Trinary Notation
Hypomodal Base 3

Around 30 years ago I was reading one of those popular science-y magazines. One of the writers stated that the ideal set of weights (masses) for use on a balance scale should be based on binary relationships. That is, they should be related as 1:2:4:8, and so on. According to the author, this sort of set would make the most efficient use of materials, and of space because there would be no need for duplicate weights.

Well, I disagreed. It occurred to me that a superior set could be constructed with the weights related as 1:3:9:27. There would still be no need for duplicates, because each weight would do double duty as a negative mass. That is to say, by placing a weight beside the unknown quantity, it becomes, in effect, a negative value. This is simple algebra:

If X + 1 = 3, we know that X = 2 even though we have no weight of 2 units.

I then went on to devise a system of notation in which a dash would represent -1. This system is a form of base three, in regards to the values of the columns (ones column, threes column, nines column, etc.), but there is no need for 2. Only 0 and the positive and negative unities (1 and – )

It seemed possible that such a system might be used to design a somewhat more compact computer. For example, 16 lines in a binary computer can carry one of 65,536 different "words" at a time. In a trinary computer, the same 12 lines have 43,046,721 possible values.

16 lines, binary:        65,536
16 lines, trinary: 43,046,721

Therefore, a 16-line address buss would access 43,046,721 locations rather than a paltry 65,536.

One quibble, though. When I put this notation together 30 years ago, it didn't occur to me that the notation would be used by people (only by computers) so I didn't consider that subtraction would be a problem. For example, how would we read 1– - – ? That is (2) - (-1).

      Trinary      
    27
9
3
1
27   1 0 0 0
26   1 0 0
25   1 0 1
24   1 0 0
23   1 0
22   1 1 1
21   1 1 0
20   1 1
19   1 0 1
18   1 0 0
17   1 0
16   1 1
15   1 0
14   1
 
    27
9
3
1
13     1 1 1
12     1 1 0
11     1 1
10     1 0 1
9     1 0 0
8     1 0
7     1 1
6     1 0
5     1
4       1 1
3       1 0
2       1
1         1
0         0
 
    27
9
3
1
-13    
-12     0
-11     1
-10     0
-9     0 0
-8     0 1
-7     1
-6     1 0
-5     1 1
-4      
-3       0
-2       1
-1        
0         0
 
    27
9
3
1
-27   0 0 0
-26   0 0 1
-25   0 1
-24   0 1 0
-23   0 1 1
-22   1
-21   1 0
-20   1 1
-19   1 0
-18   1 0 0
-17   1 0 1
-16   1 1
-15   1 1 0
-14   1 1 1

Hypomodal notations are possible only for odd-numbered bases, because every positive value requires a negative, while 0 is alone. The next higher system would be Hypomodal base 5, or Quintinary, with the digits ||, 1, 0, – , and =.

 
25
5
1  
25
5
1  
25
5
1
1       1 11     || 1 21   1 1
2       || 12     || || 22   1 ||
3     1 = 13   1 = = 23   1 0 =
4     1 14   1 = 24   1 0
5     1 0 15   1 = 0 25   1 0 0
6     1 1 16   1 = 1 26   1 0 1
7     1 || 17   1 = || 27   1 0 ||
8     || = 18   1 = 28   1 1 =
9     || 19   1 29   1 1
10     || 0 20   1 0 30   1 1 0