4 x (2f)2, or 1/4 x 4 x f2, or f2.
Not a coincidence, in other words. It’s because humans have five fingers, and because they write their numbers in a system based on ten, which is five times two, that the fraction 1/4 is expressed by the square of five when written in decimals.
What’s more, it’s possible to generalize. If humans had six fingers on each of their two hands and counted in a 12-based system, then the fraction 1/4 would be expressed to the right of the “decimal” point by the same numerals that express 6 squared—3610, or 3012. In other words, 1/4 = 0.3012, and it wouldn’t be a coincidence that 3012 is the square of 612. And if humans had four fingers on each hand and counted in an 8-based system, the fraction 1/4 would be expressed to the right of the “decimal” point by the same numerals that express 4 squared—1610, or 208. In other words, 1/4 = 0.208, and it isn’t a coincidence that 208 is the square of 48.
You can generalize in another direction, too. If you look at the fraction 1/8, and notice that 8 is the cube of 2, you’ll see that its expression in decimals is 0.125, or 5 cubed—not a coincidence, either.
What if humans had h hands, instead of just 2, and counted in a system based on b = h x f? It would still be the case that the fraction 1/h2 will be expressed as f2 in b-decimals. For example, if humans had three hands and four fingers on each hand, and therefore chose to count in twelves, the fraction 1/9 would take the form 0.1412, and it wouldn’t be a coincidence that 9 is the square of 3, and 1412 (a.k.a. 1610) is the square of 4.
I don’t know how a three-handed human would swim, however.
Ronald Reagan is often credited, especially among historians on the right, with having defeated the Soviet Union by challenging it to an arms race so costly that the Soviet economy collapsed, taking the political authority of Communism down with it. Some historians on the left prefer to credit Reagan’s diplomacy and arms negotiations, and to tip the hat to his Soviet counterpart Mikhail Gorbachev, but in 1986 Gorbachev himself acknowledged in a speech to the Politburo that the nuclear-arms threat from America was tantamount to an economic one: “We will be pulled into an arms race that is beyond our capabilities, and we will lose it because we are at the limit of our capabilities. … If the new round [of an arms race] begins, the pressures on our economy will be unbelievable.”
As a child growing up in the 1980s, I wasn’t aware that there was a plan behind Reagan’s buildup. It seemed, rather, to be a mysterious and almost autonomous process, driven by fear and nationalist rivalry, not strategy. I was a child, of course, and maybe I didn’t understand. But I find that as an adult, I still harbor a doubt, perhaps unfair, that Reagan fully intended his strategy. I don’t doubt that he wished the Soviet Union ill, and there’s no question that he thought that stockpiling nuclear weapons would harm the Soviet Union, but I’m not quite persuaded that he understood in advance that military-induced economic stress could trigger a spontaneous collapse of political authority in the Soviet Union, of the sort that Hobbes alluded to when he wrote that “The obligation of subjects to the sovereign, is understood to last as long, and no longer, than the power lasteth, by which he is able to protect them.”
Quite possibly he did, though. I was startled, a couple of weeks ago, to discover that a very articulate explanation of Reagan’s strategy had been published two decades before Reagan even took office. In the science-fiction novel His Master’s Voice, published in Polish in 1968 (and translated into English in 1983), Stanislaw Lem accurately predicted not only Reagan’s military strategy but the economic rationale behind it.
His Master’s Voice concerns a team of American scientists secretly attempting to decode a message transmitted in a stream of neutrinos from a distant galaxy. Because the team works under threat of a takeover by the American military, who fear that the Soviet Union might also learn of the message and decipher it first, the narrator has occasion to look back, from the novel’s imagined future, on America’s military strategy in the 1970s (which in 1968, of course, had not yet happened):
In the seventies, for a while, the ruling doctrine was the “indirect economic attrition” of all potential enemies; Secretary of Defense Kayser expressed this with the maxim “The thin starve before the fat lose weight.” The competition-duel in nuclear payloads gave way to a missile race, and that in turn led to the building of more and more expensive “antimissile missiles.” The next step in the escalation was the possibility of constructing “laser shields,” a stockade of gamma lasers which would line the perimeter of the country with destroyer rays; the cost of installing such a system was set at four hundred to five hundred billion dollars. After this move in the game, one could next expect the putting into orbit of giant satellites equipped with gamma lasers, whose swarm, passing over the territory of the enemy, could consume it utterly with ultraviolet radiation in a fraction of a second. The cost of that belt of death would exceed, it was estimated, seven trillion dollars. This war of economic attrition—through the production of increasingly expensive weaponry that thereby placed a severe strain on the whole organism of government—although seriously planned, could not be carried out, because the building of super- and hyperlasers turned out to be insurmountably difficult for the current technology.
Remarkably, Lem was not only predicting that America would engage in an arms race in order to sap the Soviet Union’s economic capacity, but also predicting that someone like Reagan would come along and accelerate the arms race by adding laser defenses to missile offenses, much as Reagan did in his 1983 “Star Wars” speech, which launched what Reagan called the Stategic Defense Initiative.
While googling to see whether anyone else had already written this blog post, I discovered that Lem himself explained his clairvoyance—sort of. In a 1986 book, One Human Minute, Lem wrote that years earlier he had gained access to “several volumes on the military history of the twenty-first century,” and though at first he feared betraying his knowledge of their contents, he soon realized that “The safest way to conceal a remarkable idea . . . was to publish it as science fiction,” and therefore slipped one of the secrets into page 125 of the novel he was then working on, His Master’s Voice. (A more mundane explanation is possible, of course: he might have punked English readers by slipping into the 1983 English translation a passage that wasn’t present in the 1968 Polish original.)