Question

Let’s say I want to convert “-128” into binary.

From what I understand, I get the binary representation of “128”, invert the bits and then add 1.

So 128 = 10000000

So the “inverse” is 01111111

So and “01111111” + “1” = “10000000” which is “-0” isn’t it?

My textbook makes this seem so easy but I can’t figure out what I’m doing wrong. Thanks for the help.

Answer 1

No, that’s definitely -128 (in two’s complement anyway, which is what you’re talking about given your description of negating numbers). It’s only -0 for the sign/magnitude representation of negative numbers.

See this answer for details on the two representations plus the third one that C allows, one’s complement, but I’ll copy a snippet from there to keep this answer as self-contained as possible.

---

To get the negative representation for a positive number, you:

• invert all bits then add one for two’s complement.
• invert all bits for one’s complement.
• invert just the sign bit for sign/magnitude.

You can see this in the table below:

number | twos complement     | ones complement     | sign/magnitude
=======|=====================|=====================|====================
     5 | 0000 0000 0000 0101 | 0000 0000 0000 0101 | 0000 0000 0000 0101
    -5 | 1111 1111 1111 1011 | 1111 1111 1111 1010 | 1000 0000 0000 0101

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You should be aware that there is no 128 in 8-bit two’s complement numbers, the largest value is 127.

Where the numbers pass the midpoint is where the “clever” stuff happens:

00000000 ->    0
00000001 ->    1
: :
01111110 ->  126
01111111 ->  127
10000000 -> -128
10000001 -> -127
: :
11111110 ->   -2
11111111 ->   -1

because adding the bit pattern of (for example) 100 and -1 with an 8-bit wrap-around will auto-magically give you 99:

100+  0 0110 0100
  1-  0 1111 1111
      ===========
      1 0110 0011  99+ (without that leading 1)