[continued from previous message]   
      
   equivalent of 1960 becomes, "Oddy-dye one group, totter-pohl."   
      
   Let us now consider that we have achieved a satisfactory pronouncing   
   system for binary numbers and take up the question of whether similar   
   principles can lead us to a more compact and recognizable method of   
   graphically representing these quantities. The numbing impact on the   
   senses of even a fairly large number expressed in binary terms is   
   well known. Although conventions for setting off groups and stating   
   approximations, as outlined above, may be helpful, there would appear   
   to be intrinsic opportunities for error in writing precise   
   measurements, for example, in binary notation: One of the most common   
   writing errors in numbering arises from transposing digits, and as   
   binary numbers have in general about three times as many digits as   
   their decimal equivalents, they can be assumed to furnish three times   
   as many opportunities for error.   
      
   We have previously chosen to write binary numbers in double groups of   
   three digits each. As each group represents eight possible cases we   
   would, in the manner of the Bell workers, represent each group by its   
   decimal equivalent, so that decimal 1960, which we have given as   
   binary 011 110, 101 000, could be written in some such fashion as   
   B36,50. Yet again we may hope to find advantages in a uniquely   
   designed system for binary numbers.   
      
   The author has experimented with a system which has little in common   
   with notations intended for human use but some resemblance to records   
   prepared for, or by, machines. For example, a rather rudimentary   
   "reading" machine intended to grade school examinations or similar   
   marked papers does so by taking note not of the symbol written but of   
   its position on the paper--check marks, blacked-in squares, etc. An   
   abacus, too, denotes quantities by in effect recording the position   
   of the space between the beads at the top of the wire and the beads   
   at the bottom.   
      
   Indeed, one such system might be designed after the model of the   
   abacus, requiring the use of paper pre-printed with a design like   
   this:   
      
       * * * * * * * * * * * *   
       * * * * * * * * * * * *   
       * * * * * * * * * * * *   
       * * * * * * * * * * * *   
       * * * * * * * * * * * *   
       * * * * * * * * * * * *   
       * * * * * * * * * * * *   
       * * * * * * * * * * * *   
      
   Each vertical column of eight dots can represent one three-digit   
   group. To represent the binary equivalent of 1960, one would circle   
   the fourth dot in the first column, the seventh dot in the second,   
   the sixth dot in the third and the first dot in the fourth. The first   
   dot in each column would represent 000, the second 001 and so on.   
      
   If we permit the use of preprinted paper, a more compact design might   
   be a series of drawings of two squares, one above the other, thus:   
      
       __     __     __     __     __   
      |  |   |  |   |  |   |  |   |  |   
      `--'   `--'   `--'   `--'   `--'   
      
       __     __     __     __     __   
      |  |   |  |   |  |   |  |   |  |   
      `--'   `--'   `--'   `--'   `--'   
      
   Each pair of two squares is made up of eight lines. By assigning to   
   each side of a square a value from 000 through 111, a simple check   
   mark or dot would show the value for that group:   
      
                 010 (2)   
                +-------+   
                |       |   
        001 (1) |       | 011 (3)   
                |       |   
                +-------+   
                 100 (4)   
      
                 110 (6)   
                +-------+   
                |       |   
        101 (5) |       | 111 (7)   
                |       |   
                +-------+   
                 000 (0)   
      
   A vertical pair of squares in which the left-hand side of the lower   
   square was checked would then show that the value 101, or 5, was   
   recorded for that group.   
      
   Some readers will have remarked that the orderly system of   
   representing the groups from 000 to 111 has been abandoned, the 000   
   group coming last in this representation. The reason for this is that   
   we have a perfectly adequate symbol for a zero quantity already--that   
   is, 0, which is unambiguous in numbering to almost all bases. (The   
   sole exception is the monadic system, to the base 1. As this does not   
   permit positional notation it does not require any zero.) Using the 0   
   symbol here does not, of course, fit in with our plan of checking off   
   lines on squares, but this is itself only a way-station in the   
   attempt to find a suitable notation which will not require the use of   
   preprinted paper.   
      
   We might find such a notation by drawing the squares ourselves as   
   needed, or at least drawing such parts of them as are required. For   
   001 we would have to show only the first line of the first square.   
   For 010 we need two lines, the left-hand side and the top. For 011 we   
   required three lines. At this point we begin to find the drawing of   
   lines laborious and reflect that, as it is only the last line drawn   
   which conveys the necessary information, we may be able to find a way   
   of omitting the earlier ones. Can we do this?   
      
   We can if, for example, we explode the square and give it a clockwise   
   indicated motion by means of arrowheads:   
      
                A------->   
                |       |   
                |       |   
                |       |   
                <-------V   
      
   Each arrow has a unique meaning. A is always 001; < is always 100. In   
   order to distinguish the arrows of the first and second squares, we   
   can run a short line through the arrows of the lower square; thus V   
   is 111, not 011.  Having come this far, we discover that we are still   
   carrying excess baggage; the shaft of the arrow is superfluous, the   
   head alone will give us all the information we need. We are then able   
   to draw up this table:   
      
       Decimal  Binary, digital  Binary, graphic   
       -------  ---------------  ---------------   
             0              000                0   
             1              001                A   
             2              010                >   
             3              011                V   
             4              100                <   
             5              101               A+   
             6              110               >+   
             7              111               V+   
      
   and our binary equivalent for 1960 may be written as V>+0.   
      
   The writer does not suggest that in this discussion all problems have   
   been solved and binary notation has been given all the flexibility   
   and compactness of the decimal system. Even if this were so, the   
   decimal system possesses many useful devices which have no parallel   
   here--the distinction between cardinal and ordinal numbers, accepted   
   conventions for pronouncing fractions, etc. It seems probable,   
   however, that many nuisances found in conversion between binary and   
   decimal systems can be alleviated by the application of principles   
   such as these, and that, with relatively little difficulty,   
   additional gains can be made--for example, by adapting existing   
   computers to print and scan binary numbers in a graphic   
   representation similar to the above. Indeed, it further seems only a   
   short step to the spoken dictation to the computer of binary numbers   
   and instructions, and spoken answers in return if desired; but as the   
   author wishes not to confuse his present contribution with his prior   
   publications in the field of science fiction he will defer such   
   prospects to the examination of more academic minds.   
      
         Decimal  Binary, digital  Binary, graphic  Pronounced   
         -------  ---------------  ---------------  ----------   
               0              000                0  pohl   
               1              001                A  poot   
      
   [continued in next message]   
      
   --- SoupGate-DOS v1.05   
    * Origin: you cannot sedate... all the things you hate (1:229/2)