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 |

John Paul S. DumoghoThank you so much! I really appreciate this because it helps me a lot to understand directly about the types of complement namely the radix and diminished radix complement.In fact, this is so resourceful.

aishwarya choubeyvery easy to understand and simlified form as well.