```]]]]]]]]]]]]]]]]]      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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