Chapter 1, Problem 40CRP

Explanation of Solution

The best approximation to the value one-tenth that can be represented using 8-bit floating-point format described in figure $1.24$:

• Convert the given value in the binary representation.
• Copy the bit pattern into the mantissa field from left to right, start with 1.
• Copy the bit pattern into the mantissa field from left to right, start with 1.
• The exponential field is determined by assuming the pattern of mantissa field with a radix at its left and it obtains the original binary number by determining the number of bits and direction of radix that must be moved.

The remaining three bits represent the exponential field that is shown in the below table.

Bit pattern	value
$111$	3
$110$	2
$101$	1
$100$	0
$011$	$-1$
$101$	$-2$
$001$	$-3$
$000$	$-4$

• When the given value is positive, the bit assigned on high order bit is 0 and when the given value is negative, the bit assigned on higher bit is 1.
• The best approximation of $\frac{1}{10}$ is $\frac{1}{8}$ as $\frac{1}{10}$ cannot be expressed in binary representation accurately