Signed numbers

Binary numbers can be structured as signed or unsigned. For signed numbers or integers, the most significant bit dictates what sign the number is in. This requires a defined size of the binary such as BYTE, WORD, DWORD, and QWORD. A BYTE has a size of 8 bits. A WORD has 16 bits while a DWORD (double WORD) has 32 bits. A QWORD (quad WORD) has 64 bits. Basically, the size doubles as it progresses.

In our example, let's use a BYTE. Identifying a positive binary number is easy. In positive numbers, the most significant bit, or 8^th bit in a byte, is 0. The rest of the bits from 0 to the 7th bit is the actual value. For a negative binary number, the most significant bit is set to 1. However, the value set from 0 to the 7th bit is then calculated for a two's complement value:

01011011b = +91
11011011b = -37
10100101b = -91
00100101b = +37

The "2's complement" of a value is calculated in two steps:

1. Reverse 1s and 0s, so that 1 becomes 0 and 0 becomes 1, for example, 1010b becomes 0101b. This step is called the one's complement.
2. Add 1 to the result of the previous step, for example, 0101b + 1b = 0110b.

To write down the binary equivalent of -63, assuming it is a BYTE, we only take bits 0 to 7:

1. Convert into binary using the previous procedure:

63 = 0111111b

1. Do "1's complement" as follows:

0111111b -> 1000000b

1. Add 1 to the preceding outcome to get the "2's complement" result:

1000000b + 1 = 1000001b

1. Since this is a negative number, set the most significant bit to 1:

11000001b = -63

Here's how to write the decimal of a negative binary number:

1. Take note that the significant bit is 1, and so a negative sign:

10111011b

1. Take the "1's complement," then add 1:

  01000100b
+        1b 
  01000101b

1. Convert the result to decimal, and place the – sign at the beginning, since this is a negative number:

- 01000101b = -69