All data in computer is represented in binary i.e. in 0 or 1. Computers or machines do not understand our languages, they understand bits. Generally programmer do not care about operations at the bit level. But sometimes a programmer has to dive in a deeper level and work on bits.

# Bits representation

In programming, an n bit integer is stored as a binary number that consists of n bits. So a 32-bit integer consists of 32 bits and 64 bit integer consists of 64 bits. In C++ programming language int data type is 16-bit, 32-bit and 64-bit type. see here

Here is the bit representation of 32 bit int number 10:

00000000000000000000000000001010

In C++, `int`

is either *signed* or *unsigned* and so a bit representation is either *signed* or *unsigned*.

In a signed representation, first bit represents the sign of a number (0 for positive and 1 for negative), and remaining n-1 bits contains the magnitude of the number.

There is a connection between a signed and an unsigned representation. A signed number `-x`

equals an unsigned number 2^{n} – x.

-x (signed) = 2^{n} - x (unsigned)

```
int a = -10;
unsigned int b = a;
std::cout << a << "\n"; /* -10 */
std::cout << b << "\n"; /* 4294967286 */
```

In a signed representation, the next number after 2^{n – 1} – 1 is -2^{n} – 1, and in an unsigned representation, the next number after 2^{n} – 1 is `0`

.

# Bit Operations

We can use & operator to check if a number is even or odd. If `x & 1 = 0`

then `x`

is even and if `x & 1 = 1`

then `x`

is odd. We can also say that, `x`

is divisible by 2^{k} exactly when x & (2^{k} – 1) = 0.

`x<<k`

corresponds to multiplying `x`

by 2^{k}, and `x>>k`

corresponds to dividing `x`

by 2^{k} rounded down to an integer.

# Common Bit Tasks

**Binary representation of unsigned int:**

```
void binary(unsigned int num)
{
for(int i = 256; i > 0; i = i/2) {
if(num & i)
std::cout << "1 ";
else
std::cout << "0 ";
}
std::cout << std::endl;
}
```

**Setting Bit at position:**

```
int set_bit(int num, int position)
{
int mask = 1 << position;
return num | mask;
}
```

**Getting Bit at position:**

```
bool get_bit(int num, int position)
{
bool bit = num & (1 << position);
return bit;
}
```

**Clearing Bit at position:**

```
int clear_bit(int num, int position)
{
int mask = 1 << position;
return num & ~mask;
}
```

# Representing Sets

Bits representation of an integer are 0-indexed and the index starts from right side i.e. least significant bit. So we can represent every subset of the set `{0, 1, 2, ..., n-1}`

as an *n* bit integer and whose bits indicate which element belongs to the subset. If bit at index 3 is 1 and at index 4 is 0 in binary representation of a number, then 3 belongs to the subset and 4 does not belong to the subset.

For a 32-bit integer, the set is {0, 1, 2,…, 31} and the subset is {1, 3, 4, 8}. The binary representation of the set is:
00000000000000000000000100011010

and decimal representation is 2^{8} + 2^{4} + 2^{3} + 2^{1} = 282.

Code to form subset and add elements to it:

```
int add_elements_to_subset()
{
int subset = 0;
subset = subset | (1 << 1);
subset = subset | (1 << 3);
subset = subset | (1 << 4);
subset = subset | (1 << 8);
return subset;
}
```

Code to print elements of the subset:

```
void printing_subset(int subset)
{
for (int i = 0; i < 32; i++)
{
if (subset & (1 << i)) std::cout << i << " ";
}
}
```

# Additional functions

The g++ compiler provides the following functions for counting bits:

•` __builtin_clz(x)`

: the number of zeros at the beginning of the number

• `__builtin_ctz(x)`

: the number of zeros at the end of the number

• `__builtin_popcount(x)`

: the number of ones in the number

• `__builtin_parity(x)`

: the parity (even or odd) of the number of ones

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Reference:

Competitive Programmer’s Handbook - Antti Laaksonen