> Dear listmembers,
>
> 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 (here -
> proper fraction) ?
>
> Thanks in advance,
>
> Alexander Zenkin
I think the wording of this question reveals a modern thought-habit
that makes it impossible to answer; namely that numbers are thought of
in terms of their notations. The fact that the above two notations
represent the same number follows from the Eudoxan theory of proportion,
as developed in Euclid's Elements, but of course those notations are
not used there. The numbers came before the notations, so no "agreement"
was needed.
John Conway