The Python ** bitwise right-shift** operator

`x >> n`

shifts the binary representation of integer `x`

by `n`

positions to the right. It inserts a `0`

bit on the left and removes the right-most bit. For example, if you right-shift the binary representation `0101`

by one position, you’d obtain `0010`

. Semantically, the bitwise right-shift operator is the same as performing integer division by `2**n`

.Here’s a minimal example:

print(8 >> 1) # 4 print(8 >> 2) # 2 print(-3 >> 1) # -2

Let’s dive deeper into the details next!

## Video Explainer

As you go over the article, you can watch my explainer video here:

## Example

In this example, you apply the ** bitwise right-shift **operator to integer 32 shifting it by one position:

x = 32 # Shift by one position to the right res = x >> 1 print(res) # 16 # Shift by two positions to the right res = x >> 2 print(res) # 8

The bit representation of decimal `32`

is `"00100000"`

. If you shift it by one position to the right, you obtain binary `"`

(decimal 16). If you shift by two positions to the right, you obtain binary `00010000`

"`"`

(decimal 8). Here’s the tabular explanation:`00001000`

"

| 0 | 0 | 1 | 0 | 0 | 0 | 0 | 0 |

| 0 | 0 | 0 | 1 | 0 | 0 | 0 | 0 |

| 0 | 0 | 0 | 0 | 1 | 0 | 0 | 0 |

Each row represents the resulting shifted binary representation of the original integer 32.

## Representing Negative Integers in Binaries

Python uses so-called ** complementary binaries** to represent negative integers. The first bit of a complementary binary is the sign (0: positive, 1: negative). All remaining bits encode the number. You write a negative number

`-x`

as the bit pattern for `(x-1)`

and flip all bits from 1 to 0 and from 0 to 1 (complement).Here are two simple examples:

- To represent
`x = -1`

using 8 bits you first calculate`(1-1) = 0`

and then flip all bits to calculate`"11111111"`

. - To represent
`x = -10`

using 8 bits you first calculate`(10-1) = 9`

which is`"00001001"`

in binary format. Then, you complement all bits to determine the negative (complementary) binary`"11110110"`

.

Let’s use this knowledge in a couple of examples to showcase the working of the *bitwise XOR operator* on negative integers:

## Python Bitwise Right Shift on Negative Integers

Here’s the result of the *bitwise right-shift* operator `x >> n`

when applied to example negative integer operands `x`

and `n`

:

(int) | (int) | (binary) | (binary) | (int) |
---|---|---|---|---|

-1 | 1 | 11111111 | 11111111 | -1 |

-2 | 1 | 11111110 | 11111111 | -1 |

-4 | 1 | 11111100 | 11111110 | -2 |

-4 | 2 | 11111100 | 11111111 | -1 |

You can see those examples in the following script:

print(-1 >> 1) # -1 print(-2 >> 1) # -1 print(-4 >> 1) # -2 print(-4 >> 2) # -1

## How to Resolve ValueError: negative shift count?

You cannot use a negative shift count, i.e., in the expression `x >> n`

, the second operand n must be non-negative. It can be zero. If you use a negative shift count, Python raises the “`ValueError: negative shift count`

“. To resolve it, use the left-shift operation `x << n`

instead of using a negative shift count.

Here’s an example of the `ValueError`

:

>>> 2 >> -1 Traceback (most recent call last): File "<pyshell#19>", line 1, in <module> 2 >> -1 ValueError: negative shift count

And here’s an example of how to resolve it using right-shift instead of left-shift operation:

>>> 2 << 1 4

## Python Bitwise Right-Shift Operator Overloading

To enable the right-shift operator on your custom object, use Python’s ** operator overloading** functionality. Overloading works through what is called

*magic methods*or

*dunder methods*(for “

**d**ouble-

**under**score methods”)

*.*For the right-shift operator, the magic method is the

`__rshift__(self, other)`

method. It should return a new custom object that is the result of the bitwise operation.Here’s a short overview of the Bitwise operators’ magic methods:

Bitwise Operator | Magic “Dunder” Method |
---|---|

`&` | `__and__(self, other)` |

`|` | `__or__(self, other)` |

`^` | `__xor__(self, other)` |

`~` | `__invert__(self)` |

`<<` | `__lshift__(self, other)` |

`>>` | `__rshift__(self, other)` |

Here’s an example of how to accomplish these bitwise operators on a custom class `Data`

. We marked this respective operator in the code:

class Data: def __init__(self, data): self.data = data def __and__(self, other): return Data(self.data & other.data) def __or__(self, other): return Data(self.data | other.data) def __xor__(self, other): return Data(self.data ^ other.data) def __invert__(self): return Data(~self.data) def __lshift__(self, other): return Data(self.data << other.data) def __rshift__(self, other): return Data(self.data >> other.data) x = 2 y = 3 print('Operands: \n', 'x =', x, '\n', 'y =', y) print() print('Bitwise AND: ', x & y) print('Bitwise OR: ', x | y) print('Bitwise XOR: ', x ^ y) print('Bitwise NOT: ', ~x) print('Bitwise LEFT-SHIFT: ', x << y) print('Bitwise RIGHT-SHIFT: ', x >> y)

The output is:

Operands: x = 2 y = 3 Bitwise AND: 2 Bitwise OR: 3 Bitwise XOR: 1 Bitwise NOT: -3 Bitwise LEFT-SHIFT: 16 Bitwise RIGHT-SHIFT: 0

## Bitwise Operators

Bitwise operators perform operations on the binary (bit) representation of integers. The following table gives a short overview of all existing bitwise operators. Note that we also provide the binary representation `100`

for the decimal integer `4`

, and `101`

for the decimal integer `5`

as a comment in the right column.

Operator | Name | Description | Example |
---|---|---|---|

`x = 4, y = 5` | |||

& | Bitwise AND | Performs logical AND on a bit-by-bit basis | `x & y` |

| | Bitwise OR | Performs logical OR operation on a bit-by-bit basis | `x | y` |

~ | Bitwise NOT | Performs logical NOT on a bit-by-bit basis, inverting each bit so that 0 becomes 1 and 1 becomes 0. Same as `-x-1` . | `~x` |

^ | Bitwise XOR | Performs logical “exclusive or” operation on a bit-by-bit basis | `x ^ y` |

>> | Bitwise right shift | Shifts binary of left operand to the right by the number of positions specified in right operand | `x >> 2` |

<< | Bitwise left shift | Shifts binary of left operand to the left by the number of positions specified in right operand | `x << 2` |

While working as a researcher in distributed systems, Dr. Christian Mayer found his love for teaching computer science students.

To help students reach higher levels of Python success, he founded the programming education website Finxter.com that has taught exponential skills to millions of coders worldwide. He’s the author of the best-selling programming books Python One-Liners (NoStarch 2020), The Art of Clean Code (NoStarch 2022), and The Book of Dash (NoStarch 2022). Chris also coauthored the Coffee Break Python series of self-published books. He’s a computer science enthusiast, freelancer, and owner of one of the top 10 largest Python blogs worldwide.

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