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64 CHAPTER 2 / NUMBER SYSTEMS, BINARY ARITHMETIC, AND CODES

(2) Represent each operand in BCD form.
(3) Carry out steps (1) through (4) of Algorithm 2.14. If the result is negative, the true
value is found by negation: [(resM/OiocJioc = (resultJeco- If the result is positive, that
result is the true value,

2.9.7 Floating-Point Arithmetic
Up to this point the arithmetic operations have involved fixed-point representation in which
all bits of a binary number were represented. In many practical applications the numbers may
require many bits for their representation. In Section 2.8 the floating-point number (FPN)
system was discussed for just that reason. Now it is necessary to deal with the arithmetic
associated with the FPN system, namely addition, subtraction, multiplication, and division.

FPN Addition and Subtraction Before two numbers can be added or subtracted one from
the other, it is necessary that they have the same exponent. This is equivalent to aligning
their radix points. From Eq. (2.17) for radix 2, consider the following two FPNs:

X = M x • 2 Ex

and

Ey
Y = M Y -2 .
E
Now, if for example E x > E Y, then Y is represented as M' Y • 2 '?, where

•/-„, and E' Y = E Y + (E x - E Y) = E x,

E
Ex
so that X + Y = (M x + M Y)- 2 * or X - Y = (M x - M' Y)- 2 , etc. Here, M Y =
.f-if-2 • • • f-m originally, but is now adjusted so that the exponents for both operands are
the same. The addition or subtraction of the fractions M x and M' Y is carried out according
to Algorithm 2.8 or Algorithm 2.9, respectively.
Consider the following examples of FPN addition:
EXAMPLE 2.34 —Addition

10010001.100 2 =X
145.500i 0
+27.62510 ~* 00011000.101 2 =Y
173.125,0

Comparing and equalizing the exponents E x and E Y gives

145.500,0 = .10010001100 x 2 8
8
27.625 m = .00011011101 x 2 .