Computer_Arithmetic - Computer Arithmetic Hector D Ceniceros 1 Floating Point Numbers Floating point numbers are based on scientific notation in

Computer_Arithmetic - Computer Arithmetic Hector D...

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Computer Arithmetic*Hector D. Ceniceros1Floating Point NumbersFloating point numbers are based on scientific notation in binary (base 2).For example(1.0101)2×22= (1·20+ 0·2-1+ 1·2-2+ 0·2-3+ 1·2-4)×22= (1 +14+116)×4 = 5.2510.We can write any non-zero real numberxin normalized, binary, scientificnotation asx=±S×2E,1S <2,(1)whereSis called thesignificantormantissaandEis the exponent.IngeneralSis an infinite expansion of the formS= (1.b1b2· · ·)2.(2)In a computer a real number is represented in scientific notation but us-ing a finite number of digits (bits). We call thesefloating point numbers.In single precision (SP), floating point numbers are stored in 32-bit wordswhereas in double precision (DP), used in most scientific computing appli-cations, a 64-bit word is employed: 1 bit is used for the sign, 52 bits for*These are lecture notes for Math 104 A. These notes and all course materials areprotected by United States Federal Copyright Law, the California Civil Code. The UCPolicy 102.23 expressly prohibits students (and all other persons) from recording lecturesor discussions and from distributing or selling lectures notes and all other course materialswithout the prior written permission of the instructor.1
S, and 11 bits forE. This memory limits produce a large butfinite set offloating point numberswhich can be represented in a computer. Moreover,the floating points numbers are not uniformly distributed!

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