Numerical Analysis_Computer Representation of Numbers.docx
- 1. Computer Representation of Numbers Digital computers are the principal means of calculation in numerical analysis. Computers have an integer mode and a floating-point mode for representing numbers. The integermode isused onlyto representintegers,The floating-pointformisusedtorepresentrealnumbers.The numbersallowedcanbe of greatlyvaryingsize,butthere are limitationsonboththe magnitude of the numberand on the number of digits. The floating-point representation is closely related to what is called scientific notation. Most digital computers use the base 2 (binary) number systemorsome variantof itsuch as base 8 (octal) orbase 16 (hexadecimal) Floating-Point Number Systems Floating-point numbers are analogous to scientific notation. The decimal pointfloatstothe positionrightafterthe most significant digit. Like scientific notation, floating-point numbers have a sign(S), mantissa (M), base (B), and exponent (E), Example: 4100 = 4.1 × 103 is the decimal scientific notation a mantissa of 4.1 a base of 10 an exponent of 3 32 bitsFloating-PointNumbersused1signbit,8exponent bits, and 23 mantissa bits. Example: Show the floating-point representation of the decimal number 228. Solution: 22810 = 111001002 = 1.110012 × 27. Inbinaryfloating-point,thefirstbitof themantissa(tothe leftof the binarypoint)isalways1andthereforeneednot be stored. It is called the implicit leading one. Modified Floating-point 22810 = 1.110012 × 27 The implicit leading one is not included in the 23-bit mantissa for efficiency. Onlythe fractionbitsare stored.Thisfreesupanextrabit for useful data. Computer manufacturers used incompatible floating- point formats. Results from one computer could not directly be interpreted by another computer. In 1985, The Institute of Electrical and Electronics Engineers (IEEE) created the IEEE 754 Floating-Point Standard that defines floating-point numbers. The exponent needsto represent both positive and negative exponents. floating-point uses a biased exponent, which is the original exponent plus a constant bias. 32-bit floating-pointusesa biasof 127. For example, for the exponent 7, the biasedexponent is 7 + 127 = 134 = 100001102. For the exponent −4, the biased exponent is: −4 + 127 = 123 = 011110112. The IEEE 754 standard also defines 64-bit double- precision numbers (also called doubles) and 128-bit quadruple-precision numbers (also called quads) that provide greater precision and greater range. Special Cases: 0, ±∞, and NaN
- 2. The IEEE floating-point standard has special cases to represent numbers such as zero, infinity, and illegal results. Representingthe numberzerois problematicinfloating- point notation because of the implicit leading one. Special codes with exponents of all 0’s or all l’s are reserved for these special cases. NaN is used for numbers that don’t exist, such as √−1 or log2(−5). The decimal equivalent of a floating point number can be calculated using the following formula: Where s = 0 for + number and 1 for - number e = exponent between 0 and 255 f= mantissa Decimal Machine Numbers