Complements are used in digital computers for simplifying the subtraction and for logical manipulations. There are two types of complement

- Radix complements ( r complement)
- diminished radix complements (r-1 complement)

For **base 2 **or binary number system the **r’s complement **is **2’s complement **and (**r-1)’s complement **is **1’s complement**

For **base 10 **or decimal system the r’s complement is **10’s complement **and **(r-1)’s complement **is **9’s complement. **

__10’s Complement__

**Let us assume the number 3567890, for finding the 10’s complement, there is a rule**

3 | 5 | 6 | 7 | 8 | 9 | 0 |

Most significant Digit | Least Significant Digit |

10’s complement can be obtained by

- leaving all least significant 0’s unchanged
- subtracting the first non zero least significant digit from 10
- and subtracting all higher significant digits from 9

For the above example, The 10’s complement is

Given number | 3 | 5 | 6 | 7 | 8 | 9 | 0 |

Process | 9-3 | 9-5 | 9-6 | 9-7 | 9-8 | 10-9 | unchanged |

10’s Complement | 6 | 4 | 3 | 2 | 1 | 1 | 0 |

so 10’s complement of **3567890 **is **6432110**

__9’s Complement__

9’s complement is a diminished radix complement and can be easily found out by subtracting all the given digits by 9.

For example, the 9’s complement of 3567890 is

9999999 – 3567890 = 6432109

given number | 3 | 5 | 6 | 7 | 8 | 9 | 0 |

Process | 9-3 | 9-5 | 9-6 | 9-7 | 9-8 | 9-9 | 9-0 |

9’s complement | 6 | 4 | 3 | 2 | 1 | 0 | 9 |

In short,

10’s Complement = 9’s Complement + 1 |

__2’s Complement__

For binary numbers, there is 2’s complement and 1’s complement

2’s complement can be obtained by

- leaving the least significant 0’s unchanged and the first 1 unchanged
- replacing 1’s with 0’s and 0’s with 1’s in all other higher significant digits

given number | 1 | 1 | 0 | 1 | 1 | 0 | 0 |

process | 0 | 0 | 1 | 0 | unchanged | unchanged | unchanged |

2’s complement | 0 | 0 | 1 | 0 | 1 | 0 | 0 |

ie.

2’s complement of **1101100 is 0010100**

__1’s complement__

Finding 1’s complement is just to replace all 1’s by 0’s and all 0’s by 1’s

**1’s complement of 1101100 is 0010011**

in short

2’s complement = 1’s complement + 1 |