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# 2â€™s complement

Just like 1â€™s complement, 2â€™s complement is also used to represent the signed binary numbers. For finding 2â€™s complement of the binary number, we will first find the 1â€™s complement of the binary number and then add 1 to the least significant bit of it.

For example, if we want to calculate the 2â€™s complement of the number 1011001, then firstly, we find the 1â€™s complement of the number that is 0100110 and add 1 to the LSB. So, by adding 1 to the LSB, the number will be (0100110)+1=0100111. We can also create the logic circuit using OR, AND, and NOT gates. The logic circuit for finding 2â€™s complement of the 5-bit binary number is as follows:

**Example 1: 110100**

For finding 2â€™s complement of the given number, change all 0â€™s to 1 and all 1â€™s to 0. So the 1â€™s complement of the number 110100 is 001011. Now add 1 to the LSB of this number, i.e., (001011)+1=001100.

**Example 2: 100110**

For finding 1â€™s complement of the given number, change all 0â€™s to 1 and all 1â€™s to 0. So, the 1â€™s complement of the number 100110 is 011001. Now add one the LSB of this number, i.e., (011001)+1=011010.

### 2â€™s Complement Table

Binary Number | 1â€™s Complement | 2â€™s complement |
---|---|---|

0000 | 1111 | 0000 |

0001 | 1110 | 1111 |

0010 | 1101 | 1110 |

0011 | 1100 | 1101 |

0100 | 1011 | 1100 |

0101 | 1010 | 1011 |

0110 | 1001 | 1010 |

0111 | 1000 | 1001 |

1000 | 0111 | 1000 |

1001 | 0110 | 0111 |

1010 | 0101 | 0110 |

1011 | 0100 | 0101 |

1100 | 0011 | 0100 |

1101 | 0010 | 0011 |

1110 | 0001 | 0010 |

1111 | 0000 | 0001 |

### Use of 2â€™s complement

2â€™s complement is used for representing signed numbers and performing arithmetic operations such as subtraction, addition, etc. The positive number is simply represented as a magnitude form. So there is nothing to do for representing positive numbers. But if we represent the negative number, then we have to choose either 1â€™s complement or 2â€™s complement technique. The 1â€™s complement is an ambiguous technique, and 2â€™s complement is an unambiguous technique. Letâ€™s see an example to understand how we can calculate the 2â€™s complement in signed binary number representation.

**Example 1: +6 and -6 **

The number +6 is represented as same as the binary number. For representing both numbers, take the 5-bit register.

So the +6 is represented in the 5-bit register as 0 0110.

The -6 is represented in the 5-bit register in the following way:

- +6=0 0110
- Now, find the 1â€™s complement of the number 0 0110, i.e. 1 1001.
- Now, add 1 to its LSB. When we add 1 to the LSB of 11001, the newly generated number comes out 11010. Here, the sign bit is one which means the number is the negative number.

**Example 2: +120 and -120 **

The number +120 is represented as same as the binary number. For representing both numbers, take the 8-bit register.

So the +120 is represented in the 8-bit register as 0 1111000.

The -120 is represented in the 8-bit register in the following way:

- +120=0 1111000
- Now, find the 1â€™s complement of the number 0 1111000, i.e. 1 0000111. Here, the MSB denotes the number is the negative number.
- Now, add 1 to its LSB. When we add 1 to the LSB of 1 0000111, the newly generated number comes out 1 0001000. Here, the sign bit is one, which means the number is the negative number.