Page 97 - Bebop to The Boolean Boogie An Unconventional Guide to Electronics Fundamentals, Components, and Processes

78 ChuptevEight

represented by a - b. Generating the twos complement of b results in -b,
allowing the operation to be performed as an addition: a + (-b). This means
that computers do not require different blocks of logic to add and subtract;
instead, they only require an adder and a twos complementor. As well as
being faster than a subtractor, a twos complementor requires significantly
fewer logic gates.

Binary Multiplication
One technique for performing multiplication in any number base is by
means of repeated addition; for example, in decimal, 6 x 4 = 6 + 6 + 6 + 6 = 24.
However, even though computers can perform millions of operations every
second, the repeated addition technique is time-consuming when the values to
be multiplied are large. As an alternative, binary numbers may by multiplied
together by means of a shift-and-add technique known as Booth's Algorithm
(Figure 8- 12).
Decimal
Multiplicand Multiplier equivalent
r Y 1 I I
000101 10 x 01 11001 1 22,,x 115,,
0001 01 10
0001 01 IO-
00000000~~
Partial 00000000~~~
0 L 0 00010110~~~~~ ~
products 00010110~~~~

0
~
~
0
1
1
1
~
0
~
~
0
0
~
0
0
0
0
0
0
0000100111100010 ~ ~ ~ ~ ~ ~ = 2530,,
Figure 8-1 2. Binary multiplication using Booth's Algorithm
Using Booth's algorithm, a partial product is generated for every bit in the
multiplier. If the value of the multiplier bit is 0, its corresponding partial product
consists only of Os; if the value of the bit is 1, its corresponding partial product
is a copy of the multiplicand. Additionally, each partial product is left-shifted