Test: Representation of Signed Numbers - Computer Science Engineering (CSE) MCQ

To represent a given 5-bit number using 6- bits in a 2's complement representation, we simply copy the MSB bit as it is till we get the required 6 bits, i.e.
X = 01110 = 001110 
Y = 11001 =111001
Ignoring the carry, we get the addition of the two in 2's complement number as:
∴ x + y = 000111

Concept: 
1's complement of Binary: 1's complement of a Binary number is defined by the value obtained by inverting all the bit, i.e, 0 as 1 and 1 as 0.
Calculation:
The 1's complement of the given binary digit will be:
(101100) → (010011)

Additional Information
2's complement of Binary: It is the sum of 1's complement of Binary number and 1 to the least significant bit (LSB).
∴ 2's complement = 1's complement + 1 (LSB)
Shortcut Trick
Steps to writing 2’s complement to any binary number:

• Start from right to left and search for the first ‘1’
• Write down the bits until that first ‘1’ as it is.
• Write down the remaining left bits with their respective complement.

Concept:
If number is positive; MSB = 0
Then 2’s complement will be the same
If number is negative; MSB = 1
Then 2’s complement will be different from its obtained result
Calculation:
Given,
(10010)₂ - (10111)₂
(10010)₂ + 2’s complement of (10111)₂

i.e. (11011)₂ = -[2’s complement of 11011]
= -[00101]
∴ (10010)₂ - (10111)₂ = -(00101)₂

The smallest negative number is the largest binary value.
1111 is -1, 1110 is -2, 1101 is -3, etc down to 1000 which represents -8.

Important Points
Using two's complement for negative numbers:

• Find the positive binary value for the negative number you want to represent.
• Add a 0 to the front of the number, to indicate that it is positive
• Invertor finds the complement of each bit in the number.
• Add 1 to this number.

Example:
4 using two's complement numbers,

• 4 = 100
• Adding 0 to the front becomes 0100
• 'inverted' becomes 1011
• Add 1 = 1100 (-8 + 4 = -4)

X = 00110
since, the MSB = 0
∴ it is a positive number.
Decimal equivalent: 0 + 1 × 2² + 1 × 2¹ + 0 × 2⁰ = + 6
Y = 10011
since, the MSB = 1 
∴ it is a negative number,
We need to take the 2's complement of Y, that is.
1's complement (Y) + 1
01100 + 1
⇒ 01101
Decimal equivalent → 0 + 1 × 2³ + 1 × 2² + 0 + 1 × 2⁰ → -13
The sum of X and Y is
+6 - 13 = - 7
The 2's complement of - 7
→ 1's complement of 7 + 1
→ 1's complement of 00111 + 1
→ 11000 + 1 → 11001

Concept:
1’s complement representation of a binary number is obtained by toggling all the bits, i.e. replacing 1 with 0, and 0 with 1.
2’s complement representation of a binary number is obtained by adding 1 to the 1’s complement representation.

Application:
(127)₁₀ = (01111111)₂
1’s complement representation will be:
1’s complement = 10000000
Number of 1’s is the 1’s complement is, n = 1
Now, the two (2’s) complement representation will be:
2’s complement = 10000000 + 1
= 10000001
Number of 1’s in 2’s complement is, m = 2
∴ The required ratio is m : n = 2 : 1

Signed magnitude representation uses the most significant bit (MSB) a sign bit.

• If the sign bit is ‘0’ then the number is positive.
• If the sign bit is ‘1’ then the number is negative.

The remaining bits represent the magnitude of the binary number.
Example:

• 1000101 represents a negative number as the MSB bit is '1'
• 0101001 represents a positive number as the MSB bit is '0'

Important Point:

• 1’s complement representation: It is a representation of a binary number obtained by toggling all bits in it i.e. transforming the 0 bit to 1 and the 1 bit to 0.
• 2’s complement representation: It is obtained by simply adding 1 to the 1’s complement of that binary number.

Given Number is 10100
The Right shift of the content in register is same as the content divided by 2 
Apply Right Shift ⇒ 11010       
Operation right shift is equivalent to divided by 2