Python invert operator - visualizing the negative number

Question

In simple words, the definition given for ~n is -n - 1.

For example ~1

 1 = 0001
~1 = 1110 (which is -2) 

For even numbers

say ~2

 2 = 0010
~2 = 1101 (which is the representation of -3 in twos complement)

But the question is

1110 = -2 can be easily visualized as -2 (the right two bits are 10 and the rest all 1)
1101 = -3 can't be visualized like this (going by the above logic it should be -5)

So I am wondering is there a simple way to see and tell from twos complement binary what the negative number represents without doing much calculations .

Answer 1

~n = -n - 1 is equivalent to -n = ~n + 1. That means that to figure out what the negative of a number is, you can invert it (in your head) and add one. Pretend zeros are ones and vice versa, then add one.

Example: Pretend this

1101

is this

0010

then add 1

0011

Thus, 1101 represents -3.

Answer 2

The definition comes because of 2's complement representation of integers. So "really" ~n is n with all the bits flipped. But the result of flipping "all" the bits depends in a sense on how many bits n has in the first place. CPython uses fixed-width integers internally but the language doesn't present them to the programmer, so the only definition that makes sense in general is the arithmetic one ~n = -n - 1. But the motivation for that definition is flipping the bits of a fixed-width 2's complement integer.

1110 = ~2 can be easily visualized as -2

...1110 is not ~2, it is -2. ~2 is ...1101 because 2 is ...010.

1101 = ~3 can't be visualized like this

...1101 is not ~3, it is -3. ~3 is ...1100 because 3 is ...011.

The way I visualize this (when I visualize it at all -- as a mathematician by training I prefer not to consider specific numbers), is to know that in 2's complement, ...10... is always a negated power of two. So ...10 is -2, ...100 is -4, etc.

Then in order to know what for example ...110110 is, it's ...110000 + 110, that is to say -16 + 6, which is -10.

Of course ...110110 is also (by bit flipping) ~1001, that is to say ~9, which by the formula is -9-1, which is also -10. So the system works ;-)