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Pi a la Mode.

Mathematicians tackle the seeming randomness of pi's digits

Memorizing the digits of pi--the ratio of a circle's circumference to its diameter--presents a hefty challenge to anyone undertaking that quixotic exercise. Starting with 3.14159265, the decimal digits of pi run on forever, and there is no discernible pattern to ease the task.



From the computer trial, he also obtained statistical evidence suggesting that every digit--from 0 to 9--occurs equally often. Tossing a fair 10-sided die would generate a sequence of random numbers having the same property.


Now, there's a shard of hope that mathematicians may yet lay bare the apparent randomness of pi's infinite digits. Bailey and Richard E. Crandall of Reed College in Portland, Ore., have identified a potential link between two disparate mathematical fields--number theory and chaotic dynamics--that they suspect could lead to a proof that every digit occurs with the same frequency when pi is written out ad infinitum. Bailey and Crandall report their findings in the June EXPERIMENTAL MATHEMATICS. The characteristic would hold whether pi's in decimal form or in some other base.




Pi can be calculated to as many decimal places as desired by formulas that are sums of infinitely many terms, such as [Pi]= 4/1 - 4/3 + 4/5 - 4/7 + 4/9 -.... The larger the number of terms, the more decimal places are revealed.



Bailey and other researchers amassed statistical evidence in the 1980s supporting the notion that pi is normal. For example, one would expect the digit 7 to appear 1 million times among the first 10 million decimal digits of pi. It actually occurs 1,000,207 times--close to the expected value. Each of the other digits also turns up with approximately the same frequency, showing no significant departure from predictions.

In 1999, Yasumasa Kanada and his colleagues at the University of Tokyo computed pi to a record 206 billion decimal digits (SN: 10/16/99, p. 255). Their analysis shows that 7 appears 19,999,967,594 times among the first 200 billion decimal digits.


A number is said to be "absolutely normal" if its digits are normal not only to base 10 but also to every integer base greater than or equal to 2. In base 2, for example, [Pi] = 11.0010010000.... If pi is normal to base 2, the digits 1 and 0 would appear equally often.











Together, Bailey and Crandall established that if they could prove that the numbers of particular chaotic sequences (see box) are evenly distributed between 0 and 1, the normality to base 2 of pi and log 2 would follow automatically.

"What we have done is to translate a heretofore unapproachable problem to a more tractable question in the field of chaotic processes," Bailey says. "At the very least, we have shown why the [binary] digits of pi and log 2 appear to be random."



Some mathematicians are pessimistic about whether the Bailey-Crandall approach will eventually lead to a normality proof for pi and other fundamental constants. The pair's hypothesis about chaotic sequences may itself be too hard to prove, says Jeffrey C. Lagarias of AT&T Labs-Research at Florham Park, N.J.



A flavor of pi's normality


 1/[k2.sup.k] , starting at k = 1

This value works out to 0.6931471805599453....


[x.sub.n] = ([2x.sub.n-1] + 1/n)mod 1

Starting with [x.sub.0] = 0, each iteration, n, uses the previous result, [x.sub.n-1], as the input for calculating the next number, [x.sub.n]. The term "mod 1" is an instruction to use only the fractional remainder of each iteration's result as input for the next iteration. In other words, no input is ever larger than 1.

The process generates the following sequence: [x.sub.0] = 0, [x.sub.1] = 0, [x.sub.2] =1/2, [x.sub.3] = 1/3, [x.sub.4] = 11/12, [x.sub.5] = 1/30, [x.sub.6] = 7/30, [x.sub.7] = 64/105, [x.sub.8] = 289/840.... If it could be proved that the erratically fluctuating numbers [x.sub.n] are evenly distributed between 0 and 1, log 2 would be deemed normal to base 2.

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Title Annotation:delving into the apparent random nature of pi's digits
Publication:Science News
Geographic Code:1U9CA
Date:Sep 1, 2001
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