Billions & Billions (2 page)

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 billionaires—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 passé. 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 billion 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 milliarden.” 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 0s 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, 10
⁶
= 1,000,000; 10
⁹
= 1,000,000,000; 10
¹²
= 1,000,000,000,000; and so on. These little superscripts are called exponents or powers; for example, 10
⁹
is described as “10 to the power 9” or, equivalently, “10 to the ninth” (except for 10
²
and 10
³
, 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 × 1,000,000,000 is 10
³
× 10
⁹
= 10
¹²
. Or take some larger numbers: If there are 10
¹¹
stars in a typical galaxy and 10
¹¹
galaxies, there are 10
²²
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 10
⁹
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 microbes in a teaspoon of soil (10
⁸
); of grains of sand on all the beaches of the Earth (maybe 10
²⁰
); of living things on Earth (10
²⁹
); of atoms in all the life on Earth (10
⁴¹
); of atomic nuclei in the Sun (10
⁵⁷
); or of the number of elementary particles (electrons, protons, neutrons) in the entire Cosmos (10
⁸⁰
). This doesn’t mean you can
picture
a billion or a quintillion objects in your head—nobody can. But, with exponential notation, we can
think
about and calculate with such numbers. Pretty good for self-taught beings who started out with no possessions and who could number their fellows on their fingers and toes.

Really big numbers are part and parcel of modern science; but I don’t want to leave the impression that they were invented in our time.

Indian arithmetic has long been equal to large numbers. You can easily find references in Indian newspapers today to fines or expenditures of lakh or crore rupees. The key is: das = 10; san = 100; hazar = 1,000; lakh = 10
⁵
; crore = 10
⁷
; arahb = 10
⁹
; carahb = 10
¹¹
; nie = 10
¹³
; padham = 10
¹⁵
; and sankh = 10
¹⁷
. Before their culture was annihilated by the Europeans, the Maya of ancient Mexico devised a world timescale that dwarfed the paltry few thousand years that the Europeans thought had passed since the creation of the world. Among the decaying monuments of Coba, in Quintana Roo, are inscriptions showing that the Maya were contemplating a Universe around 10
²⁹
years old. The Hindus held that the present incarnation of the Universe is 8.6 × 10
⁹
years old—almost right on the button. And the third century
B.C
. Sicilian mathematician Archimedes, in his book
The Sand-Reckoner
, estimated that it would take 10
⁶³
grains of sand to fill the Cosmos. On the really big questions, billions and billions were small change even then.

T
he way I first heard the story, it happened in ancient Persia. But it may have been India or even China. Anyway, it happened a long time ago. The Grand Vizier, the principal advisor to the King, had invented a new game. It was played with moving pieces on a square board comprised of 64 red and black squares. The most important piece was the King. The next most important piece was the Grand Vizier—just what we might expect of a game invented by a Grand Vizier. The object of the
game was to capture the enemy King, and so the game was called, in Persian,
shahmat—shah
for King,
mat
for dead. Death to the King. In Russian it is still called
shakhmat
, which perhaps conveys a lingering revolutionary sentiment. Even in English there is an echo of this name—the final move is called “checkmate.” The game, of course, is chess. As time passed, the pieces, their moves, and the rules of the game all evolved; there is, for example, no longer a Grand Vizier—it has become transmogrified into a Queen, with much more formidable powers.