Eliminating cascading carries

In this note we describe an adder for numbers in the following decimal representation

(0)     n = 〈 Σ i : 0≤i : d_i∙10ⁱ〉

(1)     −5 ≤ d_i≤ 6 for all i   .

All integers can be represented this way, but note that the representation is not necessarily unique: for instance, 26 can be represented in two decimals by (2,6) as well as by (3,−4).

In our "adder", which can calculate ±a±b, the sum of two digits ranges from −12 through +12; its addition table represents these values as c∙10 + s —from "carry" and "sum digit"—, with c, s satisfying

(2)    −1 ≤ c≤ +1 and −4 ≤ 5 ≤ +5     ,

i.e.,	from	−12	through	−5,	c=	−1	,
	from	−4	through	+5,	c=	0	,	and
	from	+6	through	+12,	c=	+1		.

Because (2) excludes for s the extreme digit values −5 and +6 and |c| ≤ 1, each sum digit can absorb a carry from the right without generating a new one. In long parallel adders the problem of carry propagation has thus been eliminated; alternatively we can add from left to right with a "look-ahead" of only 1 position.

Remark If we so desire, we may replace (1) by the weaker, symmetric −6 ≤ d_i ≤ +6. with (2) weakened accordingly, some freedom in the addition table is introduced. (End of Remark)

Please note that (i) the representation of zero is unique, and (ii) the sign of a non-zero value is determined by the sign of its most-significant non-zero digit.

The above, which was designed decades ago, was inspired by the implementation of the end-around carry in the serial adder of the ARMAC. Since the advent of systolic arrays it must look like something familiar.