Python Bitwise NOT Operator

Python’s bitwise NOT operator ~x inverts each bit from the binary representation of integer x so that 0 becomes 1 and 1 becomes 0. This is semantically the same as calculating ~x == -x-1. For example, the bitwise NOT expression ~0 becomes -1, ~9 becomes -10, and ~32 becomes -33.

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

In this example, you apply the bitwise NOT operator to integer 32:

>>> ~32
-33

The expression ~32 operates on the bit representations "0100000" (decimal 32) and performs bitwise NOT resulting in binary "1011111". This corresponds to the negative decimal number -33.

x0100000
~x1011111

How do you transform the binary “1011111” to a decimal number again? By using the following steps:

  • Flip each bit back to 0100000.
  • Get the corresponding decimal number 32.
  • Increase it by one to 33.
  • Prefix it with the negative symbol -33.

To understand this inverse method from a negative binary to an integer, you need to learn some background first. But don’t worry, it’s just a couple of minutes! ๐Ÿ

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".

๐Ÿ’ก In fact, Python uses signed integers for its bitwise operators. You may ask: what are signed integers?

  • A signed integer, for example using 32 bits, encodes an integer in the range [-2147483648 to 2147483647].
  • An unsigned integer encodes a positive integer in the range [0 to 4294967295]. The signed integer is represented in twos complement notation.

Python Bitwise NOT Operator Example

Here’s the result of the bitwise NOT operator ~x when applied to a couple of example integer operands x:

x (int)x (binary)~x (binary)~x (int)
0'00''11'-1
1'01''10'-2
3'011''100'-4
9'01001''10110'-10
11'01011''10100'-12
256'0100000000''1011111111'-257

You can see those examples in the following Python script:

>>> ~0
-1
>>> ~1
-2
>>> ~3
-4
>>> ~9
-10
>>> ~11
-12
>>> ~256
-257

Let’s use this knowledge in a couple of examples to showcase the working of the bitwise NOT operator on negative integers:

Python Bitwise NOT Examples on Negative Integers

Here’s the result of the bitwise NOT operator ~x when applied to a negative integer operand x:

x (int)~x (int)
-0-1
-10
-32
-98
-1110
-256255

You can see those examples in the following script:

>>> ~-0
-1
>>> ~-1
0
>>> ~-3
2
>>> ~-9
8
>>> ~-11
10
>>> ~-256
255

Python Bitwise NOT Overloading

You can define your own bitwise NOT operator on a custom class by overloading the __invert__ method (dunder method, magic method) with a reference to self as an argument. This allows the expression ~x on your custom objects without raising an error.

Here’s an example:

class Data:
    def __init__(self, data):
        self.data = data

    def __invert__(self):
        return Data(~self.data)


x = Data(3)

res = ~x
print(res.data)
# -4

Note: if you forget to overwrite the __invert__ method and still try to use the expression ~x, Python will raise a TypeError: bad operand type for unary ~. You can fix it by defining the dunder method __invert__(self) in your class definition.

class Data:
    def __init__(self, data):
        self.data = data


x = Data(3)

res = ~x
print(res.data)

Output:

Traceback (most recent call last):
   File "C:\Users\xcent\Desktop\code.py", line 8, in 
     res = ~x
 TypeError: bad operand type for unary ~: 'Data'

To fix this TypeError, simply define the __invert__ method as shown in the previous working example.

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.

OperatorNameDescriptionExample
x = 4, y = 5
&Bitwise ANDPerforms logical AND on a bit-by-bit basisx & y
# b100 & b101 == b100 == 4
|Bitwise ORPerforms logical OR operation on a bit-by-bit basisx | y
# b100 | b101 == b101 == 5
~Bitwise NOTPerforms 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
# -4-1 == -5
^Bitwise XORPerforms logical “exclusive or” operation on a bit-by-bit basisx ^ y
# b100 ^ b101 == b001 == 1
>>Bitwise right shiftShifts binary of left operand to the right by the number of positions specified in right operandx >> 2
# b100 == b010 == b001 == 1
<<Bitwise left shiftShifts binary of left operand to the left by the number of positions specified in right operandx << 2
# b100 == b1000 == b10000 == 16

Here’s a short overview of the Bitwise operators’ magic methods:

Bitwise OperatorMagic “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