Astronomy is the science most involved with enormous numerals, where numbers get tossed around as though one were haggling in a Turkish bazaar. Despite the fact that there are 30 trillion cells in the human body (far more than the number of stars and planets in our galaxy), human physiology is rarely associated with vastness. Yet expressing the distances between celestial points of interest is a whole ’nother story: Even using giant units like light-years brings us to bewildering numerals containing enough zeros to recall those I got in college German.
It wasn’t always this way. The word million didn’t come into general use until the thirteenth century. Before then, the largest number was a myriad, equal to ten thousand. The Greeks, who coined the term, would occasionally resort to myriads of myriads, and that was sufficient to express the most complex concepts.
A million seemed huge when we were kids. It became less intimidating only when we realized it was possible to count to a million in a few days. It’s really not so big: A million steps takes you from San Francisco to Sacramento. A vacation lasting a million seconds gives you only an eleven-day reprieve from office misadventures.
In astronomy, we use million mainly in relation to the sun, which is nearly a million miles wide and sits 93 million miles away. Million also expresses the number of miles to the nearer planets. Venus is 26 such units, Mars 34. That’s about it.
Even less useful, astronomically, is the billion, which is a thousand million (in the United States, that is; in England it’s a million million). We might say that Saturn is nearly a billion miles away from us, and that Uranus, Neptune, and Pluto are a couple of billion. And the visible universe offers for our inspection about 50 billion galaxies. But that’s where a billion’s usefulness ends. No star has a radius as big as a billion miles, and none is as close to us as a billion miles. The unit just isn’t very applicable much beyond Earth, although it’s convenient for taking our planet’s world-wide census of 6 billion people, some of whom may contemplate the accumulated wisdom of the 60 billion to 100 billion people who have ever walked the face of this forgiving planet.
So we jump to a trillion. This is a million millions, suddenly a most valuable unit for government economists, physicists, astronomers, and more than a few fanatics caught in the grip of computer nanospeak. There are almost a trillion stars in our galaxy, and about the same number of planets. The light-year is equal to 6 trillion miles. Grasping what a trillion represents is like having a floodlight illuminate the path to understanding the cosmos (and in comprehending our national debt—some $5 trillion).
One way to appreciate the enormity of a trillion is to count it out. Unfortunately, at the rate of five numbers a second, without stopping to eat or sleep, this exercise would still require 3,000 years. So a trillion seconds ago carries us back 31,000 years—to well before the dawn of recorded history. The problem with such numbers is that their zeros blur into incomprehensibility. For example, 1,000,000,000,000 looks not much different from 1,000,000,000,000,000. That is, a trillion resembles a quadrillion, more or less. Actually, they differ to the same degree that the weight of a tadpole differs from that of an elephant or that your body’s mass compares with that of three cement trucks.
This is where we come to the realm more common to hyperbole than science. I’ve told you a million times to clean up your room leads children to think it’s okay to exaggerate. Who, then, would object when textbooks continue to repeat the long-obsolete figure that our Milky Way contains “100 billion” stars—when the actual figure is four to ten times greater?
(It’s easy to be imprecise with technological vocabulary. Case in point: People often use silicon, silicone, and silica interchangeably. But the meaning of silicon is elemental. And California’s computer center is called Silicon Valley, not silicone valley—though some women who work there have probably utilized the latter for improvements cosmetic rather than cosmic.)
A few centuries ago, when it seemed as if the universe had somehow organized itself in sync with human conceptions, including our own arbitrary numbering systems, many scientists desperately tried to make planets’ orbits and physical features match various numerological schemes. With enough searching, one that worked was sure to turn up—and it proved a real goody when it was uncovered by Daniel Titius in 1766 and popularized by Johann Bode a few years later. Eventually this layout became known as the Titius-Bode law.
Nowadays we don’t call it a law because it has no physical basis for existing at all. But back then, because the coincidence seemed so incredibly compelling, it was assumed that a scientific justification for that arrangement would eventually be found. It never happened, and, okay, here’s what we’re talking about:
Start with the numbers 0, 3, 6, 12, 24…where each number after zero doubles the preceding one. Then add 4 to each number and divide by 10 and you’ve got .4, .7, 1.0, 1.6, 2.8, 5.2, 10.0, and so on. What’s so odd about this? Just compare it with the distances from the sun to the planets, expressed in Earth-sun spans (the unit astronomers use most often, called the Astronomical Unit, or A.U.).
|
PLANET |
BODE-TITIUS |
ACTUAL DISTANCE IN A.U. |
|
Mercury |
.4 |
.4 |
|
Venus |
.7 |
.7 |
|
Earth |
1.0 |
1.0 |
|
Mars |
1.6 |
1.52 |
|
Asteroids |
2.8 |
2.8 |
|
Jupiter |
5.2 |
5.2 |
|
Saturn |
10.0 |
9.54 |
|
Uranus |
19.6 |
19.18 |
|
Pluto |
38.8 |
39.44 |
Yes, this omits Neptune from the picture, which at 30 A.U. doesn’t fit into this scheme at all, but Neptune wasn’t discovered until much later. However, before that, when Uranus and the first asteroids were found, and they fit the pattern perfectly, this was interpreted as further proof that the heavens did indeed march in sync with the tidy numbering scheme. What the arrangement really proved to be was a giant coincidence, albeit a remarkable one. And instructive. For it teaches us to be very cautious when we attempt to link events with other events or with numbers schemes; it also helps us understand the difference between correspondence and coincidence.
Many of us have yet to learn that lesson. If old Aunt Lucy dies just as lightning strikes the church and stops its clock, we’d be sorely tempted to connect the two events and assign mystical significance to the whole thing. We forget that simultaneous events occur all the time, and just because the wind knocks down that big oak tree just as we turn on the cold-water faucet to brush our teeth, we mustn’t assume that one event caused the other or that they’re linked in any way. The distinction between correspondence and coincidence is one of the most critical not only in astronomy but in all the sciences, even if our minds have a much harder time making that vital separation.
When math is reduced to pure numbers independent of linkage with events or phenomena such as distances to planets, then coincidences are more easily seen for what they are. For example, the number 37 is prime, meaning it’s an oddball, divisible only by itself and the number 1. Yet the numbers 111, 222, 333, 444, 555, 666, 777, 888, and 999 are all divisible by 37 and by nothing else.
Like Peter the Great, who had his wife’s lover beheaded and kept that head in a bottle of alcohol in her bedroom for her to contemplate, nature can also be perverse, though usually with greater subtlety. There’s no rhyme or reason for the numbers that biology and astronomy spring on us. Why does each cell in our body have 90 trillion atoms, roughly the same as the number of stars in our home cluster (Virgo Group) of galaxies? Why is there exactly the same number of Earth—sun distances in a light-year as there are inches to the mile? Why does the diameter of the sun (864,000 miles) have the same numerals as the number of seconds in a day (86,400)? We could go on and on, for these strange connections are always interesting. To the rationalist, coincidences are part of life; to the mystic, for whom everything is interconnected, coincidences simply do not exist. Both will extract immense but very different pleasure from matchups involving numbers large and tiny.
In any event, the trillion is the largest numeral we ever need to comprehend, for earthly as well as celestial use. A quadrillion—a thousand trillion—crosses the line so far beyond what the mind can handle, it becomes necessary to break it up into more digestible units. Like a giant chocolate bar, it’s more easily absorbed in smaller pieces.
Astronomers oblige by slicing segments of space into light-years—units containing 6 trillion miles. The nearest star is 4.3 light-years, or 26 trillion miles, and so far the spin of our heads is still controllable. If we picture traveling from New York to California sixty times in one second—a conceivable exercise—then we have grasped the speed of light. If we imagine moving that fast for two months, then we have understood the distance of 1 trillion miles. Going that fast for a year means 1 light-year, which, you’ll recall, is 6 trillion miles. A mental stretch, to be sure, but I think we can all do it and be left with some kind of a handle on the concepts of trillion and light-year.
But if we then insist on taking the trek farther, to, say, the nearest spiral galaxy—Andromeda—then we simply lose it all, especially if that gap is expressed as 12 million trillion miles, or 12 quintillion miles. That’s meaningless. Much better to abandon the smaller unit of mile altogether (even if it is a familiar concept) and use the more esoteric light-year from here on out. Then we simply say Andromeda is 2 million light-years away, and whether we can honestly grasp that or not, we’re at least playing with all our marbles.
The farthest distance we could possibly travel? It’s a hitchhike from here to the edge of the observable universe. A mere 14 billion light-years. That may be conceptualized by some of us, but I’ll bet that very few could make any sense or headway with that distance if it were expressed as 100 billion trillion miles, or 100 sextillion—a 1 followed by twenty-three zeros. Totally off the chart. We might as well use inches.
Still, those zero-rich numbers are tiny compared with the largest number of things in the universe—the sum total of all subatomic particles such as electrons. That figure is a 1 followed by eighty-one zeroes. Even if you counted electrons at the rate of a trillion a second starting when the present universe was born, you couldn’t come close to tallying them. Yet, despite such vastness, any concept involving things has a limit, since the total mass of the universe is reasonably well known and is finite. If all the atomic nuclei of the universe, plus every electron, could be compressed so that no space remained between them, the entire universe would be easily contained within our solar system. Squeeze out the emptiness within and between each atom, and you’re left with a ball weighing 10 followed by fifty zeros tons. Placed in the sun’s present position, its surface wouldn’t quite stretch to the planet Jupiter.
In other words, the actual material in the universe is infinitesimally tiny when compared with the size or vastness of the cosmos. If the universe were a cube ten miles wide, ten miles long, and ten miles high, all the mass it contained, including even the mysterious dark matter, would be as a single grain of sand.
In the days before calculators, Sir Isaac Newton worked out many methods for dealing with large numbers. One recurring problem was multiplying binomials by themselves several times, for example, (x + y)4. Newton reduced the problem to adding up a series of smaller multiplication products, such as x4 + x3y + x2y2 + xy3 + y4, but each of these terms needed to be multipled by a coefficient, and this is the interesting part. Newton determined that the coefficients would look like this:
The denominators are factorials (3! = 1 × 2 × 3) increasing step by step to the exponent factorial, while the numerators are factorials counting backward from the exponent. When both are factorials of the whole exponent, they cancel each other out. For the exponent 4, the coefficients are 1 + 4 + 6 + 4 + 1. Hence (x + y)4 = x4 + 4x3y + 6x2y2 + 4xy3 + y4.
Later, Blaise Pascal put these coefficients in a pyramid and found another relationship. His pyramid is called Pascal’s triangle.
Each coefficient is the sum of the two numbers adjacent to it in the row above. If you have this diagram worked out to as high an exponent as you’ll ever use, it can save the work of doing all those factorials.
Another interesting relationship arises if we add up the numbers that lie diagonally in the triangle.
This new series of numbers is the Fibonacci series that turns up in golden ratios, spirals of seashells, the leafing of tree branches, pinecones, sunflower seed heads, and multiplying rabbits.
Since you’ve probably forgotten your high school algebra, we’ll define our terms:
BINOMIAL: A number equal to the sum or difference of two other numbers (e.g., x + y).
COEFFICIENT: The number by which another number is multiplied (e.g., the 2 in the product 2y).
DENOMINATOR: The number you divide by (e.g., the 2 in x/2).
NUMERATOR: The number to be divided (e.g., the x in x/2).
EXPONENT: The power to which a number is raised (e.g., the 3 in y3).
Here is how a googol looks in eight-point type:
10,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,-000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
On a typical computer you could type 4,536 zeros per page, so that it would take 221 pages of zeros to type out 10106 (1 followed by a million zeros). It would take 220,459 pages to type out a 1 followed by a trillion zeros. How many pages to type out the number represented by the word googolplex? Answer: 2.2 times 1096 pages, over a quadrillion times more pages than there are subatomic particles in the visible universe.
Yet even this, the number of subatomic particles in the entire universe, winds up being ten million trillion times less than a googol—a 1 followed by one hundred zeros—the largest existing number. (Unless you include the humorously contrived and almost infinitely larger googolplex, which is a 1 followed by a googol of zeros.) A googol can have meaning only in possibilities or permutations, since it far surpasses the number of possible things in the entire universe.
The reason possibilities will always involve far vaster numbers than things is this: To calculate permutations we have to use factorials. Let me explain. The number of ways four books can be arranged on a shelf is 4 factorial, meaning 4 times 3 times 2 times 1, or 24. (From this point on, let’s ignore that “times 1” part of the formula.) Add just one more book, and the possibilities become 5 × 4 × 3 × 2, or 120. With a mere ten books, it’s 10 factorial, or 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2, or 3,628,800! You see how the number of permutations gets out of hand very quickly. Now, what if the trillion cells of the human brain could each interact with any other? How many different connections would be possible? The answer is 1 trillion factorial, or 1,000,000,000,000 times 999,999,999,999 times 999,999,999,998 times 999,999,999,997…and on and on. This figure is almost infinitely greater than the number of subatomic particles in the universe, and some 10 billion times greater than a googol.
Now, what if each of the electrons in the universe could “communicate” with any of the others? How many possibilities then? You’d have to perform 1081 factorial. The answer would be inconceivably vaster than a googol (but much, much less than a googolplex).
The purpose of all this is to make the lowly light-year, and the number of stars in our galaxy (a paltry trillion), and the distances from any point A to any point B seem very easy and workable when expressed in light-years.
Yes, astronomical sums are manageable. Our materialistic minds can come to grips with any manner of things. But when it comes to the myriad possible routes to discovering the universe, let us not count the ways.