Third Base (2001)
7 comments
·January 23, 2025chungy
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
Another fun one is base phi = (1 + sqrt(5))/2 - see https://en.wikipedia.org/wiki/Golden_ratio_base
andrewla
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
>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
"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.)
PaulHoule
I like balanced ternary a lot.
The book from 1950 mentioned in the article https://web.archive.org/web/20100505110621/https://bitsavers...