Complementary Numbers MCQ Quiz - Objective Question with Answer for Complementary Numbers - Download Free PDF

Explanation:

Two's Complement Representation

Definition: Two's complement is a mathematical operation used in computer science to represent signed integers in binary. It is widely used because of its simplicity in handling arithmetic operations and representing negative numbers. In this system, positive numbers are represented in standard binary form, while negative numbers are obtained by inverting all the bits of the absolute value (in binary) and adding 1 to the least significant bit (LSB).

Steps to Find Two's Complement:

1. Start with the binary representation of the absolute value: Represent the number in binary. For example, the absolute value of -8 is 8, which in binary (for a 4-bit system) is 1000.
2. Invert all the bits: Flip all the bits of the binary representation. For 1000, the inverted bits become 0111.
3. Add 1 to the result: Adding 1 to the inverted binary number gives 0111 + 1 = 1000.

Thus, the two's complement representation of -8 (in a 4-bit system) is 1000.

Correct Option Analysis:

The correct option is:

Option 1: 1000

This is the correct two's complement representation of -8 in a 4-bit system. Following the steps outlined above, the binary value 1000 is derived by inverting the bits of the binary representation of 8 and adding 1.

Additional Information

To further understand the analysis, let’s evaluate the other options:

Option 2: 1010

This option is incorrect because 1010 is not derived using the two's complement method for -8 in a 4-bit binary system. The inverted bits of 1000 are 0111, and adding 1 results in 1000, not 1010.

Option 3: 110

This option is incorrect because 110 does not represent -8 in two's complement form within a 4-bit binary system. It neither matches the expected binary format nor adheres to the two's complement operation rules.

Option 4: 1110

This option is incorrect as it does not represent the two's complement of -8 in a 4-bit binary system. The correct result, as derived earlier, is 1000. 1110 represents a different value altogether in two's complement.

Conclusion:

Understanding two's complement representation is essential for dealing with signed integers in binary arithmetic. The process involves inverting the bits of the absolute value and adding 1 to the least significant bit. This method simplifies arithmetic operations and provides a consistent way to handle negative numbers in binary systems.

In the context of the given problem, -8 is correctly represented as 1000 in two's complement form for a 4-bit binary system. This representation is derived systematically by following the steps outlined above, ensuring accuracy in binary arithmetic operations and signed integer representation.

The correct option is 3
Concept:

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

Calculation:
Converting 50 into 8-bit binary form
50₁₀ = (00110010)2

Taking the 1's complement of the above, we get 11001101

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

11001101 + 1 = 11001110 

(-50)10 in 2's complement is written as:

11001110 

Concept:

2’s complement representation:

1) For positive numbers, the 2's complement representation of a number is simply the binary equivalent of the number.

2) For negative numbers, the 2's complement representation is obtained by taking the 2's complement of the binary equivalent of the decimal number.

Analysis:

20 can be represented in binary number system as given below.

20 = 010100

1’s complement = 101011

2’s complement = 1’s complement of the given number + 1

2’s complement = 101011 + 000001 = 101100

Tips and Tricks:

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.

Given number is: 010100

2’s complement representation: 101100

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 a sign bit 

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

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

The correct answer is –128 to +127

Key Points

• In a binary system, numbers are typically represented using a fixed number of bits.
• In 2's complement representation, the most significant bit (MSB) is used as the sign bit, where 0 indicates a positive number and 1 indicates a negative number.
• The remaining bits represent the magnitude of the number.
• For an 8-bit system:
  • The range for positive numbers is from 0 to 2^(n-1) - 1, where n is the number of bits. In this case, it's 2⁷ - 1, which is 127.
  • The range for negative numbers is from -2^(n-1) to -1. In this case, it's -2⁷, which is -128.

So, combining both ranges, the overall range for an 8-bit system using 2's complement representation is from -128 to +127.

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.

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)

Analysis:

The 1's complement of the given binary digit will be:

11011111 → 00100000

Adding 1 to the above, we get the 2's complement as:

00100000 → 00100001

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.

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 a sign bit 

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

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

Concept:

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.

2. 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.

3. 2’s complement representation:

It is obtained by simply adding 1 to the 1’s complement of that binary number.

Calculation:

The binary form of 7 ↔ 0111

1's complement of -7 = 1(for sign) (000)(1's complement of 7)

= 1000

2's complement of -7 = 1's complement + 1 = 1000 + 1 = 1001

Concept:

1’s complement:

• 1’s complement simply inverted every bit of input.
• Can be implemented using only NOT gate for each bit of binary number input.  

Application:

The ones' complement of a binary number is defined as the value obtained by inverting all the bits in the binary representation of the number (swapping 0s for 1s and vice versa). Hence, 10010 would be the 1's complement of binary 01101.

Note:

Logical complement is the equivalent term used for using 1’s complement as a logical operation. “Negation” used in logic is known as “logical complement”. It is a process that carries one proposition (let us suppose p) to other propositions (not p). Conclusively, it is described true when p is false and described false when p is true.

Concept:

The inverse of the function is nothing but the complement of the given expression.

Also, De-Morgans Theorem states that:

$\left(i\right)\overline{A_1\cdot A_2\cdot A_3\dots A_n}=̅A_1+̅A_2+̅A_3+\dots ̅A_n$

$\left(ii\right)\overline{A_1+A_2+A_3+\dots +A_n}=̅A_1\cdot ̅A_2\cdot ̅A_3\dots ̅A_n$

Application:

Let X̅ = A + A̅ B

The inverse of the above function will be:

$̅X=\overline{\left(A+̅AB\right)}$

By using De-morgan’s theorem, the above can be written as:

$X=̅A\cdot \overline{(̅AB)}$

Using De-Morgans theorem:

$X=̅A\cdot (A+̅B)$

$X=̅A.A+̅A.̅B$

Since A̅ A = 0, we get:

$X=̅A.̅B$

X = $\overline{A+B}$

Concept:

According to De Morgan's law:

$\overline{A+B}=\overline{A}.\overline{B}$

$\overline{A.B}=\overline{A}+\overline{B}$

De Morgan's law helps in the simplification of Boolean algebra.

Important Points

Concept:

2’s complement representation:

1) For positive numbers, the 2's complement representation of a number is simply the binary equivalent of the number.

2) For negative numbers, the 2's complement representation is obtained by taking the 2's complement of the binary equivalent of the decimal number.

Analysis:

20 can be represented in binary number system as given below.

20 = 010100

1’s complement = 101011

2’s complement = 1’s complement of the given number + 1

2’s complement = 101011 + 000001 = 101100

Tips and Tricks:

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.

Given number is: 010100

2’s complement representation: 101100

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:

Given the binary number is 1101101

Taking the 1's complement of the above, we replace all the 1's with 0's and all the 0's with 1's to get:

1's complement of 1101101 = 0010010

Adding 1 to the 1's complement, we get the 2's complement as:

0010010 + 1 = 0010011

Two's complement representation, or, in other words, signed notation - the first bit tells about the sign.

The convention is that a number with a leading 1 is negative, while a leading 0 denotes a positive value.

In an 8-bit representation, we can write any number from -128 to 127.

The name comes from the fact that a negative number is a two's complement of a positive one.

Concept:

Steps to find 2's complement of a negative number:

Step 1: For the given negative binary number, first write the binary equivalent of the positive number of the same number.

Step 2: Then find 1's complement.

Step 3: Add 1 to it.

Calculation:

Step 1: (+9)10 = (1001)2

Step 2: 1's complement of (1001)2 = (0110)2

Step 3: (0110)2 + 1

2's complement of (-9) is (0111)2

For 8-bit notation, putting 1 four times before MSB does not affect the 2's complement.

Therefore, 2's complement of (-9) in 8-bit notation is 11110111.