Signed Number Representation
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In
computing Computing is any goal-oriented activity requiring, benefiting from, or creating computing machinery. It includes the study and experimentation of algorithmic processes, and development of both hardware and software. Computing has scientific, e ...
, signed number representations are required to encode
negative number In mathematics, a negative number represents an opposite. In the real number system, a negative number is a number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed m ...
s in binary number systems. In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, negative numbers in any base are represented by prefixing them with a minus sign ("−"). However, in
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or CPU registers, numbers are represented only as sequences of
bit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s, without extra symbols. The four best-known methods of extending the
binary numeral system A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method of mathematical expression which uses only two symbols: typically "0" (zero) and "1" ( one). The base-2 numeral system is a positional notatio ...
to represent
signed number In mathematics, the sign of a real number is its property of being either positive, negative, or zero. Depending on local conventions, zero may be considered as being neither positive nor negative (having no sign or a unique third sign), or it ...
s are: sign–magnitude,
ones' complement The ones' complement of a binary number is the value obtained by inverting all the bits in the binary representation of the number (swapping 0s and 1s). The name "ones' complement" (''note this is possessive of the plural "ones", not of a sing ...
,
two's complement Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
, and
offset binary Offset binary, also referred to as excess-K, excess-''N'', excess-e, excess code or biased representation, is a method for signed number representation where a signed number n is represented by the bit pattern corresponding to the unsigned numbe ...
. 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. For integers, the representation used in most current computing devices is two's complement, although the Unisys ClearPath Dorado series mainframes use ones' complement.


History

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. One camp supported
two's complement Two's complement is a mathematical operation to reversibly convert a positive binary number into a negative binary number with equivalent (but negative) value, using the binary digit with the greatest place value (the leftmost bit in big- endian ...
, 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 systemsa key concern when the cost and packaging of discrete transistors were critical. IBM was one of the early supporters of sign–magnitude, with their
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,
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and 709x 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 represent
negative zero Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...
(−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 an
end-around borrow The ones' complement of a binary number is the value obtained by inverting all the bits in the binary representation of the number (swapping 0s and 1s). The name "ones' complement" (''note this is possessive of the plural "ones", not of a sin ...
(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. The
PDP-1 The PDP-1 (''Programmed Data Processor-1'') is the first computer in Digital Equipment Corporation's PDP series and was first produced in 1959. It is famous for being the computer most important in the creation of hacker culture at Massachusetts ...
,
CDC 160 series The CDC 160 series was a series of minicomputers built by Control Data Corporation. The CDC 160 and CDC 160-A were 12-bit minicomputers built from 1960 to 1965; the CDC 160G was a 13-bit minicomputer, with an extended version of the CDC 160-A ins ...
,
CDC 3000 The CDC 3000 series ("thirty-six hundred" of "thirty-one hundred") computers from Control Data Corporation were mid-1960s follow-ons to the CDC 1604 and CDC 924 systems. Over time, a range of machines were produced - divided into * the 48-bit upp ...
series,
CDC 6000 series The CDC 6000 series is a discontinued family of mainframe computers manufactured by Control Data Corporation in the 1960s. It consisted of the CDC 6200, CDC 6300, CDC 6400, CDC 6500, CDC 6600 and CDC 6700 computers, which were all extremely rapid a ...
,
UNIVAC 1100 The UNIVAC 1100/2200 series is a series of compatible 36-bit computer systems, beginning with the UNIVAC 1107 in 1962, initially made by Sperry Rand. The series continues to be supported today by Unisys Corporation as the ClearPath Dorado Series. ...
series, and
LINC The LINC (Laboratory INstrument Computer) is a 12-bit, 2048-word transistorized computer. The LINC is considered by some the first minicomputer and a forerunner to the personal computer. Originally named the "Linc", suggesting the project's o ...
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. 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 the
IBM System/360 The IBM System/360 (S/360) is a family of mainframe computer systems that was announced by IBM on April 7, 1964, and delivered between 1965 and 1978. It was the first family of computers designed to cover both commercial and scientific applica ...
, the
GE-600 series The GE-600 series was a family of 36-bit mainframe computers originating in the 1960s, built by General Electric (GE). When GE left the mainframe business the line was sold to Honeywell, which built similar systems into the 1990s as the division ...
, and the
PDP-6 The PDP-6, short for Programmed Data Processor model 6, is a computer developed by Digital Equipment Corporation (DEC) during 1963 and first delivered in the summer of 1964. It was an expansion of DEC's existing 18-bit systems to use a 36-bit da ...
and
PDP-10 Digital Equipment Corporation (DEC)'s PDP-10, later marketed as the DECsystem-10, is a mainframe computer family manufactured beginning in 1966 and discontinued in 1983. 1970s models and beyond were marketed under the DECsystem-10 name, especi ...
use two's complement, as did minicomputers such as the
PDP-5 The PDP-5 was Digital Equipment Corporation's first 12-bit computer, introduced in 1963. History An earlier 12-bit computer, named LINC has been described as the first minicomputer and also "the first modern personal computer." It had 2,048 12 ...
and
PDP-8 The PDP-8 is a 12-bit computing, 12-bit minicomputer that was produced by Digital Equipment Corporation, Digital Equipment Corporation (DEC). It was the first commercially successful minicomputer, with over 50,000 units being sold over the model's ...
and the
PDP-11 The PDP-11 is a series of 16-bit minicomputers sold by Digital Equipment Corporation (DEC) from 1970 into the 1990s, one of a set of products in the Programmed Data Processor (PDP) series. In total, around 600,000 PDP-11s of all models were sold, ...
and
VAX VAX (an acronym for Virtual Address eXtension) is a series of computers featuring a 32-bit instruction set architecture (ISA) and virtual memory that was developed and sold by Digital Equipment Corporation (DEC) in the late 20th century. The V ...
machines. The architects of the early integrated-circuit-based CPUs (
Intel 8080 The Intel 8080 (''"eighty-eighty"'') is the second 8-bit microprocessor designed and manufactured by Intel. It first appeared in April 1974 and is an extended and enhanced variant of the earlier 8008 design, although without binary compatibil ...
, etc.) also chose to use two's complement math. As IC technology advanced, two's complement technology was adopted in virtually all processors, including
x86 x86 (also known as 80x86 or the 8086 family) is a family of complex instruction set computer (CISC) instruction set architectures initially developed by Intel based on the Intel 8086 microprocessor and its 8088 variant. The 8086 was introd ...
,
m68k The Motorola 68000 series (also known as 680x0, m68000, m68k, or 68k) is a family of 32-bit complex instruction set computer (CISC) microprocessors. During the 1980s and early 1990s, they were popular in personal computers and workstations and w ...
,
Power ISA Power ISA is a reduced instruction set computer (RISC) instruction set architecture (ISA) currently developed by the OpenPOWER Foundation, led by IBM. It was originally developed by IBM and the now-defunct Power.org industry group. Power IS ...
, MIPS,
SPARC SPARC (Scalable Processor Architecture) is a reduced instruction set computer (RISC) instruction set architecture originally developed by Sun Microsystems. Its design was strongly influenced by the experimental Berkeley RISC system developed ...
,
ARM In human anatomy, the arm refers to the upper limb in common usage, although academically the term specifically means the upper arm between the glenohumeral joint (shoulder joint) and the elbow joint. The distal part of the upper limb between th ...
,
Itanium Itanium ( ) is a discontinued family of 64-bit Intel microprocessors that implement the Intel Itanium architecture (formerly called IA-64). Launched in June 2001, Intel marketed the processors for enterprise servers and high-performance computin ...
,
PA-RISC PA-RISC is an instruction set architecture (ISA) developed by Hewlett-Packard. As the name implies, it is a reduced instruction set computer (RISC) architecture, where the PA stands for Precision Architecture. The design is also referred to as ...
, and
DEC Alpha Alpha (original name Alpha AXP) is a 64-bit reduced instruction set computer (RISC) instruction set architecture (ISA) developed by Digital Equipment Corporation (DEC). Alpha was designed to replace 32-bit VAX complex instruction set computers ...
.


Sign–magnitude

In the ''sign–magnitude'' representation, also called ''sign-and-magnitude'' or ''signed magnitude'', a signed number is represented by the bit pattern corresponding to the sign of the number for the
sign bit In computer science, the sign bit is a bit in a signed number representation that indicates the sign of a number. Although only signed numeric data types have a sign bit, it is invariably located in the most significant bit position, so the term ...
(often the
most significant bit In computing, bit numbering is the convention used to identify the bit positions in a binary number. Bit significance and indexing In computing, the least significant bit (LSB) is the bit position in a binary integer representing the binary 1 ...
, set to 0 for a positive number and to 1 for a negative number), and the magnitude of the number (or
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
) for the remaining bits. For example, in an eight-bit
byte The byte is a unit of digital information that most commonly consists of eight bits. Historically, the byte was the number of bits used to encode a single character of text in a computer and for this reason it is the smallest addressable unit ...
, 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 is 10101011 while 4310 is 00101011. Using sign–magnitude representation has multiple consequences which makes them more intricate to implement: # There are two ways to represent zero, 00000000 (0) and 10000000 (
−0 Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...
). # Addition and subtraction require different behavior depending on the sign bit, whereas one's 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. # Comparison also require 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. # The minimum negative number is −127 instead of −128 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 The IBM 7090 is a second-generation transistorized version of the earlier IBM 709 vacuum tube mainframe computer that was designed for "large-scale scientific and technological applications". The 7090 is the fourth member of the IBM 700/7000 ser ...
) use this representation, perhaps because of its natural relation to common usage. Sign–magnitude is the most common way of representing the
significand The significand (also mantissa or coefficient, sometimes also argument, or ambiguously fraction or characteristic) is part of a number in scientific notation or in floating-point representation, consisting of its significant digits. Depending on ...
in
floating-point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
values.


Ones' complement

In the ''ones' complement'' representation, a negative number is represented by the bit pattern corresponding to the
bitwise NOT In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
(i.e. the "complement") of the positive number. Like sign–magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 (
−0 Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. However, in computing, some number representations allow for the existence of two zeros, often denoted by ...
). As an example, the ones' complement form of 00101011 (4310) becomes 11010100 (−4310). The range of signed numbers using ones' complement is represented by to and ±0. A conventional eight-bit byte is −12710 to +12710 with zero being either 00000000 (+0) or 11111111 (−0). To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to do an ''end-around carry'': that is, add any resulting
carry Carry or carrying may refer to: People *Carry (name) Finance * Carried interest (or carry), the share of profits in an investment fund paid to the fund manager * Carry (investment), a financial term: the carry of an asset is the gain or cost of h ...
back into the resulting sum. 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   ← Not the correct answer
          1     +1   ← Add carry
───────────     ──
   00000001      1   ← Correct answer
In the previous example, the first binary addition gives 00000000, 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 the
negation In logic, negation, also called the logical complement, is an operation that takes a proposition P to another proposition "not P", written \neg P, \mathord P or \overline. It is interpreted intuitively as being true when P is false, and false ...
of a positive value x (represented as the
bitwise NOT In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
of x) can also be formed by subtracting x 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 of x by subtracting x from a single large power of two that is
congruent Congruence may refer to: Mathematics * Congruence (geometry), being the same size and shape * Congruence or congruence relation, in abstract algebra, an equivalence relation on an algebraic structure that is compatible with the structure * In mod ...
to +0. 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 by bitwise complementing the magnitude (inverting all the bits after the first). For example, the decimal number −125 with its sign–magnitude representation 11111101 can be represented in ones' complement form as 10000010.


Two's complement

In the ''two's complement'' representation, a negative number is represented by the bit pattern corresponding to the
bitwise NOT In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
(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 the
end-around carry 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 RAM or CPU regist ...
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 as 00000000. Negating a number (whether negative or positive) is done by inverting all the bits and then adding one to that result. This actually reflects the
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structure on all integers modulo 2''N'': \mathbb/2^N\mathbb. Addition of a pair of two's-complement integers is the same as addition of a pair of unsigned numbers (except for detection of overflow, if that is done); the same is true for subtraction and even for ''N'' 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: # Invert all the bits through the number # Add one Example: for +2, which is 00000010 in binary (the ~ character is the C
bitwise NOT In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operat ...
operator, so ~X means "invert all the bits in X"): # ~00000010 → 11111101 # 11111101 + 1 → 11111110 (−2 in two's complement)


Offset binary

In the ''offset binary'' representation, also called ''excess-K'' or ''biased'', a signed number is represented by the bit pattern corresponding to the unsigned number plus K, with K being the ''biasing value'' or ''offset''. Thus 0 is represented by K, 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 representation with negated
most significant bit In computing, bit numbering is the convention used to identify the bit positions in a binary number. Bit significance and indexing In computing, the least significant bit (LSB) is the bit position in a binary integer representing the binary 1 ...
. Biased representations are now primarily used for the exponent of
floating-point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can b ...
numbers. The IEEE 754 floating-point standard defines the exponent field of a
single-precision Single-precision floating-point format (sometimes called FP32 or float32) is a computer number format, usually occupying 32 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. A floatin ...
(32-bit) number as an 8-bit excess-127 field. The
double-precision Double-precision floating-point format (sometimes called FP64 or float64) is a floating-point number format, usually occupying 64 bits in computer memory; it represents a wide dynamic range of numeric values by using a floating radix point. Flo ...
(64-bit) exponent field is an 11-bit excess-1023 field; see
exponent bias In IEEE 754 floating-point numbers, the exponent is biased in the engineering sense of the word – the value stored is offset from the actual value by the exponent bias, also called a biased exponent. Biasing is done because exponents have to be ...
. It also had use for binary-coded decimal numbers as
excess-3 Excess-3, 3-excess or 10-excess-3 binary code (often abbreviated as XS-3, 3XS or X3), shifted binary or Stibitz code (after George Stibitz, who built a relay-based adding machine in 1937) is a self-complementary binary-coded decimal (BCD) cod ...
.


Base −2

In the ''base −2'' representation, a signed number is represented using a number system with base −2. In conventional binary number systems, the base, or
radix In a positional numeral system, the radix or base is the number of unique digits, including the digit zero, used to represent numbers. For example, for the decimal/denary system (the most common system in use today) the radix (base number) is t ...
, 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 , the next bit represents , the next bit 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.


Comparison table

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":


Other systems

Google's
Protocol Buffers Protocol Buffers (Protobuf) is a free and open-source cross-platform data format used to serialize structured data. It is useful in developing programs to communicate with each other over a network or for storing data. The method involves an i ...
"zig-zag encoding" is a system similar to sign–magnitude, but uses the
least significant bit In computing, bit numbering is the convention used to identify the bit positions in a binary number. Bit significance and indexing In computing, the least significant bit (LSB) is the bit position in a binary integer representing the binary 1 ...
to represent the sign and has a single representation of zero. This allows a
variable-length quantity A variable-length quantity (VLQ) is a universal code that uses an arbitrary number of binary octets (eight-bit bytes) to represent an arbitrarily large integer. A VLQ is essentially a base-128 representation of an unsigned integer with the additio ...
encoding intended for nonnegative (unsigned) integers to be used efficiently for signed integers.Protocol Buffers: Signed Integers
/ref> A similar method is used in the Advanced Video Coding/H.264 and High Efficiency Video Coding/H.265 video compression standards to extend exponential-Golomb coding to negative numbers. In that extension, the
least significant bit In computing, bit numbering is the convention used to identify the bit positions in a binary number. Bit significance and indexing In computing, the least significant bit (LSB) is the bit position in a binary integer representing the binary 1 ...
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 each digit a sign, yielding the
signed-digit representation In mathematical notation for numbers, a signed-digit representation is a positional numeral system with a set of signed digits used to encode the integers. Signed-digit representation can be used to accomplish fast addition of integers because ...
. For instance, in 1726,
John Colson John Colson (1680 – 20 January 1760) was an English clergyman, mathematician, and the Lucasian Professor of Mathematics at Cambridge University. Life John Colson was educated at Lichfield School before becoming an undergraduate at Christ Chu ...
advocated reducing expressions to "small numbers", numerals 1, 2, 3, 4, and 5. In 1840,
Augustin Cauchy Baron Augustin-Louis Cauchy (, ; ; 21 August 178923 May 1857) was a French mathematician, engineer, and physicist who made pioneering contributions to several branches of mathematics, including mathematical analysis and continuum mechanics. He w ...
also expressed preference for such modified decimal numbers to reduce errors in computation.


See also

*
Balanced ternary Balanced ternary is a ternary numeral system (i.e. base 3 with three digits) that uses a balanced signed-digit representation of the integers in which the digits have the values −1, 0, and 1. This stands in contrast to the standard (unbalance ...
*
Binary-coded decimal In computing and electronic systems, binary-coded decimal (BCD) is a class of binary encodings of decimal numbers where each digit is represented by a fixed number of bits, usually four or eight. Sometimes, special bit patterns are used for ...
*
Computer number format A computer number format is the internal representation of numeric values in digital device hardware and software, such as in programmable computers and calculators. Numerical values are stored as groupings of bits, such as bytes and words. The e ...
*
Method of complements In mathematics and computing, the method of complements is a technique to encode a symmetric range of positive and negative integers in a way that they can use the same algorithm (hardware) for addition throughout the whole range. For a given num ...
*
Signedness In computing, signedness is a property of data types representing numbers in computer programs. A numeric variable is ''signed'' if it can represent both positive and negative numbers, and ''unsigned'' if it can only represent non-negative numbers ...


References

* Ivan Flores, ''The Logic of Computer Arithmetic'', Prentice-Hall (1963) * Israel Koren, ''Computer Arithmetic Algorithms'', A.K. Peters (2002), {{DEFAULTSORT:Signed Number Representations Computer arithmetic ca:Representació de nombres amb signe cs:Dvojková soustava#Zobrazení záporných čísel fr:Système binaire#Représentation des entiers négatifs