Why does ~True result in -2?

In Python console:

~True

Gives me:

-2

Why? Can someone explain this particular case to me in binary?

Solution 1:

int(True) is 1.

1 is:

00000001

and ~1 is:

11111110

Which is -2 in Two's complement¹

¹ Flip all the bits, add 1 to the resulting number and interpret the result as a binary representation of the magnitude and add a negative sign (since the number begins with 1):

11111110 → 00000001 → 00000010 
         ↑          ↑ 
       Flip       Add 1

Which is 2, but the sign is negative since the MSB is 1.

Worth mentioning:

Think about bool, you'll find that it's numeric in nature - It has two values, True and False, and they are just "customized" versions of the integers 1 and 0 that only print themselves differently. They are subclasses of the integer type int.

So they behave exactly as 1 and 0, except that bool redefines str and repr to display them differently.

>>> type(True)
<class 'bool'>
>>> isinstance(True, int)
True

>>> True == 1
True
>>> True is 1  # they're still different objects
False

Solution 2:

The Python bool type is a subclass of int (for historical reasons; booleans were only added in Python 2.3).

Since int(True) is 1, ~True is ~1 is -2.

See PEP 285 for why bool is a subclass of int.

If you wanted the boolean inverse, use not:

>>> not True
False
>>> not False
True

If you wanted to know why ~1 is -2, it's because you are inverting all bits in a signed integer; 00000001 becomes 1111110 which in a signed integer is a negative number, see Two's complement:

>>> # Python 3
...
>>> import struct
>>> format(struct.pack('b', 1)[0], '08b')
'00000001'
>>> format(struct.pack('b', ~1)[0], '08b')
'11111110'

where the initial 1 bit means the value is negative, and the rest of the bits encode the inverse of the positive number minus one.

Solution 3:

~True == -2 is not surprising if True means 1 and ~ means bitwise inversion...

...provided that

• True can be treated as an integer and
• integers are represented in Two's complement

Edits:

• fixed the mixing between integer representation and bitwise inversion operator
• applied another polishing (the shorter the message, the more work needed)