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Officials last week revealed that the U.S. contribution to ITER could cost $3.9 billion by 2034—roughly four times the...
An experimental hepatitis B drug that looked safe in animal trials tragically killed five of 15 patients in 1993. Now,...
Using the two high-quality genomes that exist for Neandertals and Denisovans, researchers find clues to gene activity...
A new report from the Intergovernmental Panel on Climate Change (IPCC) concludes that humanity has done little to slow...
Astronomers have discovered an Earth-sized planet in the habitable zone of a red dwarf—a star cooler than the sun—500...
Three years ago, Jennifer Francis of Rutgers University proposed that a warming Arctic was altering the behavior of the...
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Random Numbers Behaving Too Orderly?
15 April 1997 8:00 pm
Finding a random sequence is as easy as pi--or is it? Mathematicians often depend on irrational numbers like , e (the basis of natural logarithms), and 2 to give them an unpredictable stream of digits. But a paper in today's Proceedings of the National Academy of Sciences is upsetting the conventional wisdom about randomness by showing that some of these numbers are far more predictable than expected. That could sound alarms in fields that depend on random numbers, such as cryptography and the design of certain experiments.
The finding comes from a new technique for quantifying randomness. Any mathematician can tell you that 01101100 is more random than 01010101, but until last year none could tell you just how much more random. Then two researchers--Steve Pincus, a free-lance mathematician based in Guilford, Connecticut, and Burton Singer, a mathematician and demographer at Princeton University--created a method for determining a sequence's "entropy," or disorder.
Pincus and Singer's method builds on the observation that all possible digits are represented about equally in a perfectly random stream of digits. The binary sequences 01101100 and 01010101--each with four 1's and four O's--both pass this test, for example. Pincus and Singer also noted that when the digits are taken two at a time, a random sequence should have equal numbers of all possible pairs: 00, 01, 10, and 11, in this case. The sequence 01010101 fails this test miserably; there are no 00s or 11s at all. The same reasoning can be extended to larger groups of digits, taking them three at a time, four at a time, and so on. By comparing the actual frequency of groups of digits to their expected frequency, Pincus and Singer come up with the "approximate entropy" (ApEn) of the sequence--a measure of its randomness.
"[ApEn] is one of those tools that makes you say, `Hey, that's a good one!' and you put it in your tool kit," says Max Woodbury, a mathematician at Duke University. But it's a tool that is raising more questions than it answers, as the Proceedings paper shows. In it, Pincus and Rudolf Kalman, a mathematician at the Swiss Federal Institute of Technology in Zurich, Switzerland, calculated the ApEn of various irrational numbers. They included both algebraic numbers, which are solutions of polynomials with finite numbers of terms--3, for example--and "transcendental" numbers like and e, which are not algebraic. Transcendental numbers are, in a sense, more complicated than algebraic numbers, so Pincus expected that when they are written out in decimal form, their strings of digits would be more random. He was wrong.
" is the most irregular," says Pincus. "But I was very surprised that e was not next in line." In fact, 2, an algebraic number, was more random than e, a transcendental number. Mathematicians are scratching their heads over this. Says Singer, "It raises serious conceptual problems about the idea of randomness."