In C++, bitwise operators perform operations on integer data at the individual bit-level. These operations include testing, setting, or shifting the actual bits. For example,

```
a & b;
a | b;
```

Here is a list of 6 bitwise operators included in C++.

Operator | Description |
---|---|

`&` |
Bitwise AND Operator |

`|` |
Bitwise OR Operator |

`^` |
Bitwise XOR Operator |

`~` |
Bitwise Complement Operator |

`<<` |
Bitwise Shift Left Operator |

`>>` |
Bitwise Shift Right Operator |

These operators are necessary because the Arithmetic-Logic Unit (ALU) present in the computer's CPU carries out arithmetic operations at the bit-level.

**Note:** Bitwise operators can only be used alongside `char`

and `int`

data types.

## 1. C++ Bitwise AND Operator

The **bitwise AND** `&`

operator returns **1** if and only if both the operands are **1**. Otherwise, it returns **0**.

The following table demonstrates the working of the **bitwise AND** operator. Let **a** and **b** be two operands that can only take binary values i.e. **1 and 0**.

a | b | a & b |
---|---|---|

0 | 0 | 0 |

0 | 1 | 0 |

1 | 0 | 0 |

1 | 1 | 1 |

**Note:** The table above is known as the "Truth Table" for the **bitwise AND** operator.

Let's take a look at the **bitwise AND** operation of two integers 12 and 25:

12 = 00001100 (In Binary) 25 = 00011001 (In Binary) //Bitwise AND Operation of 12 and 25 00001100 & 00011001 _________ 00001000 = 8 (In decimal)

### Example 1: Bitwise AND

```
#include <iostream>
using namespace std;
int main() {
// declare variables
int a = 12, b = 25;
cout << "a = " << a << endl;
cout << "b = " << b << endl;
cout << "a & b = " << (a & b) << endl;
return 0;
}
```

**Output**

a = 12 b = 25 a & b = 8

In the above example, we have declared two variables `a` and `b`. Here, notice the line,

`cout << "a & b = " << (a & b) << endl;`

Here, we are performing **bitwise AND** between variables `a` and `b`.

## 2. C++ Bitwise OR Operator

The **bitwise OR** `|`

operator returns **1** if at least one of the operands is **1**. Otherwise, it returns **0**.

The following truth table demonstrates the working of the **bitwise OR** operator. Let **a** and **b** be two operands that can only take binary values i.e. **1 or 0**.

a | b | a | b |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 1 |

Let us look at the **bitwise OR** operation of two integers **12** and **25**:

12 = 00001100 (In Binary) 25 = 00011001 (In Binary) Bitwise OR Operation of 12 and 25 00001100 | 00011001 _________ 00011101 = 29 (In decimal)

### Example 2: Bitwise OR

```
#include <iostream>
int main() {
int a = 12, b = 25;
cout << "a = " << a << endl;
cout << "b = " << b << endl;
cout << "a | b = " << (a | b) << endl;
return 0;
}
```

**Output**

a = 12 b = 25 a | b = 29

The **bitwise OR** of `a = 12`

and `b = 25`

gives `29`

.

## 3. C++ Bitwise XOR Operator

The **bitwise XOR** `^`

operator returns **1** if and only if one of the operands is **1**. However, if both the operands are **0**, or if both are **1**, then the result is **0**.

The following truth table demonstrates the working of the **bitwise XOR** operator. Let **a** and **b** be two operands that can only take binary values i.e. **1 or 0**.

a | b | a ^ b |
---|---|---|

0 | 0 | 0 |

0 | 1 | 1 |

1 | 0 | 1 |

1 | 1 | 0 |

Let us look at the **bitwise XOR** operation of two integers 12 and 25:

12 = 00001100 (In Binary) 25 = 00011001 (In Binary) Bitwise XOR Operation of 12 and 25 00001100 ^ 00011001 _________ 00010101 = 21 (In decimal)

### Example 3: Bitwise XOR

```
#include <iostream>
int main() {
int a = 12, b = 25;
cout << "a = " << a << endl;
cout << "b = " << b << endl;
cout << "a ^ b = " << (a ^ b) << endl;
return 0;
}
```

**Output**

a = 12 b = 25 a ^ b = 21

The **bitwise XOR** of `a = 12`

and `b = 25`

gives `21`

.

## 4. C++ Bitwise Complement Operator

The bitwise complement operator is a unary operator (works on only one operand). It is denoted by `~`

that changes binary digits **1** to **0** and **0** to **1**.

It is important to note that the **bitwise complement** of any integer **N** is equal to **-(N + 1)**. For example,

Consider an integer **35**. As per the rule, the bitwise complement of **35** should be **-(35 + 1) = -36**. Now, let's see if we get the correct answer or not.

35 = 00100011 (In Binary) // Using bitwise complement operator ~ 00100011 __________ 11011100

In the above example, we get that the bitwise complement of **00100011** (**35**) is **11011100**. Here, if we convert the result into decimal we get **220**.

However, it is important to note that we cannot directly convert the result into decimal and get the desired output. This is because the binary result **11011100** is also equivalent to **-36**.

To understand this we first need to calculate the binary output of **-36**. We use 2's complement to calculate the binary of negative integers.

### 2's Complement

The 2's complement of a number **N** gives **-N**.

In binary arithmetic, 1's complement changes **0 to 1** and **1 to 0**.

And, if we add **1** to the result of the 1's complement, we get the 2's complement of the original number.

For example,

36 = 00100100 (In Binary) 1's Complement = 11011011 2's Complement : 11011011 + 1 _________ 11011100

Here, we can see the 2's complement of **36** (i.e. **-36**) is **11011100**. This value is equivalent to the **bitwise complement of 35** that we have calculated in the previous section.

Hence, we can say that the bitwise complement of 35 = -36.

### Example 4: Bitwise Complement

```
#include <iostream>
int main() {
int num1 = 35;
int num2 = -150;
cout << "~(" << num1 << ") = " << (~num1) << endl;
cout << "~(" << num2 << ") = " << (~num2) << endl;
return 0;
}
```

**Output**

~(35) = -36 ~(-150) = 149

In the above example, we declared two integer variables `num1` and `num2`, and initialized them with the values of `35`

and `-150`

respectively.

We then computed their bitwise complement with the codes `(~num1)`

and `(~num2)`

respectively and displayed them on the screen.

The bitwise complement of 35 = - (35 + 1) = -36 i.e. ~35 = -36 The bitwise complement of -150 = - (-150 + 1) = - (-149) = 149 i.e. ~(-150) = 149

This is exactly what we got in the output.

## C++ Shift Operators

There are two shift operators in C++ programming:

- Right shift operator
`>>`

- Left shift operator
`<<`

### 5. C++ Right Shift Operator

The **right shift operator** shifts all bits towards the right by a certain number of **specified bits**. It is denoted by `>>`

.

When we shift any number to the right, the **least significant bits** are discarded, while the **most significant bits** are replaced by zeroes.

As we can see from the image above, we have a **4-bit number**. When we perform a **one-bit** right shift operation on it, each individual bit is shifted to the right by 1 bit.

As a result, the right-most bit is discarded, while the left-most bit remains vacant. This vacancy is replaced by a **0**.

### 6. C++ Left Shift Operator

The **left shift operator** shifts all bits towards the left by a certain number of **specified bits**. It is denoted by `<<`

.

As we can see from the image above, we have a **4-bit number**. When we perform a **1 bit** left shift operation on it, each individual bit is shifted to the left by 1 bit.

As a result, the left-most bit is discarded, while the right-most bit remains vacant. This vacancy is replaced by a **0**.

### Example 5: Shift Operators

```
#include <iostream>
int main() {
// declaring two integer variables
int num = 212, i;
// Shift Right Operation
cout << "Shift Right:" << endl;
// Using for loop for shifting num right from 0 bit to 3 bits
for (i = 0; i < 4; i++) {
cout << "212 >> " << i << " = " << (212 >> i) << endl;
}
// Shift Left Operation
cout << "\nShift Left:" << endl;
// Using for loop for shifting num left from 0 bit to 3 bits
for (i = 0; i < 4; i++) {
cout << "212 << " << i << " = " << (212 << i) << endl;
}
return 0;
}
```

**Output**

Shift Right: 212 >> 0 = 212 212 >> 1 = 106 212 >> 2 = 53 212 >> 3 = 26 Shift Left: 212 << 0 = 212 212 << 1 = 424 212 << 2 = 848 212 << 3 = 1696

From the output of the program above, we can infer that, for any number **N**, the results of the shift right operator are:

N >> 0 = N N >> 1 = (N >> 0) / 2 N >> 2 = (N >> 1) / 2 N >> 3 = (N >> 2) / 2

and so on.

Similarly, the results of the shift left operator are:

N << 0 = N N << 1 = (N << 0) * 2 N << 2 = (N << 1) * 2 N << 3 = (N << 2) * 2

and so on.

Hence we can conclude that,

N >> m = [ N >> (m-1) ] / 2 N << m = [ N << (m-1) ] * 2

### Bitwise Shift in Actual Practice

In the above example, note that the `int`

data type stores numbers in **32-bits** i.e. an `int`

value is represented by **32 binary digits**.

However, our explanation for the bitwise shift operators used numbers represented in **4-bits**.

For example, the base-10 number **13** can be represented in 4-bit and 32-bit as:

4-bit Representation of 13 = 1101 32-bit Representation of 13 = 00000000 00000000 00000000 00001101

As a result, the **bitwise left-shift** operation for **13** (and any other number) can be different depending on the number of bits they are represented by.

Because in **32-bit** representation, there are many more bits that can be shifted left when compared to **4-bit** representation.