Assessing irrational irregularity.

Pi, the ratio of a circle's circumference to its diameter, is known as an irrational number because it can't be exactly expressed as a ratio of two whole numbers. It would take an infinite number of digits to write it out in full as a decimal or in binary form as a string of 1s and 0s. The square root of 2, the square root of 3, and the constant e (the base of the natural logarithms) fall into the same category.

The known digits of these numbers appear patternless. According to a recently developed method of assessing the randomness of a sequence of numbers, however, the digits of pi turn out to be more irregular than the digits of the other irrational numbers. "The results were very surprising," says Steven M. Pincus, a freelance mathematician in Guilford, Conn., who invented the technique. Pincus' method produces a measure of irregularity that he calls approximate entropy.

Pincus and Rudolf E. Kalman of the Swiss Federal Institute of Technology in Zurich describe the findings in the April 15 Proceedings of the National Academy of Sciences.

To measure the approximate entropy of the binary digits of pi, [square root of 2], [square root of 3], and e, Pincus determined how often each of eight blocks of three consecutive digits--000, 001, 010, 011, 100, 101, 110, and 111--comes up in the first 280,000 digits of the number. In the case of pi, for example, the most frequently occurring block is 000, which appears 35,035 times, and the least common block is 111, which appears 34,944 times. The maximum possible irregularity occurs when all eight blocks appear equally often.

For [square root of 3], the block 000 occurs most often (35,374 times) and 010 (34,615) least often. The greater divergence from exactly 35,000 occurrences means that the first 280,000 binary digits of [square root of 3] are further from maximum irregularity than the digits of pi. Putting the four irrationals in order, starting with the most irregular, gives pi, [square root of 2], e, and [square root of 3]. Curiously, the differences in irregularity among these four numbers are much less pronounced when their decimal digits are compared.

Whether such differences in the irregularity of irrationals have any implications for number theory remains an open question for mathematicians, Pincus says.

Because the approximate entropy method does not depend on any assumptions about the process involved in generating a sequence of numbers, it can be applied to biological and medical data and to physical measurements, such as the number of alpha particles emitted by a radioactive nucleus in specified time intervals, as readily as to the digits of irrational numbers.

Pincus has even looked at stock market performance, as measured by Standard and Poor's index of 500 stocks. His calculations show that fluctuations in the index's value are generally quite far from being completely irregular, or random. One striking exception occurred during the 2-week period immediately preceding the stock market crash of 1987, when the approximate entropy indicated nearly complete irregularity.

"What's relevant is the fact that [the approximate entropy] was remarkably different right before the crash," Pincus says. It served to flag the incipient collapse.