Three Ideas of Signed Binary Numbers
Big trilemma!
In early years of digital computing, the computer industry faced a challenge - how to implement signed binary numbers. Competing ideas emerged both at the hardware and mathematical levels. Scientists have split into three camps. The first camp implemented the idea of ones' complement, the second - the idea of two's complement (this idea is the main one today), the third - the idea of sign-with-magnitude.
Comparison Number Systems
8-bit numbers representation
| Binary Value | Unsigned | Sign and Magnitude | ones' complement | Two's complement |
|---|---|---|---|---|
| 00000000 | 0 | +0 | +0 | 0 |
| 00000001 | 1 | 1 | 1 | 1 |
| .. | .. | .. | .. | .. |
| 01111110 | 126 | 126 | 126 | 126 |
| 01111111 | 127 | 127 | 127 | 127 |
| 10000000 | 128 | -0 | -127 | -128 |
| 10000001 | 129 | -1 | -126 | -127 |
| 10000010 | 130 | -2 | -125 | -126 |
| .. | .. | .. | .. | .. |
| 11111110 | 254 | -126 | -1 | -2 |
| 11111111 | 255 | -127 | -0 | -1 |
Signed number representations (Wikipedia)