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Incomputing,signed number representations are required to encodenegative numbers in binary number systems.

Inmathematics, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, inRAM or CPUregisters, numbers are represented only as sequences ofbits, without extra symbols. The four best-known methods of extending thebinary numeral system to representsigned numbers are:sign–magnitude,ones' complement,two's complement, andoffset binary. Some of the alternative methods use implicit instead of explicit signs, such as negative binary, using thebase −2. Corresponding methods can be devised forother 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. For integers, the representation used in most current computing devices is two's complement, although theUnisys ClearPath Dorado series mainframes use ones' complement.

The early days of digital computing were marked by 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 top experts expressing very strong and differing opinions.^{[citation needed]} One camp supportedtwo's complement, the system that is dominant today. Another camp supported ones' complement, where a negative value is formed by inverting all of the bits in its positive equivalent. A third group supported sign–magnitude, where a value is changed from positive to negative simply by toggling the word's highest-order bit.

There were arguments for and against each of the systems. Sign–magnitude allowed for easier tracing of memory dumps (a common process in the 1960s) as small numeric values use fewer 1 bits. These systems did ones' complement math internally, 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 were critical. IBM was one of the early supporters of sign–magnitude, with their704,709 and709x series computers being perhaps the best-known systems to use it.

Ones' complement allowed for somewhat simpler hardware designs, as there was no need to convert values when passed to and from the math unit. But it also shared an undesirable characteristic with sign–magnitude: the ability to representnegative zero (−0). Negative zero behaves exactly like positive zero: when used as an operand in any calculation, the result will be the same whether an operand is positive or negative zero. The disadvantage is that the existence of two forms of the same value necessitates two comparisons when checking for equality with zero. Ones' complement subtraction can also result in anend-around borrow (described below). It can be argued that this makes the addition and subtraction logic more complicated or that it makes it simpler, as a subtraction requires simply inverting the bits of the second operand as it is passed to the adder. ThePDP-1,CDC 160 series,CDC 3000 series,CDC 6000 series,UNIVAC 1100 series, andLINC computer use ones' complement representation.

Two's complement is the easiest to implement in hardware, which may be the ultimate reason for its widespread popularity.^[1] Processors on the early mainframes often consisted of thousands of transistors, so eliminating a significant number of transistors was a significant cost savings. Mainframes such as theIBM System/360, theGE-600 series,^[2] and thePDP-6 andPDP-10 use two's complement, as did minicomputers such as thePDP-5 andPDP-8 and thePDP-11 andVAX machines. The architects of the early integrated-circuit-based CPUs (Intel 8080, etc.) also chose to use two's complement math. As IC technology advanced, two's complement technology was adopted in virtually all processors, includingx86,^[3]m68k,Power ISA,^[4]MIPS,SPARC,ARM,Itanium,PA-RISC, andDEC Alpha.

In thesign–magnitude representation, also calledsign-and-magnitude orsigned magnitude, a signed number is represented by the bit pattern corresponding to the sign of the number for thesign bit (often themost significant bit, set to 0 for a positive number and to 1 for a negative number), and the magnitude of the number (orabsolute value) for the remaining bits. For example, in an eight-bitbyte, only seven bits represent the magnitude, which can range from 0000000 (0) to 1111111 (127). Thus numbers ranging from −127₁₀ to +127₁₀ can be represented once the sign bit (the eighth bit) is added. For example, −43₁₀ encoded in an eight-bit byte is10101011 while 43₁₀ is00101011. Using sign–magnitude representation has multiple consequences which makes them more intricate to implement:^[5]

1. There are two ways to represent zero,00000000 (0) and10000000 (−0).
2. Addition and subtraction require different behavior depending on the sign bit, whereas ones' complement can ignore the sign bit and just do an end-around carry, and two's complement can ignore the sign bit and depend on the overflow behavior.
3. Comparison also requires inspecting the sign bit, whereas in two's complement, one can simply subtract the two numbers, and check if the outcome is positive or negative.
4. The minimum negative number is −127, instead of −128 as in the case of two's complement.

This approach is directly comparable to the common way of showing a sign (placing a "+" or "−" next to the number's magnitude). Some early binary computers (e.g.,IBM 7090) use this representation, perhaps because of its natural relation to common usage. Sign–magnitude is the most common way of representing thesignificand infloating-point values.

In theones' complement representation,^[6] a negative number is represented by the bit pattern corresponding to thebitwise NOT (i.e. the "complement") of the positive number. Like sign–magnitude representation, ones' complement has two representations of 0:00000000 (+0) and11111111 (−0).^[7]

As an example, the ones' complement form of00101011 (43₁₀) becomes11010100 (−43₁₀). The range ofsigned numbers using ones' complement is represented by−(2^N−1 − 1) to(2^N−1 − 1) and ±0. A conventional eight-bit byte is −127₁₀ to +127₁₀ with zero being either00000000 (+0) or11111111 (−0).

To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to do anend-around carry: that is, add any resultingcarry back into the resulting sum.^[8] To see why this is necessary, consider the following example showing the case of the addition of −1 (11111110) to +2 (00000010):

    binary    decimal   11111110     −1+  00000010     +2───────────     ── 1 00000000      0   ← Incorrect answer          1     +1   ← Add carry───────────     ──   00000001      1   ← Correct answer

In the previous example, the first binary addition gives00000000, which is incorrect. The correct result (00000001) only appears when the carry is added back in.

A remark on terminology: The system is referred to as "ones' complement" because thenegation of a positive valuex (represented as thebitwise NOT ofx) can also be formed by subtractingx from the ones' complement representation of zero that is a long sequence of ones (−0). Two's complement arithmetic, on the other hand, forms the negation ofx by subtractingx from a single large power of two that iscongruent to +0.^[9] Therefore, ones' complement and two's complement representations of the same negative value will differ by one.

Note that the ones' complement representation of a negative number can be obtained from the sign–magnitude representation merely bybitwise complementing the magnitude (inverting all the bits after the first). For example, the decimal number −125 with its sign–magnitude representation11111101 can be represented in ones' complement form as10000010.

In thetwo's complement representation, a negative number is represented by the bit pattern corresponding to thebitwise NOT (i.e. the "complement") of the positive number plus one, i.e. to the ones' complement plus one. It circumvents the problems of multiple representations of 0 and the need for theend-around carry of the ones' complement representation. This can also be thought of as the most significant bit representing the inverse of its value in an unsigned integer; in an 8-bit unsigned byte, the most significant bit represents the 128ths place, where in two's complement that bit would represent −128.

In two's-complement, there is only one zero, represented as00000000. Negating a number (whether negative or positive) is done by inverting all the bits and then adding one to that result.^[10] This actually reflects thering structure on all integersmodulo2^N:${\displaystyle \mathbb{Z}/2^N\mathbb{Z}}$. Addition of a pair of two's-complement integers is the same as addition of a pair ofunsigned numbers (except for detection ofoverflow, if that is done); the same is true for subtraction and even forN lowest significant bits of a product (value of multiplication). For instance, a two's-complement addition of 127 and −128 gives the same binary bit pattern as an unsigned addition of 127 and 128, as can be seen from the 8-bit two's complement table.

An easier method to get the negation of a number in two's complement is as follows:

Method two:

1. Invert all the bits through the number. This computes the same result as subtracting from negative one.
2. Add one

Example: for +2, which is00000010 in binary (the ~ character is theCbitwise NOT operator, so ~X means "invert all the bits in X"):

1. ~00000010 →11111101
2. 11111101 + 1 →11111110 (−2 in two's complement)

In theoffset binary representation, also calledexcess-K orbiased, a signed number is represented by the bit pattern corresponding to the unsigned number plusK, withK being thebiasing value oroffset. Thus 0 is represented byK, and −K is represented by an all-zero bit pattern. This can be seen as a slight modification and generalization of the aforementioned two's-complement, which is virtually the excess-(2^N−1) representation withnegatedmost significant bit.

Biased representations are now primarily used for the exponent offloating-point numbers. TheIEEE 754 floating-point standard defines the exponent field of asingle-precision (32-bit) number as an 8-bitexcess-127 field. Thedouble-precision (64-bit) exponent field is an 11-bitexcess-1023 field; seeexponent bias. It also had use for binary-coded decimal numbers asexcess-3.

In thebase −2 representation, a signed number is represented using a number system with base −2.

In conventional binary number systems, the base, orradix, is 2; thus the rightmost bit represents 2⁰, the next bit represents 2¹, the next bit 2², and so on. However, a binary number system with base −2 is also possible. The rightmost bit represents(−2)⁰ = +1, the next bit represents(−2)¹ = −2, the next bit(−2)² = +4 and so on, with alternating sign. The numbers that can be represented with four bits are shown in the comparison table below.

The range of numbers that can be represented is asymmetric. If the word has an even number of bits, the magnitude of the largest negative number that can be represented is twice as large as the largest positive number that can be represented, and vice versa if the word has an odd number of bits.

The following table shows the positive and negative integers that can be represented using four bits.

Same table, as viewed from "given these binary bits, what is the number as interpreted by the representation system":

Google'sProtocol Buffers "zig-zag encoding" is a system similar to sign–magnitude, but uses theleast significant bit to represent the sign and has a single representation of zero. This allows avariable-length quantity encoding intended for nonnegative (unsigned) integers to be used efficiently for signed integers.^[11]

A similar method is used in theAdvanced Video Coding/H.264 andHigh Efficiency Video Coding/H.265 video compression standards toextend exponential-Golomb coding to negative numbers. In that extension, theleast significant bit is almost a sign bit; zero has the same least significant bit (0) as all the negative numbers. This choice results in the largest magnitude representable positive number being one higher than the largest magnitude negative number, unlike in two's complement or the Protocol Buffers zig-zag encoding.

Another approach is to give eachdigit a sign, yielding thesigned-digit representation. For instance, in 1726,John Colson advocated reducing expressions to "small numbers", numerals 1, 2, 3, 4, and 5. In 1840,Augustin Cauchy also expressed preference for such modified decimal numbers to reduce errors in computation.

• Balanced ternary
• Binary-coded decimal
• Computer number format
• Method of complements
• Signedness

• Ivan Flores,The Logic of Computer Arithmetic, Prentice-Hall (1963)
• Israel Koren,Computer Arithmetic Algorithms, A.K. Peters (2002),ISBN 1-56881-160-8

• Computer arithmetic

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