Tumblers: A Flexible Numbering 
System Imagine a row of books on a 
shelf, numbered  1, 2, 3, ...  
Chapters in each book could be 
similarly numbered, so that chapter 
five in volume three would be 
represented as 3.5.  This could be 
extended arbitrarily, so that 
3.5.10.6 represents volume three, 
chapter five, section ten, paragraph 
six. If you had volume three open to 
the place mentioned above, and were 
told to move forward two volumes, 
sixteen chapters, and three sections, 
you would doubtless put volume three 
back, take down volume five, then 
count forward to sixteen chapters and 
three sections. This may be 
represented as a non-commutative 
arithmetic operation: 
                3. 5.10. 6 
           + 2.16. 3 
                --------- 
                5.16. 3 

This arithmetic gives different roles 
to the two numbers:  the first 
specifies where something is, the 
second specifies an offset, a 
distance to move forward.  Only the 
first digit of the original location 
matters, because in counting ahead by 
volumes, the intitial paragraph, 
section, and chapter make no 
difference.  A similar example, but 
with an offset remaining within the 
first volume, is: 
                25. 6.46.93 
           +  0. 0. 3. 1. 21 
                --------------- 
                25. 6.49. 1. 21 The 
following rule describes this sort of 
offset procedure on this sort of 
numbering system: Evaluating from the 
most to the least significant 
entries, the elements of the final 
position are equal to the 
corresponding elements of the initial 
position as long as the corresponding 
elements of the offset are zero. The 
first non-zero offset element is 
added to the corresponding element of 
the initial position. The remaining 
elements of the final position are 
equal to the remaining elements of 
the offset. The numbers in this 
system are called tumblers, and this 
addition property is the foundation 
of tumbler arithmetic.  This sort of 
numbering system is widely used in 
military documents, to permit 
insertion of material at any point 
without renumbering subsequent parts. 
Note that if one does not identify 
particular parts of a tumbler with 
any particular level of text, then 
one can insert a book's worth of 
material (with appropriately numbered 
chapters, etc.) between any two 
characters.  This merely involves 
appending to the number for the first 
character a "." followed by a "1" 
(for the first item inserted between 
those two characters). The body of 
the book could be numbered by 
appending further tumbler elements. 
