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From: dupuy@amsterdam.UUCP
Newsgroups: comp.arch
Subject: number bases (was Question: on-chip or off-chip MMU?)
Message-ID: <4656@columbia.UUCP>
Date: Wed, 3-Jun-87 14:35:04 EDT
Article-I.D.: columbia.4656
Posted: Wed Jun  3 14:35:04 1987
Date-Received: Sat, 6-Jun-87 03:58:39 EDT
References: <5635@shemp.UCLA.EDU> <3460004@hpsrla.HP.COM>
Sender: nobody@columbia.UUCP
Reply-To: dupuy@amsterdam.columbia.edu (Alexander Dupuy)
Followup-To: comp.arch
Organization: Columbia University Computer Science Dept.
Lines: 30
Summary: 3 is closer to e than 2

In article <3460004@hpsrla.HP.COM> roger@hpsrla.HP.COM (Roger Petersen) writes:
>
>Actually, if you care to get theoretical, I believe it can be shown that
>the ideal number base for a computer is e (2.7182818...).  Interesting,
>perhaps, but not very practical.

I once worked this out independently; it's been a while, but I recall that the
information ``efficiency'' for base x is $\ln X / X$ and is maximal at X = e.
This assumes that the hardware cost of providing n-valued symbols is n/2 times
the cost of bivalued symbols, which may or may not hold in real electronic
systems.

Even though there's no way to use a transcendental base in hardware, this does
argue in favor of ternary logic, since $\ln 3 / 3 \approx .3662$ is only
slightly less than $\ln e / e \approx .3679$, while $\ln 2 / 2 \approx .3466$.
It can be said however, that ternary logic would be wasteful for boolean
operations, which are heavily used in most systems (e.g. branches).

There was an interesting Martin Gardner column in Scientific American in the
late 60's or early 70's on ternary logic.  For example, there are several ways
of representing signed ternary numbers.  The most interesting is using the
symbols -1, 0 and +1, rather than 0, 1, and 2.  This avoids wasting a ternary
symbol on the (bivalued) sign bit.  Another possibility is three's complement,
although it makes the sign harder to determine.

@alex

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