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Counting the Personalities of Numbers

March 16, 2000|K.C. COLE

What could be more dry than a statistic? More indifferent than a number?

To be treated like a number, in common parlance, is to become an entirely replaceable part. An object lesson in depersonalization.

It makes you wonder what mathematician Richard Friedberg could have had in mind when he wrote, in his book, "An Adventurer's Guide to Number Theory":

"Two is solid and tingly, like the Liberty Bell. . . . Eight is rough and hard like a stone, and 10 is smooth like a pebble on the beach. Nine--seems ready not only to ring but to shatter and burst like a fruit."

Friedberg's numerical affections may come as a surprise to laypeople, but not to mathematicians, who know there's more to numbers than simple counting. Numbers don't just line up like well-ordered rows of dominoes, keeping track of quantities. They inspire love, hate, fear, amusement, friendship.

Sometimes, numbers are to die for.

Consider, for example, the oft-told tale of the discovery of so-called irrational numbers. Irrational numbers cannot be expressed as the ratio of two whole numbers--which is to say, in some sense, that they can't be expressed exactly at all. Numbers like pi (3.14159--ad infinitum) and the square root of 2 (1.41421--ad infinitum) just trail on endlessly.

Such fuzzy indefiniteness didn't sit well with Pythagoras, the ancient Greek geometer who made a religion out of the seeming perfection of numbers. Pythagoras was so upset by the discovery of this obvious flaw, the story goes, that his disciples were pledged to secrecy. A rebel intent on taking the word out to the larger world was drowned at sea (by the gods, or his jealous colleagues--depending on who's telling the tale).

The discovery (or invention, if you will) of the number zero was greeted with similar horror. Zero was regarded as the creation of the devil, and for a time, edicts were issued in Florence that forbade the use of the number and the new "Arabic" system that ushered it in.

"And, as usual, prohibition did not succeed in abolishing, but merely served to spread bootlegging," says Tobias Dantzig in "Number, the Language of Science." He notes that despite the edict, 13th century merchants continued to use zero and its fellow numerals as a kind of secret code.


Others merely scoffed at zero, putting down the new digit as a puffed-up pretender to the status of number. Sniffed one unidentified 15th century source: "Just as the rag doll wanted to be an eagle, the donkey a lion and the monkey a queen, the cifra [zero] put on airs and pretended to be a digit."

And zero's alter ego, infinity, has been irritating mathematicians and physicists ever since it was discovered. Infinity, being without end, was widely regarded as a direct encroachment on holy territory. "The last number," writes Dantzig, "belonged to God."

Some numbers even had what amounted to moral qualities. The number 1 was reasonable; 2 opinionated. Followers of Pythagoras prayed to the number 4 (which stood for justice).

These days, mathematicians don't pray to numbers, but they do put them on the couch to analyze their personalities. "The number theorist strikes up a closer acquaintance and soon learns intimate details," writes Friedberg, including "likes and dislikes." Numbers, like people, can be complex, perfect, imaginary, amicable, surreal, transcendental, excessive, square, prime.

A perfect number, for example, is one that is equal to the sum of its divisors--like 6. (Six can be divided by 1, 2 and 3, and 1 plus 2 plus 3 add up to 6.) Excessive numbers, like 14, add up to more than the sum of their divisors. Amicable numbers are pairs in which each is the sum of the divisors of the other like 220 and 284. The number 64 is both a square (of 8) and a cube (of 4).

These are things you might never know, Friedberg points out, if you merely regard numbers as a kind of alphabet for counting.

So forget all that mystical stuff about the number 666 belonging to the devil. Next time you want to know about transcendental qualities of numbers, don't go to a psychic. Ask a number theorist. And don't worry if you disagree with Friedman's characterization of the number 7 as "dark and full of liquid, like oil when it oozes from the ground." Or 5 as "pale but round like a ball."

The good mathematician leaves room for personal interpretation. "Perhaps you see 7 as a pincushion," he says, "and 5 as a bright spot of light."

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