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Nine and Ten's Complement

I am using Kumar for this.
- But the wikipedia article is ok.
The problem is, subtraction is hard.
- If you look, you can find the logic for a subtractor.
- So it is not THAT hard.
But also, remember, we had limited transistors
- So if we can reuse something like the adder, life would be better.
- AND WE CAN!
It turns out, we can perform perform subtraction by changing the representation and performing addition.
There are two related methods we will discuss in relationship to decimal: 9's complement and 10's complement.
Later we will discover some encoding methods that make it easy to use these methods
And we will discuss 1's complement and 2's complement, which will simplify binary subtraction.
The 9's complement of a number is found by subtracting each digit from 9
- Digit 9's Compliment
  
  0 9
  
  1 8
  
  2 7
  
  3 6
  
  4 5
  
  5 4
  
  6 3
  
  7 2
  
  8 1
  
  9 0
- Find the 9's complement of 354
```
 999
-354
====
 645
```
- Note there will never be a "borrow"

Digit	9's Compliment
0	9
1	8
2	7
3	6
4	5
5	4
6	3
7	2
8	1
9	0

To subtract,

Form the 9's complement of the second number 
Add the two numbers
If there is a carry out
   Add it to the LSB of the sum
Else
   The result is negative, find the 9's complement of the sum and add a - sign

Let's try a few simple ones


  9 - 3 =>  9 + 6 = 15 => 6  
  5 - 6 =>  5 + 3 => 08 => -1   The 9's complement of 8 is 1
  7 - 7 =>  7 + 2 = 09 => 0


  521   =>  521             521  =>  521
- 167      +832            -634     +365
           ----                     ----
           1353 => 354              0886 => (-) 113

do 1000-999

If the second number is not large enough, pad with 0.

 200  => 200 =>  200
-  1    -001    +998 
                ----
                1198 => 199

In this method, we always have to add numbers the same size, but that is a good thing in a computer!

With 10's complement there are a few changes.

To find the 10's complement
- Find the 9's complement and add 1

Subtraction

If the carry out is 0, the sum is negative, take the 10's complement.
Otherwise the result is fine

Find the 10's complement of

125 => 874 + 1 => 875    Test    125
                                +875
                                ----
                                1000 => 000 

5298 =>  4701 + 1 => 4702  Test  5298
                                +4702
                                -----
                                10000

Let's try some subtraction

  269              269
- 128 => 871+1 => +872
                   ---
                  1141 => 141

  3421 =>       => 3421 
- 4237   5762+1   +5763
                  -----
                   9184 => 0815+1 => -816

With 9's complement if there is a carry, take the 9's complement
With 10's complement, if there is no carry take the 10's complement