New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | SMALL | MEDIUM | LARGE Below is the OCR-scanned text from this page: 28. Scales of Notation Representation of Numbers Any positive real number x can be uniquely represented in the scale of some integer b > l as X= (Am . . . AiAo. 6 - 1 0 ~ 2 . . .)(a), where every At and a-j is one of the integers 0, 1, . . ., b-1, not all A,, are zero, and A,>O if x 2 1. There is a one-to-one correspond- ence between the number and the sequence m X=A,b"+ . . . +Aib+Ao+C a-jb-' 1 where the infinite series converges. The integer b is 'called the base or radix of the scale. The sequence for x in the scale of b may ter- minate, i.e., a-n-l=a-n-,= . . . =O for some n 2 l so that x=(A, . . . A,AO*U-~U-~ . . . ~ - ~ ) ( a ) ; then x is said to be a finite b-adic number. A sequence which does not terminate may have the property that the infinite sequence a-l, u - ~ , . . . becomes periodic from a certain digit a-,(n>l) on; according as n=l or n>l the sequence is then said to be pure or mixed recurring. A sequence which neither terminates nor recurs represents an irrational number. Names of Scales I I Base I Scale I Base -I I- Binary Ternary Quaternary Quinary Senary Septenary 8 9 10 11 12 16 Scale Octal Nonary Decimal Undenary Duodenary Hexadecimal General Conversion Methods Any number can be converted from the scale of b to the scale of some integer \$#b, Z > l , by using arithmetic operations in either the b-scale or the 8-scale. Accordingly, there are four methods of conversion, depending on whether the number to be converted is an integer or a proper fraction. IntegersX=(A, . . . AIAO)(R (I) b-scale arithmetic. Convert 8 to the b-scale and define xp=x, +z\$) - - Xlp=X2+A:/b , -t where z, xi, . . ., A; are the remainders and XI, X2, . . ., X;;; the quotients (in the b-scale) where X, XI, . . ., XK-,, respectively are divided by 8 in the b-scale. Then convert the remainders to the 8-scale, (z;)(;)=Flo) (&)=&) . . ., (z)(;)=& and obtain - -- X= (A;;; . . . AiAo)(;). (11) h e a l e arithmetic. Convert b and Aol A,, . . ., A, to the &scale and define, using arithmetic operations in the 8-scale, Xm-l=Amb+Am-l, Xm-z= Xm-,b+Am.-,, Xi = Xd + Ai , X= Xlb +Ao. then Proper fractions X = (0.a-la-1 . . . ) ( b ) To convert a proper fraction x, given to n digits in the b-scale, to the scale of 7; f b such that inverse conversion from the %-scale may yield the same n rounded digits in the b-scale, the representation of x in the %scale must be obtained to n rounded digits where n satisfies J;>b". (111) b-scale arithmetic. Convert ? to the b-scale and define - zb = xi + ZLl x1b=22 - 27A=z;;+al. The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.