> Does anybody know who, when, and where (citations and/or
> references are desirable) was FIRST introduced, in an
> explicit form, the AGREEMENT that the two notations, say
> in the binary system, such as:
>
> (1) 0, a1 a2 . . . ak 1 0 0 0 . . . ,
> and
> (2) 0, a1 a2 . . . ak 0 1 1 1 . . . ,
>
> where all ai (i = 1, 2, ..., k) are "0" or "1",
>
> are two DIFFERENT representations of THE SAME rational number?
I do not have the answer to your question, but I have discovered a few
interesting things about repeating decimal (or binary) fractions.
Zeno's "Dichotomy" Paradox can be expressed as follows: (using binary
fractions) Achilles cannot reach the end of a field one stadium long because
first he must reach the halfway point (0.1 stadium), then the halfway point
of the remaining distance (0.11 stadium), and so on.
One of SEVERAL possible ways of interpreting the Dichotomy Paradox is that
Zeno is arguing that the number 0.11111... does NOT equal 1.0.
(John Conway writes (99-07-20 06:46:24 EDT) "The fact that the above two
notations represent the same number follows from the Eudoxan theory of
proportion, as developed in Euclid's Elements ". Does anyone know if
Eudoxus attempted to refute Zeno?)
The Borwein brothers (J. M. and P. B. Borwein "The Arithmetic-Geometric Mean
and Fast Computation of Elementary Functions" SIAM Review Vol. 26 no. 3 July
1984 page 353) found it necessary to write "...a-sub-n and L will 'agree'
through the first O(2 to the n) digits (provided we adopt the convention that
.9999...9 and 1.000...0 agree through the required number of digits)."
Any rational number can be represented by a repeating decimal fraction (e.g.
1/3 is .333333... and 1/2 can be represented by either .500000... or .4999999...)
Now take an arbitrary real number from (0, 1). What is the probability that
its decimal expansion will repeat with a period of one (as does 1/30 =
.0333333...) Start with any decimal digit. The odds that the next digit of
the arbitrary number will be the same is 1/10. The odds that the next two
will be the same is 1/10x1/10. The odds that all remaining digits will be
the same is the limit as n goes to infinity of (1/10) to the n-th, which is
zero. This is true no matter at which decimal position you start. Hence the
probability is zero that this arbitrary real number has a repeating decimal
expansion of length one. Similarly for length 2, 3, etc. Hence the
probability is zero that an arbitrary real number is rational, from which we
can deduce that the real numbers are uncountable.
(A simpler version of the above proof is that while (0, 1) has Lebesgue
measure 1, the set of all rational numbers in (0, 1) has measure zero.
However, measure theory presupposes Cantorian set theory, so this latter
proof may be circular.)
James A. Landau