Before knowing about **9’s complement** and **10’s complement** we should know why they are used and why their concept came into existence. The complements are used to make the arithmetic operations in digital system easier. In this article we will discuss about the following topics

**9’s complement****10’s complement****9’s complement subtraction****10’s complement subtraction**

Now first of all let us know what 9’s complement is and how it is done. To obtain the 9’s complement of any number we have to subtract the number with (10^{n} – 1) where n = number of digits in the number, or in a simpler manner we have to divide each digit of the given decimal number with 9. The table given below will explain the 9’s complement more easily.

Decimal digit | 9s complement |

0 | 9 |

1 | 8 |

2 | 7 |

3 | 6 |

4 | 5 |

5 | 4 |

6 | 3 |

7 | 2 |

8 | 1 |

9 | 0 |

Now coming to **10’s complement**, it is relatively easy to find out the 10’s complement after finding out the 9’s complement of that number. We have to add 1 with the **9’s complement** of any number to obtain the desired 10’s complement of that number. Or if we want to find out the 10’s complement directly, we can do it by following the following formula, (10^{n} – number), where n = number of digits in the number. An example is given below to illustrate the concept of obtaining 10’s complement.

Let us take a decimal number 456, 9’s complement of this number will be

10’s complement of this no

9’s complement subtraction

We will understand this method of subtraction via an example

A = 215

B = 155

We want to find out A-B by 9’s complement subtraction method

First we have to find out 9’s complement of B

Now we have to add **9’s complement** of B to A

The left most bit of the result is called carry and is added back to the part of the result without it

Another different type of example is given

A = 4567

B = 1234

We need to find out A – B

9’s complement of B

8765

Adding 9’s complement of B with A

Adding the carry with the result we get

3333

Now the answer is – 3333

NB: If there is no carry the answer will be – (9’s complement of the answer)

Subtraction by 10’s complement

Again we will show the procedure by an example

Taking the same data

A = 215

B = 155

10’s complement of B = 845

Adding 10’s complement of B to A

In this case the carry is omitted

The answer is 60

Taking the other example

A = 4567

B = 1234

10’s complement of B = 8766

Adding 10’s complement of B with A

To get the answer the carry is ignored

So, the answer is – 3333

NB: If there is a carry then the answer is – (10’s complement of the sum obtained)