Page 89 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

70 ChapterEight

Binary Subtraction
Unsigned binary numbers may be subtracted from each other using an
identical process to that used for decimal subtraction. However, for reasons of
efficiency, computers rarely perform subtractions in this manner; instead, these
operations are typically performed by means of complement techniques.
There are two forms of complement associated with every number system,
the radix complement and the diminished radix complement, where radix refers to
the base of the number system. Under the decimal (base-10) system, the radix
complement is also known as the tens complement and the diminished radix
complement is known as the nines complement. First, consider a decimal sub-
traction performed using the nines complement technique-a process known
in ancient times as “Casting out the nines” (Figure 8-3).

Standard subtraction Nines complement equivalent
r----------------
1
6 4 7 I 999 647 I
I
-283 1-283 + 716 I
I I
=364 I=716 = 1363
1 -
I - 1
I Takenines 364 I

I complement -I
I Add nines comple- I
I ment to minuend I

Figure 8-3. Nines complement decimal subtraction

The standard way of performing the operation would be to subtract the
subtrahend (283) from the minuend (647)) which, as in this example, may
require the use of one or more borrow operations. To perform the equivalent
operation in nines complement, each of the digits of the subtrahend is first
subtracted from a 9. The resulting nines complement value is added to the
minuend, and then an end-around-carry operation is performed. The advantage
of the nines complement technique is that it is never necessary to perform a
borrow operation.2

2 This made this technique extremely popular in the days of yore, when the math skills of the
general populace weren’t particularly high.