2's Complement MCQ Quiz - Objective Question with Answer for 2's Complement - Download Free PDF

2's Complement - It is a type of mathematical and logical (binary) representation that helps in representing signed numbers and performing arithmetic operations such as subtraction, addition, etc.

To perform 2's complement of (1000)₂ we will perform the following steps -

1. We will perform 1's complement on (1000)₂ by flipping 1s to 0s and 0s to 1s.
   (1000)₂ ===> (0111)₂
2. Now we will add 1 to the resultant value, that is, (0111)2.
   (0111)₂ + (1)₂ ===> (1000)₂
3. Hence, we get (1000)2 back after 2's complement.

Key Points

•  In a 2's complement number system, the range of integers that can be represented depends on the number of bits, n.
• The most significant bit (MSB) in a 2's complement representation indicates the sign of the number.
• If the MSB is 0, the number is positive or zero. If the MSB is 1, the number is negative.
• The range of integers that can be represented with n bits in 2's complement is from -2^n-1 to 2^n-1 - 1.
• This range allows for an equal number of positive and negative values, with one additional value for negative numbers.

Important Points

•  For example, with 8 bits (n=8), the range is from -128 to 127.
• Understanding the range of 2's complement numbers is crucial for tasks involving binary arithmetic and computer architecture.

Additional Information

•  2's complement representation simplifies the implementation of arithmetic operations in computer systems.
• It allows for the use of the same addition and subtraction circuits for both signed and unsigned integers.
• It is the standard method for representing signed integers in modern computing systems.

Range of 4-bit in 2’s complement

= (-2³ to 2³ – 1) = -8 to 7

Since number is in 2’s complement

1011 = -2³×1 + 2² × 0 + 2¹× 1 + 1 = -5

0101 = -2³×0 + 2² × 1 + 2¹× 0 + 1 = 5

1011 + 0101 = -5 + 5 = 0

0 falls in the range of -8 to 7

∴ no overflow

1011 + 0101 = 1 0000

Note:

28₁₀ = (11100)₂ = (0000 0000 0001 1100)₂

-28₁₀ = 2’s complement of 0000 0000 0001 1100

2’s complement of 0000 0000 0001 1100 = 1111 1111 1110 0100

Note:

Tricks to find: 2’s complement

Concept:

The procedure for representing a negative decimal in 2's complement representation is as shown:

Application:

The binary representation of 17

17₁₀ = (10001)₂

Taking the 1's complement of the above, we get 01110

Adding 1 to the 1's complement, we get the 2's complement representation of the number, i.e.

11101110 + 1 = 01111

(-17)₁₀ in 2's complement is written as:

101111

Calculation:

14 in binary form is represented as:

1410 = (00001110)2

Taking the 1's complement of the above, we get 11110001

Adding 1 to the 1's complement, we get the 2's complement representation of the number, i.e. 11110010

Since there is a 1 in the MSB, the number is a negative number with value -14.

∴ The 2's complement of -6410 contains 7 bits.

28₁₀ = (11100)₂ = (0000 0000 0001 1100)₂

-28₁₀ = 2’s complement of 0000 0000 0001 1100

2’s complement of 0000 0000 0001 1100 = 1111 1111 1110 0100

Note:

Tricks to find: 2’s complement

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.

∴ 1's complement of 1100 0110 = 0011 1001

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)

Calculation:

Given Binary Number,

0011 0101 1001 1100

1's complement = 1100 1010 0110 0011

2's complement = 1's complement + 1 (LSB)

 Alternate Method

Note: A shortcut method of forming the 2's complement of a binary number is to copy bits from the right until a one-bit has been copied, then invert the remaining bits i.e, 0 as 1 and 1 as 0.

Find the 2s complement of (-19)₁₀

Step 1 :Convert given decimal number into binary

(19)₁₀ = (010011)₂

Step 2: Take 1s complement of the binary number by converting 0 with 1 and vice-versa

1s complement : (101100)₂

Step 3: Add 1 to 1s complement

101100 + 1 = 101101

Hence correct option is "4"

Important PointsTo find 2s complement quickly

Move from the right of the binary number to left , keep bits till first "1" as it is then complement each bit.

ex : (19)₂ = 010011

2s complement : 101101

The correct answer is option 1

Concept:

When we multiply a number by 2 this means a binary representation of this number is shifted left

shifting depends upon how many time the number is multiplied by 2

Explanation:

Given a hexadecimal number in 2's complement and required in 2's complement

So, nothing want to change it

P = (F87B)16 = (1111 1000 0111 1011)₂

8P =2³*P

23*P this means that the binary number that is represented by P is shifted 3 times left

So, it becomes

(1111 1000 0111 1011)2 = (1100 0011 1101 1000)₂ = (C3D8)₁₆

P = (F87B)16 = (1111 1000 0111 1011)2

Most significant bit is 1 here its mean number is negative

To get the value 2's complement of this is (0000 0111 1000 0101)₂

(0000 0111 1000 0101)2 =(-1925)₁₀

8P =8 × -1925 =-15400

now need to find 2's complement of -15400

(15400)₁₀= (0011 1100 0010 1000)2

take 2's complement of (0011 1100 0010 1000)2

(1100 0011 1101 1000)2 = (C3D8)16

Addition of integers in One’s complement-

One’s complement of 011000 → 100111

One’s complement of 011000 → 100111

Now, addition-

 100111 + 100111 = 1001110

Therefore, in this 1001110 the digit 1-001110 is an overflow.

Addition of integers in Two’s complement-

Two’s complement of 011000 → 101000

Two’s complement of 011000 → 101000

Now, addition-

101000 + 101000 = 1010000

Therefore, in this 1010000 the digit 1-010000 is an overflow.

Addition of integers in sign and magnitude format

If start with 0 that means +ve.

If start with 1 that means –ve.

So, given numbers 011000 and 011000 both are positive

Now, addition

11000 + 11000 = 110000

Therefore, in this 110000 the digit 1-10000 is an overflow.

Data:

number of bits = n = 8 

Formula:

Range of 2's compliment number with n bits is (–2^n-1 ) to +(2^n-1 – 1)

Smallest integer = –2n-1

largest integer = 2n-1 – 1

Calculation:

Smallest integer = –28-1 = –128 

Concept:

It is easy to change a negative integer in base ten into binary by using the method of 2's compliment

• Step 1: Write the absolute value of the given number in binary form. Prefix this number with 0 to indicate that it is positive.
• Step 2: Take the complement of each bit by changing zeroes to ones and ones to zero.
• Step 3: Add 1 to your result. This is the 2's complement representation of the negative integer.

Calculation:

Step 1:

(541)₁₀ = (0010 0001 1101)₂

Step 2:

Take the compliment : (1101 1110 0010)₂

Step 3:

Add 1 : (1101 1110 0010 + 1)₂ = (1101 1110 0011)₂

(1101 1110 0011)₂ ⇒  (DE3)₂

Concept:

The range of n bit word in 2’s complement representation is,

\(\left( {-{2^{n - 1}} } \right)\;to\;\left( { {2^{n - 1}-1}} \right)\)

(n-1) is used here because out of n bits 1 bit is used as sign bit 

Analysis:

There is one extra negative number because "0" has only single representation in 2's complement form

For 8 bit word the range will be -128 to 127. 

Concept:

The procedure for representing a negative decimal in 2's complement representation is as shown:

Application:

The binary representation of 17

17₁₀ = (10001)₂

Taking the 1's complement of the above, we get 01110

Adding 1 to the 1's complement, we get the 2's complement representation of the number, i.e.

11101110 + 1 = 01111

(-17)₁₀ in 2's complement is written as:

101111