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)

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as the msb is 1, so the decimal no is negative.

the actual no is 10101 which is in 2's complement

so, - 01011 which is -11 .

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🔍 Step-by-step:

1. The first bit is 1, so the number is negative.

2. To find its magnitude:

   • Invert all bits: becomes 0000 0000 0000 1010

   • Add 1: becomes 0000 0000 0000 1011

3. Convert to decimal: 1011 is 11

4. Apply the negative sign: final answer is −11

Answer 1

$1111 \; 1111\; 1111\; 0101$

$2's$ complement of $1111 \; 1111\; 1111\; 0101 = 0000 \; 0000\; 0000\; 1011 =+11$

$2's$ complement of $-11 = +11,$ in $2's$ complement representation

$2's$ complement of $+11 = -11,$ in $2's$ complement representation

So $-11$ it should be.

Comments

it can also solve by using booth algorithm.

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@Praveen Saini

in 2's C representation ..we can ignore leading 1's till first 1 ..

so it will be 10101 shortcut to find decimal value of number in 2's C representation is

-16   8   4    2    1 

  1    0    1    0    1  ==> -16 +(4+1) = -11

is this right to use sir ?

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calling it a ‘shortcut’ would be an understatement.

It is the very essence of sign extension.

Answer 2

Answer $= -11$.

for $2's$ complement  1 111 1111 1111 0101  $→$$2′s$ complement$→$$2′s$ complement  $1$ $000$  $0000$  $0000$  $1011$

1bit$^{st}$ is same not involved in $2's$ complement same with $1's$ complement. since msb bit for sign.

Take one's complement and add $1$ we get $11$, and as it is negative number we get answer as $-11$.

Reference: http://www.electronics-tutorials.ws/binary/signed-binary-numbers.html.

Comments

@Hira Thakur How you are applying booths algo here Please explain

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i am nt getting ur solution ...

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Another way-
In $2's \ complement - 11111..\color{RED}10101$ represents the same number as $10101.$
$\Rightarrow -2^4 + 2^2+2^0 =-16+4+1=-11 $

Answer 3

   1111111111110101  -> ignore all 1's, except one 1 from left side(because they have no effect on final decimal value)

=  10101

=  (-2⁴)*1 + (2³)*0 + (2²)*1 + (2¹)*0 + (2⁰)*1  

= -16 + 4 + 1

= -11

Answer 4

Procedure:

Consider the power place values ofthe number from right to left is 0,1,2,3,4,5........ so on

Calculate the place value of first 1 of the number from left where contineous 1s are there to the left of it till end(Let called the place i).

Then do this formula: -(2^i)+(sum of all 2^j)   Here j is from i-1  to 0

Here Number is:1111111111110101

  -(2^4) +2^2+2^0=-16+4+1=-11

Answer 5

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 6

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