DIGITAL DESIGN

Section 2.4 Addition and Subtraction of Nondecimal Numbers 29
DO NOT COPY
1 1 1 1 1 1 1
X 190 1 0 1 1 1 1 1 0
X 173 1 0 1 0 1 1 0 1
Y + 141 + 1 0 0 0 1 1 0 1
Y + 44 + 0 0 1 0 1 1 0 0
X + Y 331 1 0 1 0 0 1 0 1 1 X + Y 217 1 1 0 1 1 0 0 1
DO
Figure 2-1
NOT COPY
Examples of decimal and corresponding binary additions.
examples from the figure are repeated below along with two more, this time
showing the borrows as a bit string B:
DO NOT COPY
B
001111100
B
011011010
X 229 11100101
X 210 11010010
Y − 46 − 00101110
Y −109 − 01101101
DO
X − Y 183 10110111
NOT
X − Y 101
COPY
01100101
B
010101010
B
000000000
X 170 10101010
X 221 11011101
Y − 85 − 01010101
Y − 76 − 01001100
DO NOT COPY
X − Y 85 01010101 X − Y 145 10010001
A very common use of subtraction in computers is to compare two numbers. For comparing numbers
example, if the operation X − Y produces a borrow out of the most significant bit
position, then X is less than Y; otherwise, X is greater than or equal to Y. The relationship
DO
between carries and
NOT
borrow in adders and subtractors
COPY
will be explored
in Section 5.10.
Addition and subtraction tables can be developed for octal and hexadecimal
digits, or any other desired radix. However, few computer engineers bother
to memorize these tables. If you rarely need to manipulate nondecimal numbers,
DO NOT COPY
Must borrow 1, yielding
the new subtraction 10–1 = 1
Figure 2-2
Examples of decimal
After the first borrow, the new
DO
subtraction
NOT
for this column is
COPY
and corresponding
0–1, so we must borrow again.
binary subtractions.
The borrow ripples through three columns
to reach a borrowable 1, i.e.,
100 = 011 (the modified bits)
+ 1 (the borrow)
DO NOT COPY
0 10 1 1 10 10
0 10 10 0 1 10 0 10
minuend X 229 1 1 1 0 0 1 0 1
X 210 1 1 0 1 0 0 1 0
subtrahend Y – 46 – 0 0 1 0 1 1 1 0
Y – 109 – 0 1 1 0 1 1 0 1
difference X – Y 183 1 0 1 1 0 1 1 1 X – Y 101 0 1 1 0 0 1 0 1
Copyright © 1999 by John F. Wakerly Copying Prohibited