GATE | Gate IT 2005 | Question 8

Using Booth’s Algorithm for multiplication, the multiplier -57 will be recoded as
(A) 0 -1 0 0 1 0 0 -1
(B) 1 1 0 0 0 1 1 1
(C) 0 -1 0 0 1 0 0 0
(D) 0 1 0 0 -1 0 0 1


Answer: (A)

Explanation: Booth’s algorithm: first take 2’s complement of given number if number is negative, then append 0 into LSB.

Then, for each pair from LSB to MSB (add 1 bit at a time):
00 = 0, 01 = +1, 10 = -1, 11 = 0

Therefore, given number in signed representation (2’s complementation) of -57

= 2's complement of (00111001) = 11000110+1 = 11000111
= append 0 into LSB of (11000111) = 110001110
Now Booth's code (add 1 bit at a time, from LSB to MSB):
= 11, 10, 00, 00, 01, 11, 11, 10 = 0 -1 0 0 1 0 0 -1

Alternative way –
You can calculated decimal values in given options:
(A) 0 -1 0 0 1 0 0 -1 = 0*(2^6) -1*(2^6) + 0*(2^5) + 0*(2^4) + 1*(2^3) + 0*(2^2) + 0*(2^1) -1*(2^0) = -2^6 + 2^3 – 2^0 = -64 +8 -1 = -57
(B) 1 1 0 0 0 1 1 1 = 2^7 + 2^6 + 2^2 2^1 + 2^0 = 199
(C) 0 -1 0 0 1 0 0 0 = -2^6 + 2^3 = -56
(D) 0 1 0 0 -1 0 0 1 = 2^6 – 2^3 + 2^0 = 57



So, option (A) is correct.

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