[HN Gopher] Third Base (2001)
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       Third Base (2001)
        
       Author : picafrost
       Score  : 30 points
       Date   : 2025-01-23 14:43 UTC (8 hours ago)
        
 (HTM) web link (www.americanscientist.org)
 (TXT) w3m dump (www.americanscientist.org)
        
       | chungy wrote:
       | One of my favorite parts in The Art of Computer Programming is
       | when Knuth goes into alternative numeric bases. He makes mention
       | there of "base 2i" as well, though he details it in a much older
       | paper: https://dl.acm.org/doi/10.1145/367177.367233
       | 
       | Base 2i is particularly fascinating for allowing a numeric system
       | that can represent all complex numbers without signs and without
       | addition.
        
         | madcaptenor wrote:
         | Another fun one is base phi = (1 + sqrt(5))/2 - see
         | https://en.wikipedia.org/wiki/Golden_ratio_base
        
       | andrewla wrote:
       | Almost all of these analyses of ternary hinge on the idea of
       | efficiency of representation, being the (number of possible
       | symbols) * (number of digits to represent).
       | 
       | But it seems way more useful to use the log of the number of
       | possible symbols, in which case all bases (except unary, which is
       | non-positional) have the exact same efficiency, and the argument
       | becomes moot. It becomes largely a question of representation in
       | reality, and binary wins on that front.
       | 
       | More exotic non-positional number systems seem way more
       | interesting -- Fibonacci bases or Gray bases. Even balanced
       | ternary if we have to break out of the binary paradigm.
        
         | variaga wrote:
         | >Almost all of these analyses of ternary hinge on the idea of
         | efficiency of representation, being the (number of possible
         | symbols) * (number of digits to represent). >But it seems way
         | more useful to use the log of the number of possible symbols
         | 
         | This always bothered me as well. Hardware cost does not scale
         | linearly with the number of possible states of each digit.
         | 
         | My personal theory is that the real hardware cost is less
         | related to the number of possible states/symbols, and more
         | related to the amount of effort required to reliably
         | distinguish one state from another. Taking a cue from
         | information theory, that implies the complexity per digit
         | should track the Eb/No (SNR per information bit) of the symbol
         | set.
        
         | madcaptenor wrote:
         | "Gray bases"? What do you have in mind? (I'm assuming you mean
         | as in Gray codes, but I'm having trouble making sense of that.)
        
           | andrewla wrote:
           | I can't seem to find a reference at the moment.
           | 
           | I think I saw a talk on this years ago. The idea is that the
           | circuit complexity of adders is high because of carry cascade
           | but since Gray codes only ever flip a single bit on
           | increments, then a Gray code counter can be made very simple
           | and have no glitch states. Might have showed up in the
           | context of asynchronous VLSI designs.
        
       | PaulHoule wrote:
       | I like balanced ternary a lot.
        
       | sitkack wrote:
       | The book from 1950 mentioned in the article
       | https://web.archive.org/web/20100505110621/https://bitsavers...
        
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