Volume 11, No. 1—February 2001  
Table of contents for this issue  
 
HYPOTHESES

Larger Than Life
Can numbers become too big?
By Jim Holt

HOW BIG IS A KILLION? THAT, in case you didn't know, is a legendary number so enormous that the mere apprehension of it is fatal to humans. It stands to reason that such a number exists. Humans are finite; numbers go on forever; eventually numbers must become lethal in their sheer immensity.

But how do we come to know truly huge numbers? One way is by naming them. The first really impressive number that most of us encounter as children is 1 million. Now, a million is no killion. It is easily domesticated, as Maria Drew, a fifty-one-year-old housewife from Waterloo, Iowa, showed some years ago. After her son's teacher assured him that counting to 1 million was impossible, Mrs. Drew typed out all the numbers from 1 to 1 million. This took her five years and 2,473 sheets of typing paper.

Far larger numbers, however, can easily be named. In English, we can get up to a primo-vigesimo-centillion (10³⁶⁶) or even a milli-millillion (10^3000003). These two numbers greatly exceed a googol (1 followed by a 100 zeros), a neologism famously coined by the nine-year-old nephew of the mathematician Edward Kasner. Yet the googol system of nomenclature can be effortlessly extended by the term "googolplex" (1 followed by a googol of zeros, or 10^googol), which leapfrogs far beyond a milli-millillion. Outside the decimal system, the largest number with an independent name is the Buddhist asankhyeya (on the order of 10¹⁴⁰).

These numbers seem big by any standard. But if we can manipulate a number—take it apart—then it can't really be regarded as big. Manipulability is relative to two things: the power of our computers, and the efficiency of our algorithms. Taking a number apart means analyzing it into its prime factors; to take apart the number 6, for example, you write it as the product of 2 and 3, both of which are prime and hence not further reducible. In the past three decades, number theorists have become quite deft at this. Drawing on a great deal of deep mathematics, like the theory of elliptic curves, they have invented ingenious algorithms for factoring large numbers. (Such algorithms, once beautifully useless, are now critical in code breaking.) Numbers with as many as 160 digits are today routinely dismantled. If all the computer power in the world were marshaled a decade or two from now, even a number with 200 digits—one the size of a googol squared—could conceivably be taken to pieces.

Perhaps a better way to get one's hands around a big number is by counting. Archimedes, in "The Sand Reckoner," used a clever series of multiplications to count the grains of sand on all the beaches of Sicily, then extended his calculation to estimate the number of grains of sand required to fill the entire universe, which came to 10⁶³. (Indian Jain mathematicians arrived at their own considerable magnitudes by thinking of mustard seeds.) However, neither that figure nor the estimated number of elementary particles in the universe (around 10⁸⁰) inspires deadly awe.

The problem is that we're merely counting objects. Suppose we instead try counting possibilities, which are far more numerous. The number of possible chess moves is 10 raised to the 10⁵⁰ power—not bad, but still a long chalk from a googolplex. So let's turn the entire cosmos into a chessboard, with elementary particles as the "pieces," any switch in the positions of two particles a "move." Then the number of possible games becomes something like 10 to the 10 to the 10 to the 34th. Curiously, this tally of cosmic chess games is very close to another gargantuan number that was discovered by a pure mathematician in 1933: Skewes's number. A figure having to do with the way primes are distributed in the number sequence, Skewes's number was deemed by the English mathematician G.H. Hardy "the largest number which has ever served any definite purpose in mathematics."

Since then, combinatorial mathematicians—those concerned with counting possible arrangements of things—have apprehended vastly larger numbers. They have invented "exploding" functions whose mind-boggling values cannot be written down using the usual exponential notation—even if the entire universe were turned to ink and paper. Yet the mythical killion has so far eluded them.

Perhaps it always will. Sub specie aeternitatis, the most powerful intellects among us might be no different from jackdaws, which, ethologists tell us, cannot keep track of quantities past four; or from the Kalahari bushmen, who, anthropologists used to claim, could not describe quantities greater than five. As long as your cognitive grasp is finite, going for big numbers is a mug's game. Take the largest number that has been defined to date—call it N. If an infinite being were to reach into the number sequence at random and pull something out, the probability of this randomly chosen number being bigger than N would logically be 100 percent. This follows from the fact that no matter how big N is, there are only finitely many numbers behind it and infinitely many ahead.

Might we be able to apprehend ever-larger numbers in the afterlife? Not on current evidence. After being transported to heaven, Mohammed said: "I saw there an angel, the most gigantic of all created beings. It had 70,000 heads, each head had 70,000 faces, each face had 70,000 mouths, each mouth had 70,000 tongues, and each tongue spoke 70,000 languages; all were employed in singing God's praises." That comes to only 1.6807 septillion languages.

JIM HOLT

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