The Power And Beauty Of Quantification

There are some . . . who think that the number of [grains of] sand is infinite. . . . There are some who, without regarding it as infinite, yet think no number has been named which is great enough. . . . But I will try to show you [numbers that] exceed not only the number of the mass of sand equal to the Earth filled up ... but also that of a mass equal in magnitude to the Universe.

The Sand-Reckoner

I never said it. Honest. Oh, I said there are maybe 100 billion galaxies and 10 billion trillion stars. It's hard to talk about the Cosmos without using big numbers. I said "billion" many times on the Cosmos television series, which was seen by a great many people. But I never said "billions and billions." For one thing, it's too imprecise. How many billions are "billions and billions"? A few billion? Twenty billion? A hundred billion? "Billions and billions" is pretty vague. When we reconfigured and updated the series, I checked—and sure enough, I never said it.

But Johnny Carson—on whose Tonight Show I'd appeared almost thirty times over the years—said it. He'd dress up in a corduroy jacket, a turtleneck sweater, and something like a mop for a wig. He had created a rough imitation of me, a kind of Doppelganger, that went around saying "billions and billions" on late-night television. It used to bother me a little to have some simulacrum of my persona wandering off on its own, saying things that friends and colleagues would report to me the next morning. (Despite the disguise, Carson—a serious amateur astronomer—would often make my imitation talk real science.) Astonishingly, "billions and billions" stuck. People liked the sound of it. Even today, I'm stopped on the street or on an airplane or at a party and asked, a little shyly, if I wouldn't—-just for them—say "billions and billions."

"You know, I didn't actually say it," I tell them. "It's okay," they reply. "Say it anyway."

I'm told that Sherlock Holmes never said, "Elementary, my dear Watson" (at least in the Arthur Conan Doyle books); Jimmy Cagney never said, "You dirty rat"; and Humphrey Bogart never said, "Play it again, Sam." But they might as well have, because these apocrypha have firmly insinuated themselves into popular culture.

I'm still quoted as uttering this simple-minded phrase in computer magazines ("As Carl Sagan would say, it takes billions and billions of bytes"), newspaper economics primers, discussions of players' salaries in professional sports, and the like.

For a while, out of childish pique, I wouldn't utter or write the phrase, even when asked to. But I've gotten over that. So, for the record, here goes: "Billions and billions."

What makes "billions and billions" so popular? It used to be that "millions" was the byword for a large number. The enormously rich were millionaires. The population of the Earth at the time of Jesus was perhaps 250 million people. There were almost 4 million Americans at the time of the Constitutional Convention of 1787; by the beginning of World War II, there were 132 million. It is 93 million miles '(150 million kilometers) from the Earth to the Sun. Approximately 40 million people were killed in World War I; 60 million in World War II. There are 31.7 million seconds in a year (as is easy enough to verify). The global nuclear arsenals at the end of the 1980s contained an equivalent explosive power sufficient to destroy 1 million Hiroshimas. For many purposes and for a long time, "million" was the quintessential big number.

But times have changed. Now the world has a clutch of Millionaires—and not just because of inflation. The age of the Earth is well-established at 4.6 billion years. The human population is pushing 6 billion people. Every birthday represents another billion kilometers around the Sun (the Earth is traveling around the Sun much faster than the Voyager spacecraft are traveling away from the Earth). Four B-2 bombers cost a billion dollars. (Some say 2 or even 4 billion.) The U.S. defense budget is, when hidden costs are accounted for, over $300 billion a year. The immediate fatalities in an all-out nuclear war between the United States and Russia are estimated to be around a billion people. A few inches are a billion atoms side by side. And there are all those billions of stars and galaxies.

In 1980, when the Cosmos television series was first shown, people were ready for billions. Mere millions had become a little downscale, unfashionable, miserly. Actually, the two words sound sufficiently alike that you have to make a serious effort to distinguish them. This is why, in Cosmos, I pronounced "billions" with a fairly plosive "b," which some people took for an idiosyncratic accent or speech deficiency. The alternative, pioneered by TV commentators—to say, "That's billions with a b"—seemed more cumbersome.

There's an old joke about the planetarium lecturer who tells his audience that in 5 billion years the Sun will swell to become a bloated red giant, engulfing the planets Mercury and Venus and eventually perhaps even gobbling up the Earth. Afterward, an anxious member of the audience buttonholes him: "Excuse me, Doctor, did you say that the Sun will burn up the Earth in 5 billion years?" "Yes, more or less."

"Thank God. For a moment I thought you said 5 million."

Whether it's 5 million or 5 billion, it has little bearing on our personal lives, as interesting as the ultimate fate of the Earth may be. But the distinction between millions and billions is much more vital on such issues as national budgets, world population, and nuclear war fatalities.

While the popularity of "billions and billions" has not entirely faded, these numbers too are becoming somewhat small-scale, narrow-visioned, and passe. A much more fashionable number is now on the horizon, or closer. The trillion is almost upon us.

World military expenditures are now nearly $1 trillion a year. The total indebtedness of all developing nations to Western banks is pushing $2 trillion (up from $60 billion in 1970). The annual budget of the U.S. Government is also approaching $2 trillion. The national debt is around $5 trillion. The proposed, and technically dubious, Reagan-era Star Wars scheme was estimated to cost between $1 trillion and $2 trillion. All the plants on Earth weigh a trillion tons. Stars and trillions have a natural affinity: The distance from our Solar System to the nearest star, Alpha Centauri, is 25 trillion miles (about 40 trillion kilometers).

Confusion among millions, billions, and trillions is still endemic in everyday life, and it is a rare week that goes by without some such muddle on TV news (generally a mix-up between millions and billions). So, perhaps I can be excused for spending a moment distinguishing: A million is a thousand thousand, or a one followed by six zeros; a billioli is a thousand million, or a one followed by nine zeros; and a trillion is a thousand billion (or equivalently, a million million), which is a one followed by 12 zeros. This is the American convention. For a long time, the British word "billion" corresponded to the American "trillion," with the British using—sensibly enough—"thousand million" for a billion. In Europe, "milliard" was the word for a billion. As a stamp collector since childhood, I have an uncanceled postage stamp from the height of the 1923 German inflation that reads "50 mil-liarden." It took 50 trillion marks to mail a letter. (This was when people brought a wheelbarrow full of currency to the baker's or the grocer's.) But because of the current world influence of the United States, these alternative conventions are in retreat, and "milliard" has almost disappeared.

An unambiguous way to determine what large number is being discussed is simply to count up the zeros after the one. But if there are many zeros, this can get a little tedious. That's why we put commas, or spaces, after each group of three zeros. So a trillion is 1,000,000,000,000 or 1 000 000 000 000. (In Europe one puts dots in place of commas.) For numbers bigger than a trillion, you have to count up how many triplets of Os there are. It would be much easier still if, when we name a large number, we could just say straight out how many zeros there are after the one.

Scientists and mathematicians, being practical people, have done just this. It's called exponential notation. You write down the number 10; then a little number, written above and to the right of the 10 as a superscript, tells how many zeros there are after the one. Thus, 106 = 1,000,000; 109 = 1,000,000,000; 1012 = 1,000,000,000,000; and so on. These little superscripts are called exponents or powers; for example, 109 is described as "10 to the power 9" or, equivalently, "10 to the ninth" (except for 102 and 103, which are called "10 squared" and "10 cubed," respectively). This phrase, "to the power"—like "parameter" and a number of other scientific and mathematical terms— is creeping into everyday language, but with the meaning progressively blurred and distorted. In addition to clarity, exponential notation has a wonderful side benefit: You can multiply any two numbers just by adding the appropriate exponents. Thus, 1,000 X 1,000,000,000 is 103 X 109 = 1012. Or take some larger numbers: If there are 1011 stars in a typical galaxy and 10n galaxies, there are 1022 stars in the Cosmos.

But there is still resistance to exponential notation from people a little jittery about mathematics (although it simplifies, not complicates, our understanding) and from typesetters, who seem to have a passionate need to print 109 as 109 (the typesetters for Random House being a welcome exception, as you can see).

The first six big numbers that have their own names are shown in the accompanying box. Each is 1,000 times bigger than the one before. Above a trillion, the names are almost never used. You could count one number every second, day and night, and it would take you more than a week to count from one to a million. A billion would take you half a lifetime. And you couldn't get to a quintillion even if you had the age of the Universe to do it in.

Once you've mastered exponential notation, you can deal effortlessly with immense numbers, such as the rough number of ___

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