In mathematics, negative numbers in any base are represented in the usual way, by prefixing them with a "−" sign. However, on a computer, there are various ways of representing a number's sign. This article deals with four methods of extending the binary numeral system to represent signed numbers: sign-and-magnitude, ones' complement, two's complement, and excess-N. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... The NASA Columbia Supercomputer. ... The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, usually 0 and 1. ... A negative number is a number that is less than zero, such as −3. ... The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...

For most purposes, modern computers typically use the two's-complement representation, but other representations are used in some circumstances.

Sign-and-magnitude

One may first approach this problem of representing a number's sign by allocating one sign bit to represent the sign: set that bit (often the most significant bit) to 0 for a positive number, and set to 1 for a negative number. The remaining bits in the number indicate the magnitude (or absolute value). Hence in a byte with only 7 bits (apart from the sign bit), the magnitude can range from 0000000 (0) to 1111111 (127). Thus you can represent numbers from −127₁₀ to +127₁₀ once you add the sign bit (the eighth bit). A consequence of this representation is that there are two ways to represent 0, 00000000 (0) and 10000000 (−0). Decimal −43 encoded in an eight-bit byte this way is 10101011. In computer science the sign bit is the bit in a computer numbering format which indicates the sign of the number. ... This article is about the unit of information. ... The binary representation of decimal 149, with the MSB highlighted. ... In mathematics, the absolute value (or modulus[1]) of a real number is its numerical value without regard to its sign. ... In computer science a byte is a unit of measurement of information storage, most often consisting of eight bits. ... −0 is the representation of negative zero, a number that exists in computing in some signed number representations for integers and in most floating point number representations. ...

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) used this representation, perhaps because of its natural relation to common usage. (Many decimal computers also used sign-and-magnitude.) IBM 7090 console The IBM 7090 was a second-generation transistorized version of the earlier IBM 709 vacuum tube mainframe computers and was designed for large-scale scientific and technological applications. The 7090 was the third member of the IBM 700/7000 series scientific computers. ... Decimal computers represent numbers and/or addresses in decimal, and provide instructions to operate on those numbers and/or addresses directly; examples of encoding used are BCD, ASCII, and EBCDIC. Many early computers, for example the ENIAC, IBM 650, IBM 1401, IBM 1620, IBM NORC, IBM 7070, and UNIVAC 1...

Ones' complement

Alternatively, a system known as ones' complement can be used to represent negative numbers. The ones' complement form of a negative binary number is the bitwise NOT applied to it — the complement of its positive counterpart. Like sign-and-magnitude representation, ones' complement has two representations of 0: 00000000 (+0) and 11111111 (−0). In computer programming, a bitwise operation operates on one or two bit patterns or binary numerals at the level of their individual bits. ... Negation (i. ... Look up Complement in Wiktionary, the free dictionary. ... −0 is the representation of negative zero, a number that exists in computing in some signed number representations for integers and in most floating point number representations. ...

As an example, the ones' complement form of 00101011 (43) becomes 11010100 (−43). The range of signed numbers using ones' complement in a conventional eight-bit byte is −127₁₀ to +127₁₀.

To add two numbers represented in this system, one does a conventional binary addition, but it is then necessary to add any resulting carry 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 binary addition alone gives 00000000—not the correct answer! Only when the carry is added back in does the correct result (00000001) appear.

This numeric representation system was common in older computers; the PDP-1 and UNIVAC 1100/2200 series, among many others, used ones'-complement arithmetic. The PDP-1 (Programmed Data Processor-1) was the first computer in Digital Equipments PDP series and was first produced in 1960. ... 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. ...

(A remark on terminology: The system is referred to as "ones' complement" because the negation of x is formed by subtracting x from a long string of ones. Two's complement arithmetic, on the other hand, forms the negation of x by subtracting x from a single large power of two.^[1])

The IPv4 header checksum uses ones' complement arithmetic, here even on two's complement machines the inconvenience of having to add back a carry is a desirable error-checking property, because "it is equally sensitive to errors in all bit positions"^[2]. In the UDP protocol one of the two representations of 0 provided by the ones' complement arithmetic is used for indicating that the optional checksum feature has been omitted (all 0s), whereas the other (all 1s) indicates a checksumming result of 0^[3]. Internet Protocol version 4 is the fourth iteration of the Internet Protocol (IP) and it is the first version of the protocol to be widely deployed. ... A checksum is a form of redundancy check, a simple way to protect the integrity of data by detecting errors in data that are sent through space (telecommunications) or time (storage). ... The abbreviation UDP can refer to: User Datagram Protocol Usenet Death Penalty Ulster Democratic Party Uridine-diphosphate, cf. ...

Note that the ones' complement representation of a number can be obtained from the sign-magnitude representation merely by bitwise complementing the magnitude.

Two's complement

Main article: Two's complement

The problems of multiple representations of 0 and the need for the end-around carry are circumvented by a system called two's complement. In two's complement, negative numbers are represented by the bit pattern which is one greater (in an unsigned sense) than the ones' complement of the positive value. The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...

In two's-complement, there is only one zero (00000000). Negating a number (whether negative or positive) is done by inverting all the bits and then adding 1 to that result. 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). 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 above table. In computing, signedness is a property of variables representing numbers in computer programs. ...

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

Excess-N

Excess-N, also called biased representation, uses a pre-specified number N as a biasing value. A value is represented by the unsigned number which is N greater than the intended value. Thus 0 is represented by N, and −N is represented by the all-zeros bit pattern.

This is a representation that is now primarily used within floating-point numbers. The IEEE floating-point standard defines the exponent field of a single-precision (32-bit) number as an 8-bit Excess-127 field. The double-precision (64-bit) exponent field is an 11-bit Excess-1023 field. This article or section is in need of attention from an expert on the subject. ... The IEEE Standard for Binary Floating-Point Arithmetic (IEEE 754) is the most widely-used standard for floating-point computation, and is followed by many CPU and FPU implementations. ... In mathematics, exponentiation is a process generalized from repeated multiplication, in much the same way that multiplication is a process generalized from repeated addition. ... It has been suggested that this article or section be merged into IEEE floating-point standard. ... In computing, double precision is a computer numbering format that occupies two storage locations in computer memory at address and address+1. ...

See also

• Excess-3
• two's complement

Excess-3 binary coded decimal (XS-3) is a numeral system used in some old computers. ... The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...

Base −2

In conventional binary number systems, the base, or radix, 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 −2⁰=+1, the next bit represents −2¹=−2, the next bit −2²=+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.

Comparison table

The following table shows the positive and negative integers that can be represented using 4 bits.

See also

• Computer numbering formats
• Excess-3
• Method of complements
• Two's complement

The term computer numbering formats refers to the schemes implemented in digital computer and calculator hardware and software to represent numbers. ... Excess-3 binary coded decimal (XS-3) is a numeral system used in some old computers. ... In mathematics and computing, the method of complements is a technique used to subtract one number from another using only addition of positive numbers. ... The twos complement of a binary number is the value obtained by subtracting the number from a large power of two (specifically, from 2N for an N-bit twos complement). ...

References

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

1. ^ Donald Knuth: The Art of Computer Programming, Volume 2: Seminumerical Algorithms, chapter 4.1
2. ^ R Braden et al., Computing the Internet Checksum, RFC 1071, [[1]]
3. ^ RFC 768