1.6 Representation of Negative Numbers

There are two common methods for representing negative numbers using binary digits: signed-magnitude and complement number systems. These methods are widely used in computer systems.

Part 1.6.1 Signed-magnitude System

In binary numbers, an extra bit is used to indicate the sign, often referred to as the sign bit. This sign bit is typically located at the most significant bit (MSB) position, where 1 represents a positive value and 0 represents a negative value. The remaining bits hold the magnitude of the number.

For Examples

01010101₂= -85

11010101₂= 85

01111111₂ = -127

11111111₂ = 127

Part 1.6.2 Complement Number System

There are two types of complements used in number systems: radix complement and radix minus one complement.

Part 1.6.2.a Diminished Radix Complement or Radix-Minus-One Complement or (r-1)’s Complement System

Radix minus one complement, also known as (r − 1)'s complement, is a method of representing negative numbers in a positional numeral system. In this system, to find the negative representation of a number, each digit is subtracted from the radix (base) minus one and then the entire result is complemented (all 0s changed to 1s and vice versa).

For example, in the decimal system (radix 10), the 9's complement of 25 is 74.

Solution:

9 – 2 = 7

9 – 5 = 4

Let’s read it from top to bottom, the answer is 74.

In binary (radix 2), the 1's complement of 1011 is 0100.

1 – 1 = 0

1 – 0 = 1

1 – 1 = 0

Let’s read it from top to bottom, the answer is 0100.

Radix minus one complements are used in some older computer architectures and are still relevant in certain applications like checksum calculations and error detection. However, they have largely been replaced by two's complement representation, which is more efficient and widely used in modern computers.

Part 1.6.2.b Radix Complement or True Complement System

Radix complement, also known as r's complement, is a method of representing negative numbers in a positional numeral system. In this system, to find the negative representation of a number, the entire number is subtracted from the radix (base) raised to the power of the number of digits.

For example, in the decimal system (radix 10), the 10's complement of 25 is 75.

Solution:

The radix-minus-one complement of 25 is 74. Add one in 74 = 75.

In binary (radix 2), the 2's complement of 1011 is 0101.

Solution:

The radix-minus-one of 1011 is 0100. Then add one to the 0100, the answer is 0101.

Part 1.6.3 Subtraction using Complement Number System

Part 1.6.3.1 Using Radix-minus-one Complement

Procedure:

Step 1. Copy the minuend.

Step 2. Obtain the radix-minus-one complement of the subtrahend.

Step 3. Add the radix-minus-one complemented subtrahend to the minuend.

Step 4. Handle carry-over: a. If the outmost carry-over from step 3 is '1', add this to the least significant digit obtained in step 3. b. If the outmost carry-over from step 3 is '0', take the radix-minus-one complement of the result in step 3 and prefix the negative sign.

Example No. 1

Subtract 17 from 92

Solution.

Step 1. Copy the minuend.

Step 2. Obtain the radix-minus-one complement of the subtrahend.

9 – 1 = 8

9 – 7 = 2

Let’s read it from top to bottom: 82

Step 3. Add the radix-minus-one complemented subtrahend to the minuend.

92 + 82 = 174

74 + 1 = 75 This is the Answer.

Example No. 2

Subtract 111 from 110010

Solution.

Step 1. Copy the minuend.

110010

Step 2. Obtain the radix-minus-one complement of the subtrahend.

1	1	1	1	1	1
0	0	0	1	1	1
1	1	1	0	0	0

The Radix-minus-one of 000111 is 111000

Step 3. Add the radix-minus-one complemented subtrahend to the minuend.

110010 + 111000 = 1101010

Add 1 to 101010 = 101011₂ this is our answer.

Example No. 4

Subtract 110010 from 000111

Solution:

Step 1. Copy the minuend.

000111

Step 2. Obtain the radix-minus-one complement of the subtrahend.

The radix-minus-one complement of subtrahend is 001101

Step 3. Add the radix-minus-one complemented subtrahend to the minuend.

Add 000111 and 001101, the answer is 010100.

The radix-minus-one complement of 010100 is 101011 and add negative sign.

Our answer is -101011.

Part 1.6.3.2 Using Radix Complement

Procedure:

Step 1. Copy the minuend.

Step 2. Obtain the true complement of the subtrahend.

Step 3. Add the true complemented subtrahend to the minuend.

Step 4. Handle carry-over: a. If the outmost carry-over from step 3 is '1', discard it. b. If the outmost carry-over from step 3 is '0', take the true complement of the result in step 3 and prefix the negative sign.

Step 5. The obtained result in step 4 is the difference between the two numbers.

Example No. 5

Subtract 17 from 92

Solution:

Step 1. Copy the minuend.

Step 2. Obtain the true complement of the subtrahend.

Let’s get first the radix-minus-one complement of 17

9 – 1 = 8

9 – 7 = 2

The radix-minus-one complement of 17 is 82, then add one.

Therefore, the true compliment of 17 is 83.

Step 3. Add the true complemented subtrahend to the minuend.

Add 83 and 92, the sum is 175.

Our answer is 75.

Example No. 6

Subtract 111 from 110010

Solution:

Step 1. Copy the minuend.

110010

Step 2. Obtain the true complement of the subtrahend.

The radix-minus-one complement of 000111 is 111000.

Therefore, the true compliment of 000111 is 111000 + 1 = 111001.

Step 3. Add the true complemented subtrahend to the minuend.

Add 110010 and 111001, the sum is 1101011.

Our answer is 101011₂.

Example No. 7

Subtract 110010 from 000111

Solution:

Step 1. Copy the minuend.

000111

Step 2. Obtain the true complement of the subtrahend.

The radix-minus-one complement of 110010 is 001101.

Therefore, the true compliment of 110010 is 001101 + 1 = 001110.

Step 3. Add the true complemented subtrahend to the minuend.

Add 000111 and 001110, the sum is 010101.

The Radix-minus-one complement of 010101 is 101010. Then add 1 to 101010 to get the true complement.

Our answer is – 101011₂_.