Official PYQ · 1999Q1.20Number Representations (Fixed and Floating Point)MCQ1 mark

GATE CS 1999 Q1.20 — Number Representations (Fixed and Floating Point)

Booth's coding in $8$ bits for the decimal number $-57$ is:

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Answer: B

To find the Booth's recoding of the decimal number $-57$ in $8$ bits, we follow these steps:

Step 1: Represent $-57$ in 8-bit 2's complement form

The binary representation of $+57$ in 8 bits is: $+57 = 00111001_2$
To find $-57$ in 2's complement, we invert the bits of $+57$ and add $1$ : $\text{Invert bits: } 11000110_2$ $\text{Add 1: } 11000111_2$ So, the 8-bit 2's complement representation of $-57$ is $A = a_7 a_6 a_5 a_4 a_3 a_2 a_1 a_0 = 11000111_2$ .

Step 2: Apply Booth's Recoding

In Booth's recoding, we append an implicit bit $a_{-1} = 0$ to the right of the LSB. The recoded digit $y_i$ for each position $i$ (from $7$ down to $0$ ) is calculated using the formula:

y_i = a_{i-1} - a_i

Let's align the bits and compute each $y_i$ :

Position ( $i$ )	$7$	$6$	$5$	$4$	$3$	$2$	$1$	$0$	$-1$
Bit ( $a_i$ )	$1$	$1$	$0$	$0$	$0$	$1$	$1$	$1$	$0$ (implicit)
Recoded ( $y_i = a_{i-1} - a_i$ )	$1 - 1 = 0$	$0 - 1 = -1$	$0 - 0 = 0$	$0 - 0 = 0$	$1 - 0 = +1$	$1 - 1 = 0$	$1 - 1 = 0$	$0 - 1 = -1$	—