Complement and its types. Two types are 1's complement and 2's complement.

## Complement

### Complement types

Two types-
1. 1's complement
2. 2's complement

#### 1's Complement

In a binary number changing all the zeros to ones and all the ones to zeros, we will get a number that is called 1's  Complement.

That means,

If we replace 1 instead of 0 and 0 instead of 1 in a binary number then we will get 1' s  Complement.

101011  ← Given Binary number
↓↓↓↓↓↓
010100  ←1's  Complement

#### 2's Complement

If we add 1 with the 1's  Complement of a binary number then we will get a number. That is 2's  Complement number.

Here, if we add 1 with this 1's  Complement number then we will get 2's  Complement.

101011 ← Given Binary number010100 ← 1's  Complement
+        1
ㅡㅡㅡ
010101 ← 2's  Complement

#### Importance of 2's Complement

1. The subtraction process can be completed by using additive processes.
2. A simple logical circuit building becomes easier.
3. By using this addition and subtraction process can be completed by the circuit.

Example: What will be the 2's complement of -209?

Solution:

Binary of  +209 = 11010001

[If we see here is less than 8 bit then we have to add Zeros in left. In this case, we don't need it.]

Now,

For +209

1 1 0 1 0 0 0 1(Real value)
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
0 0 1 0 1 1 1 0(1's complement)

+                   1
ㅡㅡㅡㅡㅡㅡ
0 0 1 0 1 1 1 1(2's complement)

#### Subtraction using 2's complement

By 2's complementary addition process we can do subtraction. Here from which number we are subtracting is called partitive and which number we are subtracting from that number is called minus

We have to take these steps for finding the subtraction -

1. We have to add Zeros making the number 8 bit for the 8-bit register subtraction of both numbers.
2. We have to find the 2's complement of the 2nd number(minus).
3. Then we have to add this number with the first number as like binary addition process.
4. Extra carry of sign bit will not be considered. That mean we have to skip the extra bit.

Example: Subtract (1101101)₂ from (110110₂) by using the method of 2's complement.

Solution:

Here ,

1st number(Partitive) = 01101101

2nd number (Minus) = 00110110

Now from 2nd number,

0 0 1 1 0 1 1 0  (Real value)
↓ ↓ ↓ ↓ ↓ ↓ ↓ ↓
1 1 0 0 1 0 0 1  (1's complement)
+                    1
ㅡㅡㅡㅡㅡㅡ
1 1 0 0 1 0 1 0  (2's complement)

0 1 1 0 1 1 0 1 = 0 1 1 0 1 1 0 1
0 0 1 1 0 1 1 0 = 1 1 0 0 1 0 1 0  [2's complement]
ーーーーーーーーーーーーーー
0 0 1 1 0 1 1 1 [After adding]
↗️