## Binary Fractions

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In computingsigned number representations are required to encode negative numbers in binary number systems. However, in computer hardwarenumbers are represented only as sequences of bitswithout extra symbols. The four best-known methods of extending the binary numeral system to represent signed numbers are: Corresponding methods can be devised for other baseswhether 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 Binary representation of negative fractional numbers 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.

Another camp supported ones' complement, where any positive value is made into its negative equivalent by inverting all of the bits in a word. There were arguments for and against each of the systems. 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 binary representation of negative fractional numbers to the register.

IBM was one of the early supporters of sign-magnitude, with theirand x 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 binary representation of negative fractional numbers and from the math unit. 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, however, is that the existence of two forms of the same value necessitates two rather than a single comparison when checking for equality with zero.

Ones' complement subtraction can also result in an end-around borrow described below. Two's complement is the easiest to implement in hardware, which may be the ultimate reason for its widespread popularity. The architects of the early integrated circuit-based CPUs Inteletc.

As IC technology binary representation of negative fractional numbers, virtually all adopted two's complement technology. This representation is also called "signâ€”magnitude" or "sign and magnitude" representation.

In this approach, the problem of representing a number's sign can be to allocate one sign bit to represent the sign: The remaining bits in the number indicate the magnitude or absolute value. Hence, in a byte with only seven bits apart from the sign bitthe magnitude can range from 0 to Some early binary computers e. Signed magnitude is the most common way of representing the significand in floating point values.

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, i. Like sign-and-magnitude representation, ones' complement has two representations of 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: In the previous example, the first binary addition giveswhich is incorrect.

The correct result only appears when the carry is added back in. A remark on terminology: Note that the ones' complement representation of a negative number can be obtained from the sign-magnitude representation merely by bitwise complementing the magnitude. 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 binary representation of negative fractional numbers greater in an unsigned sense than the ones' complement of the positive value.

In two's-complement, there is only one zero, represented as Negating a number whether negative or positive is done by inverting all the bits and then adding one 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 overflowif that is done ; the same is true for subtraction and binary representation of negative fractional numbers for N lowest significant bits of a product value of multiplication.

Offset binaryalso called excess- K or biased representationuses a pre-specified number K as a biasing value. A value is represented by the unsigned number which is K greater than the intended value.

Biased representations are now primarily used for the exponent of floating-point numbers. The IEEE floating-point standard defines the exponent field of a single-precision bit number as an 8-bit excess field.

The double-precision bit exponent field is an bit excess field; see exponent bias. It also had use for binary-coded decimal numbers as excess In conventional binary binary representation of negative fractional numbers systems, the base, or radixis 2; thus the rightmost bit represents 2 0the next bit represents 2 1the next bit 2 2and so on.

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's Protocol Buffers "zig-zag encoding" is a system similar to sign-and-magnitude, but uses the least significant bit to represent the sign and has a single representation of zero.

This allows a variable-length quantity encoding intended for nonnegative unsigned integers to be used efficiently for signed integers. Another approach is to give each digit a binary representation of negative fractional numbers, yielding the signed-digit representation. For instance, inJohn Colson advocated reducing expressions to "small numbers", numerals 1, 2, 3, 4, and 5. InAugustin Cauchy also expressed preference for such modified decimal numbers to reduce errors in computation.

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