How does the bitwise complement (~) operator work?

Question

Why is it that ~2 is -3?

Answer 1

~ flips the bits in the value.

Why ~2 is -3 has to do with how numbers are represented bitwise. Numbers are represented as two's complement.

So, 2 is the binary value

00000010

And ~2 flips the bits so the value is now:

11111101

Which, is the binary representation of -3.

Answer 2

Remember that digits are stored in two's complement. As an example, here's the representation of -2 in two's complement: (8 bits)

1111 1110

The way you get this is by taking the binary representation of a number, taking it's complement (inverting all the bits) and adding one. Two starts as 0000 0010, and by inverting the bits we get 1111 1101. Adding one gets us the result above. The first bit is the sign bit, implying a negative.

So let's take a look at how we get ~2 = -3:

Here's two again:

0000 0010

Simply flip all the bits and we get:

1111 1101

Well, what's -3 look like in two's complement? Start with positive 3: 0000 0011, flip all the bits to 1111 1100, and add one, 1111 1101.

So if you simply invert the bits in 2, you get the two's complement representation of -3.

The complement operator (~) JUST FLIPS BITS. It is up to the machine to interpret these bits.

Answer 3

As others mentioned ~ just flipped bits (changes one to zero and zero to one) and since two's complement is used you get the result you saw.

One thing to add is why two's complement is used, this is so that the operations on negative numbers will be the same as on positive numbers. Think of -3 as the number to which 3 should be added in order to get zero and you'll see that this number is 1101, remember that binary addition is just like elementary school (decimal) addition only you carry one when you get to two rather than 10.

 1101 +
 0011 // 3
    =
10000
    =
 0000 // lose carry bit because integers have a constant number of bits.

Therefore 1101 is -3, flip the bits you get 0010 which is two.

Answer 4

This operation is a complement, not a negation.

Consider that ~0 = -1, and work from there.

The algorithm for negation is, "complement, increment".

Did you know? There is also "one's complement" where the inverse numbers are symmetrical, and it has both a 0 and a -0.