C 207 ] 


XXXII. Of the Theory of Circulating De¬ 
cimal FraBions. By John Robertfon, 

Lib. R. S. 


Read Nov. 24,^ |' 1 HE great advantages arifing from 
5768 JL the ufe of Arithmetic, particularly 
in philofophy and commerce, is fufficiently known $ 
therefore, every Hep taken towards its perfec¬ 
tion, has always been countenanced by thofe who 
were beft acquainted with its nature and value. On 
thefe motives I have been induced to offer the annexed 
paper to this learned Society. 

Regiomontanus, it is faid, firft among Europeans, 
added to the then known arithmetic, an operation 
by decimal fractions 5 which he exemplified in his 
triangular table. Its utility was readily feen, and 
embraced in many nations, and particularly in this j 
where it appears to have been cultivated in its theory, 
and facile modes of operation, more than in other 
places. 

Many writers have remarked its excellency in nu¬ 
meral computation, and have pointed out compen- 
diums to avoid the trouble of writing down fuper- 
fluous figures j particularly in the operations with 
concrete numbers, or thofe relative to money, weight, 
and meafure j where the gradations from one deno¬ 
mination to another do not proceed in an uniform 
progreffion. 

In finding the decimal values of the fractional 
parts of concrete, and other numbers, it often hap- 
D d 2 pens. 


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[ 208 J 

pens, that thofe decimals do not terminate, or end, 
with a few figures only ■, and fometimes are infinite, 
or never end ; and among tfaefe are many which have 
one or more figures conftamly recurring; as in the 
following proportions, viz. 

3:2:: 1,0000, &c. : 0,6666, &c. 
and 12:5:; 1,0000, &c .: 0,4166, &c. 
alfo 7:3:: 1,0000, &c. : 0,428571,428571, &c. 

In operations, with fuch recurring decimal frac¬ 
tions, particularly in multiplication and divifion, the 
work will either be longer than necefiary, or be very 
inaccurate, if the numbers are not confidered as cir¬ 
culating ones: and to come at the true refults of 
fuch operations, feveral authors have given precife 
rules; and tome of them have fhewn the principles 
upon which thofe rules were founded. 

In the annexed paper thofe principles are, en¬ 
deavoured to be, exhibited in a different, and in a 
more general and concife manner, than has hitherto 
been fhewn: but the modes of working are not here 
annexed, as they are to be found in Cunn , Malcolm , 
Marjh, and others •, and may hereafter be fully ex¬ 
emplified in a treatife of Arithmetic, by the author 
hereof, confidered in a more mathematical order, 
than what has hitherto been appropriated to this moffc 
ufeful lcience. 

General Principles. 

1. Number is fuppofed to begin at unity, and from 
thence to afcend and defcend : thofe terms afcending 
above unity, are integers j and thofe defcending be¬ 
low unity* are fractions. 


When 




[ 20 9 ] 

Wlien in the afcending and defcending parts of the fcale, the gradation 
proceeds by a tenfold value from the right hand towards the left, the rank 
of numbers, thus generated, is called the decimal fcale. 

As every place in this fcale is ten times the value of its next right hand 
place; therefore the firft place in thefradtional part, is _4 of the place of units; 
and the fecond, third, fourth, 3 c. defcending places in the fradtional part, 
is t4tt> t..W» -ru^oir. & c - part of the place of units. 

Therefore every decimal fraction is equal to a feries arifing from tn ulti- 
plying the firft, fecond, third, fourth, 3 c. terms of the decreafing geome¬ 
trical progreffion ^4- + ^ -\- W-sr- + by the firft, fecond, 

third, and fourth, 3 c. terms in the given fraction refpedtively. 

Thus. Let the given fraction be 0,3587 or ^4444. 

Then 0,3 587=^4. x 3-K-U X 5 + Ww X 8 + ^ x 7 

3 4 - 4 - 8 4 - 7 

jo 1 100 l i ~6 00 * ~i"zrw~o~Q' 

- 3000 5 0 0 80 7 - J S« f 

-nr^HSTT I Too cro" J ““ooT'o* I ‘T"o'3"oT TTo“cTo* 

2. Every decimal fraction arifes from divifion, when the dividend is Iefs 
than the divifor. 

For, divifor : dividend :: 10 : firft term of the fraction j 

:: 100 : fum of the firft and fecond terms; 

:: xooo : fum of the firft, fecond, and third, terms, 

3 c. 

And according to the ratio of the divifor to the dividend, the quotient, 
or decimal fraction, will be finite or infinite. 

3. Among thofe decimal fradtions which are infinite, or do not end, fome 
of them recur, or circulate; that is, the lame figure or figures run over 
again and again ad infinitum. 

As 0,333 ( 3 c.', 0,2323 3 c.', 0,758758 3 c. j 0,999 3 c. 

Here 0,333 3 c. =A+t &c. 

°> 2 3 2 3 -To-f•’HrS'""!""Xo-S-Q - ! - 3c. 

0,785785 3 c. 

0,999, & C . =tV+t!t-KAv+t^. 

Vol. LVIII. E e 


The 



[ 210 ] 

The circulating fradion 0,999, * s e q ua l to 1 >0. 

For the difference between t.o and 0,999 &c. is lefs than can be 
affigned. 

To fave a repetition of figures, it is ufual to mark the firft and lafl of 
circulating expreflions, with points over the figures.- 

Thus,0,333 &c. is Wrote 0,3 

0,2323 &c. 0,23 

°j7 8 57 8 5 °>7 8 5 

4. Like circulating decimal fradions are thofe which have each the fame 
number of circulating places j and begin to recur .each at the fame name. 

Every finite decimal fraction may be conftdered as infinite j cyphers being 
ufed as the circulating part. 

Either place of a circulating eXpreflion may be taken as the firft; obferv- 
ing that the number and order of the circulating places be not altered. 

For as the decimal fraction arifes by divifion ; if either place of the re¬ 
curring figures be taken for the firft, the others will from thence re¬ 
gularly circulate. 

Hence feveral unlike circulating decimal fradions may be made to begin 
and end at places of like names. 


5. If in the decimal fcale io, too, 1000,10000, 100000, 1000000, (Sc. 
continued indefinitely, be feleded any rank of equi-diftant terms, fuch, 
that whatever term therein is taken for the firft term, and the firft term is 
made the common ratio to the reft; then will the fum of the reciprocals of 
thofe terms, be equal to the reciprocal of the number which is unity lefs 
than the firft term. 


Thus 


■A + 

T-0-T5- 4" 


*T o o 
1 

TTS'-o'cro* 


TToT 4“ TTTTTV? 


+ 
+ 
+ T 


*T o 'b o 

1 

TTinnr o "o’ 


7 0 0 9 0?0 O O 


4- (Sc . = 10 being the 1 ft term. 

■\-&c.z=: T ‘~ ; 100 being the 1 ft term. 
4. (Sc.zzz^-, 1000 being the 1 ft term. 


For 



[ 211 ] 


I 

+’ 

— + 

IOO 1 

T 

JL 

&c. = 

100 + lo^f-l 

10 

IOOO 

l 

IOOO 

I 




+ 

II 

29 

10000 + roo + c 

100 

i0000 t 

1000000 

. 1000000 

I 

+ 

_j— j- 

1000000 1 

I 

JL 

&c. — 

IOOOOOO + 3 000 + 1 

IOOO 

IOOOOOOOOO 

1 

IOOOOOOOOO 


IT! &C . 

iooo £?c* 
ioioi ©V. 
ibooooo &c* 

IOOIOOX 

IOOOOOOOOO 


But hi &c. x 9 ~ iooo &c. (by 3 d ) Then 

ioioi &c. x 99 =• 100000 &c-, Then 

iooioai &c. X999 = 1000000000 &c. Then 


IT! &C4 
IOOO 6 fc. 

IOIOI &c, 

1000000 &c. 

1001001 &c, 
IooOooooqo & c t 


t 


9 

X 

99 

I 

999 


Hence the reciprocal of a number confifting of n places of 9’s, is 
equal to a circulating number of n places* the right hand figure being 1, 
and the reft o’s. 


Thus, l ss 
9 


t 



9999 


ill &c. 

IOOO &C, 

IOIOI &c. 
1000000 &c, 

IOOIOOX &c, 
IOOOOOOOOO fiftf.. 

IOOO IOOI &ff. 
IOOOOOOOOOOOO &c» 


0,1 

0,01 

• ? 

0,001 

; ■» » 

0,0001 


6. If the reciprocal of a number confifting of n places of 9*s, be multi¬ 
plied by a number D, not exceeding n places; the produdt will be a cir¬ 
culating decimal fraction of n places, the right hand ones being the fame 
digits as are in the number D. 

Let D = 3 j or D = 23 j or D = 785 * or to any other number. 


E e 2 


Now 



[ «* ] 


Now | = 
Therefore 


# I 

looo Mr. * 99 

2_, or 1111 &c - 

9 iooo Me. 


l n _ 10101 Me, 

9 s joooooo Me. 


I 0 I 0 I Me, t 

looocoo Me, * 999 


x 23 
x 3 


100100 1 Me , 
1000000000 Me. 


: (by 5*). 


X 3 = = 0,3 

^ IOOO Me, 


3030 3 fefc. 
1000000 fife. 


231323 ©V. 

1000000 Me, 

30030 03 Me, 
1000000000 ©V. 


= 0,23 


iooiooi Me, 
1000000000 Me. 


x 23 


23023023 fgrV. 
1000000000 Me, 


= 0,023 


X 78c = _Z!iZ!iz!i &c - s= 0,78? 

u 1 000000000 Me . 1 J 

7. Hence every circulating decimal fraction will be equivalent to a vulgar 
fraction, wherein the numerator is thofe circulating figures, and the denomi¬ 
nator confifts of as many 9’s, as are figures in the numerator. 

Thus 0,3 sa 4. For 4 x 3=4= 0,3 (by 6th) 

°>°3 = A tV* 3 = rr ~ °>°3 

0,23 = 44 -3-V X 23 = 44 = 0,23 

8. Hence a circulating decimal fra&ion, of any number of places, being 
multiplied by a number of as many 9’s, will give a finite expreflion, having 
the fame figures as are in the circulating one. 


Thus 0,6 

X 

9 

S 3 6 . 

For 9:1: 

: 6 

: 0,6 

0,06 

X 

99 

=1 6 . 

99 : 1 ■: 

: 6 

: 0,06 

0,25 

X 

99 

= 25 . 

99 J 1 : 

*• 25 

: 0,25 

0,625 

X 

999 

= 625. 

999 '• 1 : 

: 625 

: 0,62 5 


Hence 




[ 213 ] 

Hence it appears, that, in common multiplication, the product of a cir¬ 
culating number, by its proper denominator, in 9’s, will be deficient of the 
true product by that circulating number. 

Thus 0,6 x 9 s=s 5,4 j then 5,4 ,6 =: 6. For 1 = 0,6 

0,06 x 99 = 5,94 5 then 5*9 4 + °>° 6 = 6 * For A = °>o6 
0,625 ^ 999 = 624,375 > then 624,375 4 0,625 =: 625,0 

Hence. Any finite number is in proportion to the fame number re¬ 
curring, as the proper denominator of the circulate is to that denomina¬ 
tor increafed by unity. 

i « 

Thus 9:10 : : 6 : 6. For 6 X9 = 6x10 
99 : 100 :: 25 : 25. For 25 x 99 = 25 x 100. 

SCHOLIUM. 

If to the preceding articles, be joined the compendiums of multiplying 
and dividing by any number of 9’s, they will conftitute the whole of the 
theory, upon which depend all the operations with circulating numbers: for 
as thefe have 9’s for their denominator, wanting unity in the lowed: place to 
make them io’s; therefore unity for every 9 is applied in fome additions and 
multiplications: Or, the circulating parts being reduced to finite number; 
then working with them by the common rules, will give finite refults; which 
refults are to be reduced to circulates by contrary operations to what were 
ufed to reduce the circulates to finites. 


XXXIII. A 



