GATE CSE 2016 Set 1 | Question: 07

Question

The $16\text{-bit}\;2's$ complement representation of an integer is $1111 \quad 1111 \quad 1111 \quad 0101;$ its decimal representation is ____________

Comments

I think this is easier process

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Another way to approach such type of question

we can apply what we have learnt from booths algorithm that is contiguous 1’s can be written as

example ;-

11111 can be written as 2^6 – 2^0 = 64-1 = 63 so we can apply this concept here

1111111111110101 given number

 

we will divide into three part

(1) (11111111111) (0101)

 

1 → (-2^15)

“11111111111” this block of one can be written as → (2^15-2^4)

“0101” →  this decimal is (5)

so finally we will add all up

-2^15 + 2^15 – 2^4 + 5 = -16 +5 = -11 (note: -2^15 + 2^15  this two became zero)

Answer 1

it can also b computed by booth's algorithm ....

1111 1111 1111 0101

pad with one zero

1111 1111 1111 01010

now booth's recoded code is

00000000000-11-11-1

now expand it 

(-1)*2⁰ +1*2¹+(-1)*2²+1*2³+(-1)*2⁴ 

=-11

Answer 2

There is a difference between 2 complement and 2 complement representation

In 4 bit system -

2 complement of 7 is 1001

but 2 complement representation of 7 is 0111.

Here in this question we are talking about 2 complement representation .

Hence -11 is the answer.

refer : Potential ambiguities of terminology at

https://en.wikipedia.org/wiki/Two%27s_complement

 

Answer 3

there is one simple trick to solve this question as we know in 2's complement representation when there are all 1's  the value of the number is -1 in this question all are 1 apart from 2^1 i.e  place value 2  and 2^3 i.e place value 8 they are 0 and there sum is 10 so subtract 10 from -1 i.e -1-10 =-11 the answer will be -11 it will save you a lot of time

Answer 4

Check the most significant bit (MSB): The MSB is 1, indicating a negative number in 2's complement representation. Invert all the bits (take the 1's complement): \[1111 \ 1111 \ 1111 \ 0101 \ \text{(2's complement)} \rightarrow 0000 \ 0000 \ 0000 \ 1010 \ \text{(1's complement)}\] Add 1 to the 1's complement: \[0000 \ 0000 \ 0000 \ 1010 + 1 = 0000 \ 0000 \ 0000 \ 1011 \ \text{(binary)}\] Convert the binary to decimal: \[0000 \ 0000 \ 0000 \ 1011 \ \text{(binary)} = 11 \ \text{(decimal)}\] Apply the negative sign: The original 2's complement number was negative, so the final decimal value is \(-11\). Therefore, the decimal representation of \(1111 \ 1111 \ 1111 \ 0101\) in 2's complement is \(-11\).