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From: hopkins@burdvax.UUCP (Bill Hopkins)
Newsgroups: net.arch
Subject: Re: Complement Arithmetic
Message-ID: <1425@burdvax.UUCP>
Date: Thu, 26-Jan-84 11:53:53 EST
Article-I.D.: burdvax.1425
Posted: Thu Jan 26 11:53:53 1984
Date-Received: Sat, 28-Jan-84 05:47:01 EST
Organization: System Development Corporation, Paoli PA
Lines: 39

I can recall three main arguments for two's complement arithmetic:

-  There is a unique representation for each integer in the range
represented, [-(2**(n-1))..2**(n-1)-1]. One's complement representation has
the infamous and mathematically troublesome "positive and negative zero"
problem, which has prematurely aged too many compiler writers.

-  The arithmetic is the same as for unsigned integers; only the
interpretation of carry and overflow change.  In one's complement the two
require separate ALU operations.

-  Multiple precision addition and subtraction are simply done by propagating
carries (borrows) through status bits.  It's virtually impossible to simulate
a 32-bit integer operation with a 16-bit ALU in one's complement (consider
the end-around carry).

My only argument in favor on one's complement is the symmetry of the range,
which avoids the possibility of overflow on complementing the most negative
number.  (There may be others; let's hear from the one's complement freaks
out there.)

(N.b.: we're only talking integers (really fixed point) because that's how
n's-complement arithmetic is defined; floating point representations are made
up out of an integer exponent and a fixed point mantissa.  The arguments for
integers apply to these unchanged.  Similarly, multiplication and division
are built up out of addition and subtraction.)

In the folklore department:  The CDC 6000/Cyber-whatever series designed by
Seymour Cray use one's complement arithmetic, presumably because he could
make it go faster back in the early 60's.  The Cray-1 uses two's complement;
the folk tale is the Seymour finally figured out how to do it fast.  Any
confirmation or correction to this tale would be appreciated.

	Bill Hopkins		{presby,psuvax,sdcrdcf}!burdvax!hopkins
	SDC-Burroughs		(215) 648-7578
	P.O. Box 517
	Paoli, PA  19335

And may all you complements be computable.