bitwise negate and bitwise XOR 1 not equivalent?

Question

My discrete math tells me that negate every bit of a binary number is equivalent to XOR it with 1. That is:

~1010 === 0101
1010 XOR 111 = 0101

But this doesn't hold in javascript:

~123 === -124
123 ^ 1 === 122

Why?

The generalization to arbitrary widths is to XOR with -1 (which also works for the single-bit case) — harold, Jan 21 '17 at 21:29
@Boyang because "all bits set", at any length, is the -1 for that length. Proof: adding 1 to it carries through all the bits and results in zero. "all bits set" + 1 = 0, so "all bits set" = -1. — harold, Jan 21 '17 at 21:39
@harold Ok. You can add an answer if you will, and I will accept — Boyang, Jan 21 '17 at 22:04

score 2 · Accepted Answer · answered Jan 21 '17 at 21:22

2

1 is not binary 111

Negating every bit of a number is equivalent to XOR every bit with 1, i.e., with a number of equal size where all bits are 1.

For a single byte, you'd want to xor with binary 11111111 which equals to decimal 255, not 1. Decimal 1 is binary 00000001, so in a bitwise XOR you're flipping only the last bit.

answered Jan 21 '17 at 21:22

Peteris

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Ok. How can I make sure I'm XORing with 11111...? Maybe 123 ^ (2**32)? – Boyang Jan 21 '17 at 21:34
2**32 is 100...000. Usually you'd just write the hex constant 0xFFFFFFFF which is equal to 32 bit 11...11. – Peteris Jan 21 '17 at 21:47

bitwise negate and bitwise XOR 1 not equivalent?

1 Answers1

1 is not binary 111