Bitwise operators

A bit (short for binary digit) is the smallest unit of data in computing. It can hold one of two possible values: 0 or 1.

If we have 2 bits, each bit can independently be 0 or 1. This gives $2 \times 2 = 2^2 = 4$ possible combinations: 00, 01, 10 and 11.

In general, with N bits, each bit can take 2 values, so there are:

Bitwise operators work on integers at the level of individual bits. Unlike logical operators (and, or) which treat values as simply True or False, bitwise operators examine each bit position independently.

Bitwise AND (&)

The & operator compares each pair of bits and returns 1 only if both bits are 1:

Bitwise OR (|)

The | operator compares each pair of bits and returns 1 if at least one bit is 1:

print(5 and 3)  # Output: 3 (both truthy, returns the second)
print(5 & 3)    # Output: 1 (compares bits)

Bitwise AND example: 12 & 10

First, convert both numbers to binary:

• $12$ in binary is 1100
• $10$ in binary is 1010

Then apply AND to each pair of bits:

The result 1000 in binary equals $8$ in decimal.

print(12 & 10)  # Output: 8

Bitwise OR example: 12 | 10

Using the same binary representations:

The result 1110 in binary equals $14$ in decimal.

print(12 | 10)  # Output: 14

Bitwise XOR (^)

The ^ operator (exclusive or) compares each pair of bits and returns 1 if exactly one bit is 1:

XOR is sometimes described as "one or the other, but not both."

Bitwise NOT (~)

The ~ operator inverts all bits in a number. However, due to how Python represents negative numbers internally, the result follows this pattern:

For example:

Bitwise XOR example: 12 ^ 10

Convert both numbers to binary:

• $12$ in binary is 1100
• $10$ in binary is 1010

Apply XOR to each pair of bits:

The result 0110 in binary equals $6$ in decimal.

print(12 ^ 10)  # Output: 6

Bitwise NOT example: ~5

Using the pattern $\sim x = -(x + 1)$:

print(~5)  # Output: -6

The shift operators move all bits in a number left or right by a specified number of positions.

Left shift (<<)

The << operator shifts bits to the left, filling empty positions on the right with 0s. Each left shift by one position multiplies the number by $2$:

Right shift (>>)

The >> operator shifts bits to the right, discarding bits that fall off the end. Each right shift by one position floor divides the number by $2$:

Left shift example: 5 << 2

$5$ in binary is 101. Shifting left by 2 positions:

Using the formula:

print(5 << 2)  # Output: 20

Right shift example: 20 >> 2

$20$ in binary is 10100. Shifting right by 2 positions:

Using the formula:

print(20 >> 2)  # Output: 5