Science: The Longest Root

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Since ancient times, mathematicians have been fascinated by the problem of determining the square root of 2 —that number which, when multiplied by itself, will equal 2. As early as 1750 B.C., the Babylonians computed a value that was accurate to five decimal places (1.41421). By 1967, researchers in England, working with a computer, had stretched the answer to 100,000 digits. Now a Columbia University mathematician has surpassed even that prodigious effort. In what may well be the lengthiest computation of a mathematical constant of all time, Jacques Dutka has calculated the square root of 2 to more than one million places.

Starting with a rough approximation of the root derived from the mathematically well-known Pell Equation, Dutka devised a special algorithm (mathematical procedure) that enabled a computer to refine that answer to an extraordinary degree. After 471 hours of computer time and billions upon billions of individual calculations, the electronic brain ticked off an answer that was correct to at least 1,000,082 digits.

Although some of his bemused colleagues jovially accused him of being a "numbers nut," Dutka is convinced that his 200 pages of tightly spaced computer printout, each containing 5,000 digits, has some practical value. The square root of 2 is what mathematicians call an irrational number, one that runs maddeningly on without any repetitive patterns or predictable sequence no matter how far it is carried out. Such numbers are also apparently completely random,* an important quality to mathematicians, who have contrived lengthy random numbers for use in computer studies of such chance phenomena as incidence of telephone usage, highway traffic patterns and even the lineup of shoppers in a supermarket. Dutka claims, however, that a naturally occurring random number, like the square root of 2, is better for those studies because there may be subtle, hard-to-detect biases in random numbers that are artificially generated.

Delighted by his success, Dutka is now eying more ambitious projects: calculating million-digit values for TT (3.14159 . . .), the ratio of the circumference of a circle to its diameter; and the mathematical constant e (2.71828 . . .), the base of natural logarithms and one of the most significant numbers in higher mathematics. Says Dutka: "After that, I can well afford to call it quits."