Floating-Point Representation
For modern electronic computers, mathematical computations are carried out within the arithmetic logic unit (ALU) of its central processing unit (CPU). The numbers stored and manipulated by the computer can be represented in either a fixed-point representation or a floating-point representation. Numbers expressed in fixed-point notation are allotted a fixed number of digits on either side of the decimal point (the "decimal point" is in actuality a binary point for use within the computer because modern computers use the binary number system). A simple example of a fixed-point format is "XX.XX," where the "X" characters represent digits; in this example there are two places represented on each side of the decimal point. A representation of a decimal number in that format is "87.01." However, the number "567.89" could not be represented in the fixed-point format used for this illustration because the "hundreds" component is not allowed (i.e., only the truncated "67.89" would appear). The nature of the fixed-point representation of numbers restricts the range of values that are possible, but it does allow faster arithmetic processing.
A floating-point representation is a compact way of expressing numbers over a large range of magnitudes, from extremely large to extremely small. The representation of numbers in floating-point notation has utility in a variety of computer applications, ranging from scientific research to computer graphics. However, as compared to fixed-point arithmetic, floating-point arithmetic is more complicated and (unless special provisions are made for its implementation) can take considerably longer for the CPU to perform. Early electronic computers did not generally have hardware dedicated to performing floating-point arithmetic. (The IBM 704, introduced during the late 1950s, was probably the first general-purpose, commercial computer to implement hardware designed specifically to carry out floating-point operations.) As modern electronic computers developed, some computers were designed so that their CPU could utilize a special mathematical processor, called a "math coprocessor," to perform floating-point calculations. Because the coprocessor performed its computations in parallel with the CPU's own operations, this architecture considerably sped up processes that required floating-point calculations. (Computer graphics, for example, can require intense floating-point processing.) At one time, many personal computers (PCs) had the option of containing (or not containing) a coprocessor installed to aid in floating-point operations. Nearly all general-purpose computers now come with a built-in coprocessor. For example, beginning with Intel's 80486 CPU (and continuing with its line of Pentium® CPUs) the coprocessor was integrated into the CPU.
Floating-point numbers are of the general form: ±m x Re, where ± is the sign (positive or negative), m is the mantissa, x stands for the operation of multiplication, R is the base, or radix, of the number system being used, and e is the exponent of the radix. A simple example of this form is -1.62 x 103; in this instance, the sign is negative, the mantissa is 1.62, the radix is 10 (i.e., base 10), and the exponent of 10 is 3 (i.e., the base 10 is raised to the third power).
The exponent in a floating-point number is often expressed in "excess-n notation." In that notation a number called the characteristic, or biased exponent, is stored in memory in place of the actual exponent. This procedure may be clarified with the following example: for exponents represented by the use of 8 bits, the exponents can have values (expressed in decimal notation) ranging from -127 to +128. Using excess-n notation, the stored values would range between 0 and 255 (this procedure is performed to eliminate negative numbers). When needed, the characteristic is subtracted from the stored (excess-n) exponent to return to the actual value of the exponent. The characteristic for the eight-bit exponent (which can have values of -127 to +128) is 127 (because the characteristic of 127 added to the low (-127) and high (+128) range yields 0 to 255).
The floating-point representations used in computers have been standardized by an organization called the IEEE (Institute of Electrical and Electronics Engineers). The IEEE is an officially sanctioned body for composing and promoting standards for the telecommunications and computer industries in the United States. The "IEEE 704" code specifies floating-point formats. Floating-point formats are usually expressed within the computer in either single precision (i.e., 32 bits) or double precision (i.e., 64 bits); however, there are less-used formats available. The great majority of programming languages support both the single-precision and double-precision expressions of floating-point numbers.
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