A Sentimental Journey to the Googolplex
Chapter 1 from Universe Down To Earth, © 1994 Neil deGrasse Tyson. Columbia University Press
The methods of science hold their deepest roots in the origin of numbers. We owe it to the “number line” to give due attention to all numbers big and small. As one who studies the universe, however, you can expect that I have slanted the coverage toward the big ones.
The last time somebody asked you, “How big is a billion?”, what was your response? If you had nothing much to say, then one can infer that you are probably not an astronomer. Of all the natural and physical sciences, astronomers dominate the “big numbers” market. Indeed, it is nearly impossible to conduct a conversation with an astronomer without greeting numbers that contain more zeroes than you would bother to count. This chapter may help you appreciate the evolution of this phenomenon.
A Sentimental Journey to the Googolplex
It is generally agreed among historians that economics played a role in the birth of mathematics. For example, if I breed chickens and you breed sheep, and I want some of your sheep, it would be natural for us to swap chickens for sheep. But first we must answer the question: How many chickens equal one sheep? This seemingly simple question, in fact, requires the invention of a logical scheme for counting. Some of the earliest evidence for the ability to count comes to us on a 30,000 year old wolf bone excavated in eastern Europe that was deeply etched with fifty-five notches in groups of five. But the ability to count, undeniably a sophisticated concept, is still not sufficient to deal with all our problems. Suppose I have only five chickens, but you think a sheep is worth ten. I cannot afford to “buy” a whole sheep. With this predicament, a revolutionary concept of numbers is needed to help us consummate our trade: the concept of a half a sheep. But we need not stop with only half a sheep. Suppose my unit of barter is not a chicken but a pea from my vegetable garden. Certainly five peas will buy much less than half a sheep—perhaps only one-thousandth of a sheep. Apart from the logistical difficulty of actually trading a fraction of a (living) sheep, it is clear that advanced barter requires one to be comfortable with numerical quantities borrowed from the world of fractions.
For most of the 5,000 years of recorded history, little scientific significance was attached to extremely small numbers. It was not until the late 1600s when the Dutch naturalist Anton van Leeuwenhoek introduced the microscope to the world of biology. With the desire to measure precisely the sizes of cells, protozoa, and bacteria, there proliferated tiny fractions of the measuring unit such as one-thousandth, one-millionth, and one thousand-millionth.
Meanwhile, astronomers actively explored in the opposite direction with the help of Galileo Galilei’s introduction of the telescope to the world of astronomy in the year 1610. The telescope heralded a new scientific era that allowed astronomers to worry about, and subsequently estimate the sizes of objects in the universe, and the distances to them. Equally bulky numbers then emerged, such as one million-billion, one billion-trillion, and one trillion-trillion.
Astronomers and biologists alike were faced with the same problem: how can one cleanly and neatly talk about extreme quantities of the universe without polluting the conversation with countless “-illions” or “-illionths”? And European travelers need not be reminded that a billion in England and most of Europe is a thousand times bigger than a billion in the United States and France. This numerical dilemma was compounded in the early 20th century after atoms and sub-atomic particles were discovered by the physicists J. J. Thompson, Ernest Rutherford, and James Chadwick at the Cavendish Laboratory of the University of Cambridge.
What follows is a journey through some awkward “-illionths” and “illions” that once bedeviled the scientific community.
|one thousandth||one over one with three zeroes||This is the approximate radius of a peppercorn.|
|one millionth||one over one with six zeroes||The head of a human sperm typically has this radius.|
|one billionth||one over one with nine zeroes||A common radius for the tiniest bacteria.|
|one trillionth||one over one with twelve zeroes||Fifty three of these is the classical radius of the hydrogen atom.|
|one quadrillionth||one over one with fifteen zeroes||About three of these will get you the classical radius of the electron.|
|one quintillionth||one over one with eighteen zeroes||This slice of a sheep would not buy much in any economy.|
|one thousand||A one followed by three zeroes||This is about the number of times per minute that Earth is struck by lightning.|
|one million||A one followed by six zeroes||When counting people, there are about eight of these piled into New York City.|
|one billion||A one followed by nine zeroes||If you never go to sleep, it will take you 32 years to count this high. And cows are dismayed to learn that, when last checked, McDonald’s hamburger food chain has sold about one hundred of these. Laid end-to-end, this many hamburgers would go around the Earth 200 times, and, with what remains after you have eaten a few, would bridge three round trips to the Moon.|
|one trillion||A one followed by twelve zeroes||This is about how many seconds of time have passed since the Neanderthal roamed Europe, Asia, and northern Africa.|
|one quadrillion||A one followed by fifteen zeroes||If the human population ever reaches this number then everybody will have to stand in order to fit on the surface of the Earth.|
|one quintillion||A one followed by eighteen zeroes||This is the sum of all sounds and words ever uttered since the dawn of the human species. The tally does include congressional debates and filibusters. Coincidentally, this is also about the number of grains of sand on an average beach.|
|one sextillion||A one followed by twenty-one zeroes||This is the estimated number of stars in the universe.|
|one bezillion||In spite of what your friends may tell you, this number does not really exist.|
The scientific community, unhappy with such awkward terminology, was in need of a more elegant method of numerical organization. Hence, a sensible system of prefixes was formalized by the International Union of Pure and Applied Physics to be used in conjunction with the metric system. With such a scheme, physical quantities are described in units of thousands such that every three zeroes appended to a number yields a new prefix. Additionally, scientific notation was introduced so that writer’s cramp would not arise if, for some reason, you chose to write out the number. A large number like 2,000,000 (two million) would be written in scientific notation as 2.0 × 106, where the “6” in the “106” tells you how many places the decimal hops to the right. For a tiny number such as .000002 (two millionths), scientific notation represents it as a 2.0 x 10-6, where the “-6” in the “10-6” tells you how many places the decimal moves to the left.
The officially accepted prefixes are listed below.
From what appears to be a melodic quartet in pentameter, one just locates the correct prefix and appends it to whatever quantity is measured. Some common examples: centimeter (one hundredth of a meter), kilogram (one thousand grams), and megahertz (one million hertz).
A romp through the scientific offerings of the universe can occasionally tempt you to invent new units that will make measurement simpler for the intended task. This, of course, has already been done for the smallest to the largest length scales. The branch of physics known as quantum mechanics dictates that the structure of space, itself, is discontinuous on scales of what is called the “Planck length”, which is about 1.6 x 10-33 centimeters. The role of the German physicist Max Planck, in the dawn of quantum mechanics, is discussed further in Chapter 8. Atomic distances and wavelengths of visible light are commonly measured in units of “Ångstroms”, which is defined to be 10-8 centimeters. Yellow light has a wavelength of about 5000 Ångstroms.
Distances among planets in the solar system are conveniently measured in “astronomical units”, which is defined to be the average distance between Earth and the Sun—about 93 million miles. On this scale, for example, the average distance of Pluto from the Sun is just under 40 astronomical units. The distance that light travels in one year is enormous. This is the famous “light year”, which is about 5.8 trillion miles. It forms a convenient yardstick to measure distances between the stars. The nearest star to the Sun, Proxima Centauri, is about 4.1 light years away.
For obscure historical reasons, most astronomers use the “parsec” rather than the light year as the yardstick of choice. One parsec equals 3.26 light years. There is no widely used unit of distance that is larger than the parsec, although one could, in principle, define the size of the entire universe (about 15 billion light years in diameter) as a new yardstick, but it would not be very useful—what else could you find to measure with it?
In any adopted set of units, there is no doubt that astronomers monopolize the big numbers. But the biggest number of them all—the one that signifies the physical limit of measurable Nature—is a very clean, compact-looking number within which all of astronomy is contained:
This unsuspecting quantity represents the estimated number of atoms in the Universe, yet it has no name. How about totillion? If you are worried that each atom contains subatomic particles that can each stand up and be counted, then you needn’t worry too much. Over ninety percent of all atoms in the universe are hydrogen atoms. Hydrogen in its most common form contains no neutrons—only a single proton and a single electron. For a better estimate we should then double our newly named number: 2 x 1081
Does this mean that we cannot discuss numbers bigger than 2 x 1081? Certainly not. We must simply remember that such numbers have no relationship with physically countable quantities in Nature. Let’s take 10100, for example. It is a one followed by 100 zeroes. This rounded, neat-looking number, which is ten quadrillion times larger than the number of atoms in the universe, actually has a name. It was christened a “googol” by a nine-year-old nephew of the mathematician Edward Kasner. Though it is a worthy, even lovable number, it is not my favorite. That distinction goes to the number ten raised to the googol power:
10(googol) = 1010100
It has the immortal name of the “googolplex”. This number was originally supposed to be a one followed by as many zeroes as it would take for someone to get tired of writing them. Since different people obviously get tired at different rates, the googolplex was re-defined in terms of the googol. Thus, a googolplex is so big that it cannot be written without the aid of scientific notation. It has more zeroes than can fit in Universe. Actually, this is not surprising since a googolplex is a one followed by a googol zeroes, and a googol is a number bigger than the sum of all particles in the Universe. Even if you could write your zeroes small enough to place one on every existing atom, the googolplex still could not be written out in the space of the Universe. It is sobering to see the dynamic range of astronomy humbled by the imagination of a nine-year-old just as it is enlightening to realize that one’s imagination can extend beyond the limits of astronomical perspectives.
Just for the record, there exists a named number that dwarfs even the googolplex. This is Skewes’ number, written as:
It is said that this number gives mathematicians information about the distribution of prime numbers. Skewes’ number can also be discussed abstractly even though it obviously has no measurable application to Nature. For example, the mathematician G. H. Hardy pointed out that if the entire Universe were a giant cosmic chessboard, and the interchange of protons between any two atoms were legal, then Skewes’ number would represent the total possible number of moves!
Is this the whole story of science and numbers, or can there be another level of investigation? One cloudy night, when I had nothing better to do, I decided to look more closely at our revered international system of metric prefixes.
In these days of inflated modifiers, a thank-you note might get more attention if you signed it “Thanks × 109”, provided, of course, you really do mean “Thanks a billion.”
What happens if you are 10-6 biologist? That must make you a microbiologist.
How about if you just read a copy of 2 x 103 Mockingbird? That must have been Harper Lee’s undiscovered classic novel, Two Kilo Mockingbird.
If you have ever played 10-12 boo with a child, then it was probably the metric version called pico-boo.
If the facial beauty of Helen of Troy was sufficient to launch a thousand ships, then the “10-3 Helen”, better known as the “milli-Helen”, must be the beauty required to launch just one ship.
Suppose you owned 101 cards? That would be your personal deka-cards.
What happens if you live in a 106 lopolis? This is none other than a megalopolis.
And finally, if you just had a 10-2 mental journey to the googolplex, what kind of journey was it?