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Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55)
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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
X=A,b"+ . . . +Aib+Ao+C a-jb-'
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
Base I Scale I Base
General Conversion Methods
Any number can be converted from the scale
of b to the scale of some integer $#b, Z > l , by
using arithmetic operations in either the b-scale
or the 8-scale. Accordingly, there are four
methods of conversion, depending on whether the
number to be converted is an integer or a proper
IntegersX=(A, . . . AIAO)(R
(I) b-scale arithmetic. Convert 8 to the b-scale
xp=x, +z$) - -
-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,
Xi = Xd + Ai ,
X= Xlb +Ao.
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
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