Signed number representation

In computing, signed number representations are required to encode negative numbers in binary number systems.

In mathematics, negative numbers in any base are represented by prefixing them with a minus ("−") sign. However, in computer hardware, numbers are represented only as sequences of bits, without extra symbols. The four best-known methods of extending the binary numeral system to represent signed numbers are: sign-and-magnitude, ones' complement, two's complement, and excess-K. Some of the alternative methods use implicit instead of explicit signs, such as negative binary, using the base −2. Corresponding methods can be devised for other bases, whether positive, negative, fractional, or other elaborations on such themes.

There is no definitive criterion by which any of the representations is universally superior. The representation used in most current computing devices is two's complement, although the Unisys ClearPath Dorado series mainframes use ones' complement.

The early days of digital computing were marked by a lot of competing ideas about both hardware technology and mathematics technology (numbering systems). One of the great debates was the format of negative numbers, with some of the era's most expert people having very strong and different opinions. One camp supported two's complement, the system that is dominant today. Another camp supported ones' complement, where any positive value is made into its negative equivalent by inverting all of the bits in a word. A third group supported "sign & magnitude" (sign-magnitude), where a value is changed from positive to negative simply by toggling the word's sign (high-order) bit.

There were arguments for and against each of the systems. Sign & magnitude allowed for easier tracing of memory dumps (a common process 40 years ago) as small numeric values use fewer 1 bits. Internally, these systems did ones' complement math so numbers would have to be converted to ones' complement values when they were transmitted from a register to the math unit and then converted back to sign-magnitude when the result was transmitted back to the register. The electronics required more gates than the other systems – a key concern when the cost and packaging of discrete transistors was critical. IBM was one of the early supporters of sign-magnitude, with their 704, 709 and 709x series computers being perhaps the best known systems to use it.

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