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Introduction to Two’s Complement and how to calculate it?

Alex James by Alex James
October 25, 2022
in Education
0
Introduction to Two’s Complement and how to calculate it?

Two’s complement is a mathematical operation on binary numbers and is an example of radix complement (10’s complement). It is majorly used in computing. In this post, we will discuss 2’s complement definition, rules to evaluate 2s complement, and a lot of examples to perfectly understand the concept.

Table of Contents show
1 What is two’s complement?
2 How to find two’s complement?
3 Rules of two’s complement
3.1 Binary to two’s complement:
3.2 Decimal to two’s complement:
3.3 Positive integer
3.4 Negative integer
4 Table of 4-bit two’s complement values
5 Table of 8-bit two’s complement some values

What is two’s complement?

Two’s complement is a method of placing integers so that common math problems become simple to implement.

Two’s complement is a technique that encodes positive and negative numbers in a binary number representation. In case of positive numbers 0 is in the most significant position while in case of negative 1 is in the most significant position.

To get two’s complement of binary number, first convert all the 0’s to 1 and all the 1’s into 0 in other words transpose that binary number or invert that binary number. After inverting the binary number add 1 to LSB (Least significant bit).

How to find two’s complement?

In this section, we will use several examples to depict the calculation process of 2’s complement. However, you can also find 2s complement using online 2’s complement calculator.

Example:

Convert 00110010 into two’s complement?

Solution:

First of all, keep the bits in mind each bit system has different rules.

In the given example we have 8-bits.

Firstly, invert the binary number

00110010 change into 11001101 after inverting

Now add 1 in the LSB.

 

The two’s complement of 00110010 is 11001110.

Example:

Convert 0111 into two’s complement?

Solution:

First of all, keep the bits in mind each bit system has different rules.

In the given example we have 4-bits.

Firstly, invert the binary number

0111 change into 1000 after inverting

Now add 1 in the LSB.

1000 + 1 = 1001

The two’s complement of 0111 is 1001

Example:

Convert 9 to Two’s complement

Solution:

Step 1:

First of all, convert the decimal number into binary number. 

We have

(9)2 = (1001)2

Step 2:

Now invert the binary number

1001 becomes 0110

Step 3:

Now add 1 to LSB

                             0110 + 1 = 0111

The two’s complement of 9 is 0111

Example:

Convert 11 to Two’s complement

Solution:

Step 1:

First of all, convert the decimal number into binary number. 

We have

(11)2 = (1011)2

Step 2:

Now invert the binary number

1011 becomes 0100

Step 3:

Now add 1 to LSB

                             0100 + 1 = 0101

The two’s complement of 11 is 0101.

 

Rules of two’s complement

We generally find the two’s complement of 

  1. Binary
  2. Decimal

 

Binary to two’s complement:

To get two’s complement of binary number, first convert all the 0’s to 1 and all the 1’s into 0 in other words transpose that binary number or invert that binary number. After inverting the binary number add 1 to LSB (Least significant bit).

Example:

Convert 00110010 into two’s complement?

Solution:

First of all, keep the bits in mind each bit system has different rules.

In the given example we have 8-bits.

Firstly, invert the binary number

00110010 change into 11001101 after inverting

Now add 1 in the LSB.

 

The two’s complement of 00110010 is 11001110.

Example:

Convert 0111 into two’s complement?

Solution:

First of all, keep the bits in mind each bit system has different rules.

In the given example we have 4-bits.

Firstly, invert the binary number

0111 change into 1000 after inverting

Now add 1 in the LSB.

1000 + 1 = 1001

The two’s complement of 0111 is 1001

Decimal to two’s complement:

The best way to discover two’s complement conversion is to start with small numbers of bit e.g., let’s start with 4-bits which can represent 16 decimal number range -8 to 7.

  1. Positive integer

Example:

Convert 5 to Two’s complement

Solution:

Step 1:

First of all, convert the decimal number into binary number. 

We have

(5)2 = (0101)2

Step 2:

Now invert the binary number

0101 becomes 1010

Step 3:

Now add 1 to LSB

                             1010 + 1 = 1011

The two’s complement of 5 is 1011

Example:

Convert 7 to Two’s complement

Solution:

Step 1:

First of all, convert the decimal number into binary number. 

We have

(7)2 = (0111)2

Step 2:

Now invert the binary number

0111 becomes 1000

Step 3:

Now add 1 to LSB

                             1000 + 1 = 1001

The two’s complement of 7 is 1001.

  1. Negative integer

Negative number are formed by taking the positive number complementing it (bit-by-bit inversion) and adding 1 to the results

Example:

Convert -7 to Two’s complement

Solution:

Step 1:

First of all, convert the positive number 7 into binary number. 

We have

(7)2 = (0111)2

Step 2:

Now invert the binary number

0111 becomes 1000

Step 3:

Now add 1 to LSB

                             1000 + 1 = 1001

We have the two’s complement of 7 is 1001

Now invert the number

1001 becomes 0110

Now add 1 to LSB

0110 + 1 = 0111

Hence the two’s complement of -7 is 0111

Table of 4-bit two’s complement values

Decimal Two’s Complement
-8 1000
-7 1001
-6 1010
-5 1011
-4 1100
-3 1101
-2 1110
-1 1111
0 0000
1 0001
2 0010
3 0011
4 0100
5 0101
6 0110
7 0111

 

In case of 8-bit we have the range -128 to 127.

In case of 12-bit we have the range -2048 to 2047

In case of 16-bit we have the range of -32,768 to 32,767

 

Table of 8-bit two’s complement some values

Decimal Two’s Complement Decimal Two’s Complement
0 0000 0000
1 1111 1111 -1 0000 0001
2 1111 1110 -2 0000 0010
3 1111 1101 -3 0000 0011
4 1111 1100 -4 0000 0100
5 1111 1011 -5 0000 0101
6 1111 1010 -6 0000 0110
7 1111 1001 -7 0000 0111
8 1111 1000 -8 0000 1000
9 1111 0111 -9 0000 1001
10 1111 0110 -10 0000 1010
11 1111 0101 -11 0000 1011

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