PO SRl W = T SRR T R e A =
e e R R e TR L
< oy —— . ST g

 

S i

ST L e P T i 8, B+ S B A

44 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

We may also write our final result in the following form:
a=ao+ (a1 (@a+(as+ - - 4+ (@n2+ (@1 +ar)r)r- - )r)r)r
or in the form:

@ = An? + Qn1)? + Qna)? + -+ + a2)r + ar)r + a,

For example, if » = 3, we have

g2 = ast + ax
@ = @r + a
a = gl'r + Qg
= (g7 + a))r + ao
= ((asr + ag)r + a)r + ao

102. The Radix Theorem. If a is any positiv or zero integer
and r, an integer greater than 1, then one and only one set of
integers ao, @1, Qs, ***, Qn, all positiv or zero and all less than r
can be found, such that

n+1

a =2 apar*t=atar+ar’t- -+ a4 aurn
k=1
n+1
=2 Qu ™ * VD =g+ a1+ - +asr?+arr +a,
k=1

As in § 101 a;_;, where 1 = k= #n + 1, is the remainder
corresponding to the exact or lower approximate quotient g;
obtaind by dividing the predecessor of g; in the series a, ¢,
go, Q3y *°* by 7.

That is, a, is the positiv or zero remainder got by dividing
a by r; if ¢ is the quotient obtaind in this division, a; is the
positiv or zero remainder got by dividing ¢; by 7; if g. is the
quotient obtaind in this division, a, is the positiv or zero
remainder got by dividing ¢, by 7; and so on.

Ex.1. a=71,r=2. Wefindn=2,60=1,01=1,03= 1.

Hence 7=1-224+1-2+41

Ex.2. a=29,r=3. 29 =1-334+0-3240-342
Ex.3. a=287,r=4. 87 =1-434+1-424+1-4+43
Ex. 4. 639 =2-6+5-62+4-6+3

Ex. 5. 86347 =8-(ten)*+ 6-(ten)*+ 3 (ten)2+4- (ten) + 7