Encyclopedia of Computer Science and Technology

338 numeric dataselect customer where (today - customer.lastpurchasedate) > 30Programming packages such as Mathematica are alsononprocedural in that they allow for problems to be statedusing the same symbolic notation that mathematiciansemploy, and many standard procedures for solving or transformingequations are then carried out automatically.Other examples of relatively nonprocedural languagesinclude logic-programming languages (see Prolog andexpert systems) and languages where the desired resultsare built up from defining functions rather than through aseries of procedural steps (see lisp and functional languages).Further ReadingAbraham, Paul W., et al. Functional, Concurrent and Logic ProgrammingLanguages. Vol. 4 of Handbook of Programming Languages,edited by Peter H. Salus. Indianapolis: MacmillanTechnical Publishing, 1998.Gilmore, Stephen, ed. Trends in Functional Programming. Portland,Ore.: Intellect, 2005.Truitt, Thomas D., Stuart B. Mindlin, and Tarralyn A. Truitt. AnIntroduction to Nonprocedural Languages: Using NPL. NewYork: McGraw-Hill, 1983.numeric dataText characters and strings can be stored rather simply incomputer memory, such as by devoting 8 bits (one byte) or16 bits to each character. The storage of numbers is morecomplex because there are both different formats and differentsizes of numbers recognized by most programminglanguages.Integers (whole numbers) have the simplest representation,but there are two important considerations: the totalnumber of bits available and whether one bit is used to holdthe sign.Since all numbers are stored as binary digits, an unsignedinteger has a range from 0 to 2 bits where “bits” is the totalnumber of bits available. Thus if there are 16 bits available,the maximum value for an integer is 65535. If negativenumbers are to be handled, a signed integer must be used(in most languages such as C, C++, and Java, an integer issigned unless unsigned is specified). Since one bit is usedto hold the sign and each bit doubles the maximum size, itfollows that a signed integer can have only half the rangeabove or below zero. Thus, a 16-bit signed integer can rangefrom -32,768 to 32,767.One complication is that the available sizes of integersdepend on whether the computer system’s native data sizeis 16, 32, or 64 bits. In most cases the native size is 32 bits,so the declaration “int” in a C program on such a machineimplies a signed 32-bit integer that can range from - 2 31 or-2,147,483,647 to 2 31 -1, or 2,147,483,647. However, if one isusing large numbers in a program, it is important to checkthat the chosen type is large enough. The long specifier isoften used to indicate an integer twice the normal size, or64 bits in this case.Floating Point NumbersNumbers with a fractional (decimal) part are usually storedin a format called floating point. The “floating” means thatthe location of the decimal point can be moved as necessaryto fit the number within the specified digit range. A floatingpoint number is actually stored in four separate parts. Firstcomes the sign, indicating whether the number is negativeor positive. Next comes the mantissa, which contains theactual digits of the number, both before and after the decimalpoint. The radix is the “base” for the number systemused. Finally, the exponent determines where the decimalpoint will be placed.For example, the base 10 number 247.35 could be representedas 24735 × 10 -2 . The -2 moves the decimal pointat the end two places to the left. However, floating-pointnumbers are normalized to a form in which there is just onedigit to the left of the decimal point. Thus, 247.35 wouldactually be written 2.4735 × 10 2 . This system is also knownas scientific notation.As noted earlier, actual data storage in modern computersis always in binary, but the same principle applies.According to IEEE Standard 754, 32-bit floating-point numbersuse 1 bit for the sign, 8 bits for the exponent, and 23bits for the mantissa (also called the significand, since itexpressed the digits that are significant—that is, guaranteednot to be “lost” through overflow or underflow in processing).The double precision float, declared as a “double”in C programs, uses 1, 11, and 52 bits respectively.Programmers who use relatively small numbers (such ascurrency amounts) generally don’t need to worry about lossof precision. However, if two numbers being multiplied arelarge enough, even though both numbers fit within the 32-bit size, their product may well generate more digits thancan be held within the 23 bits available for the mantissa.This means that some precision will be lost. This can beavoided to some extent by using the “double” size.Since floating-point calculations use more processorcycles (see microprocessor) than integer calculations,processor designers have paid particular attention toimproving floating-point performance. Indeed, processorsare often rated in terms of “megaflops” (millions of floatingpointoperations per second) or even “gigaflops” (billions offlops).Further Reading“IEEE Standard for Floating Point Arithmetic.” Available online.URL: http://www.psc.edu/general/software/packages/ieee/ieee.html. Accessed August 16, 2007.“Numeric Data Types and Expression Evaluation [in C].” Availableonline. URL: http://www.psc.edu/general/software/packages/ieee/ieee.html. Accessed August 16, 2007.Sebesta, Robert W. Concepts of Programming Languages. 8th ed.Boston: Addison-Wesley, 2007.“Type Conversion and Conversion Operators in C#.” Availableonline. URL: http://www.psc.edu/general/software/packages/ieee/ieee.html. Accessed August 16, 2007.“xmL Schema Numeric Data Types.” Available online. URL: http://www.psc.edu/general/software/packages/ieee/ieee.html.Accessed August 16, 2007.