Page 90 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

Binary Arithmetic w 71

Now consider the same subtraction performed using the tens complement
technique (Figure 8-4).

5tandard subtraction Tens complement equivalent
r_______--------_
1
6 4 7 I 1000 647 I
I
-283 1 - 283 + 717 I
I
=364 I= 717 = 1364 Drop
I - 364 I 2%
I Take tens - 1
I complement Add tens comple- I
I ment to minuend I
L ________________ J
Figure 8-4. Tens complement decimal subtraction

The advantage of the tens complement is that it is not necessary to
perform an end-around-carry; any carry-out resulting from the addition of the
most significant digits is simply dropped from the final result. The disadvantage
is that, during the process of creating the tens complement, it is necessary to
perform a borrow operation for every digit in the subtrahend. This problem
could be solved by first taking the nines complement of the subtrahend,
adding one to the result, and then performing the remaining operations as
for the tens complement.
Similar techniques may be employed with binary (base-2) numbers, where
the radix complement is known as the twos complement and the diminished
radix complement is known as the ones complement. First, consider a binary
subtraction performed using the ones complement technique (Figure 8-5).

Once again, the standard way of performing the operation would be to
subtract the subtrahend (0001 11 10,) from the minuend (001 11001,),
which may require the use of one or more borrow operations. To perform the
equivalent operation in ones complement, each of the digits of the subtrahend
is first subtracted from a 1. The resulting ones complement value is added to
the minuend, and then an end-around-carry operation is performed. The
advantage of the ones complement technique is that it is never necessary to
perform a borrow operation. In fact, it isn’t even necessary to perform a