]]]]]]]]]]]]]]]]] PI IS AN IRRATIONAL NUMBER [[[[[[[[[[[[[[[ July 7, 1988 Dear Dr. Beckmann: Because of your interest in pi, I'll ask you this question. The value of pi has been calculated to 134 million places, it said in yesterday's paper. Of course, the mysterious number is one which has been generated in the numerical system which we have arbi- trarily based on a decimal system. What would pi be like in a different numerical system, such as one with a base of twelve? Might it repeat at some point?" C.M.H., Cranford, N.J. * * * Dear Mr. H.: If the number pi were rational, meaning capable of being ex- pressed as a fraction of two integers, then it could be written as a geometric series regardless of the number base. For example, the deci- mal number 0.123123123123... equals 123/1000 + 123/1000**2 + 123/1000**3 + ... (where "**" stands for "to the power" so it can be typed in the line). That is a geometric series, which can be summed by high-school algebra and results in 123/999. In ANY numbering system, the number 0.abcabcabcabc, where the letters stand for digits in any numbering system, can similarly be expressed as abc/B**3 + abc/(B**3)**2 + abc/(B**3)**3 + ... where B is the number base (12 in the duodecimal system). This is again a geometric series resulting in the rational number abc/(1 - (1/B)**3). (The power of 3, by the way, occurs because the sample number repeated in groups of 3; if it had been 0.abcdef abcdef abcdef, the power would have been 5.) But as proved by Adrien Marie Legendre in 1794, pi is NOT ratio- nal; that is, it is irrational or incapable of being expressed as a fraction of two integers, and hence not expressible as a geometric series, and hence cannot have repeating groups in ANY number system. Pi is not merely irrational, but also transcendental, that is, it cannot be the root of an algebraic equation with a finite number of terms. However, to answer your question, only its irrationality is needed. A number that is ONLY irrational, but not transcendental, such as the squared root of 2, can likewise not have repeating groups of digits in ANY numbering system. For more details, see my "History of Pi,", Golem Press, Box 1342, Boulder, CO 80306, $12.95. What would pi look like in a system based on 12? I am too lazy to do the arithmetic, but it can quickly be done to the base 16, because hexadecimal calculators are readily available to every computer pro- gramer. It starts off as 3.243F..., where the point is a "hexadecimal" point and F stands for "fifteen." Cordially, P.B.
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