[ 2^ ] 
e (by 5*). 
°>3 
°,°*3 
°> 2 3 
0,003 
0,023 
0,785 
7. Hence every circulating decimal fraction will be equivalent to a vulgar 
iradtion, wherein the numerator is thofe circulating figures, and the denomi- 
lator confifts of as many 9’s, as are figures in the numerator. 
Thus 0,3 = For 2. x 3 = ~ = 0,3 (by 6th) 
°)°3 = tV tV* 3 = tV = °>°3 
0,23 = tV X 23 = 44- = 0,23 
8. Hence a circulating decimal fradtion, of any number of places, being 
multiplied by a number of as many 9’s, will give a finite expreffion, having 
:he fame figures as are in the circulating one. 
Thus 0,6 
X 
9 = 6 . 
For 9:1:16 : 0,6 
• • 
0,06 
X 
99 = 6 . 
• • 
99 : 1 : : 6 : 0,06 
0,25 
X 
99 — 25 • 
99 : 1 :: 25 : 0,25 
0,625 
X 
999 — 6z J- 
• • 
999 : 1 : : 625 : 0,625 
Now l == 
9 
Therefore 
I 1 I ©C. 
jooo ©*. 7 99 
JL, or 111 &e -. 
9 JOOO ©c. 
1 IOJOI ©C<. 
■ , or 
99' IOOOOOO ©c. 
— , or 
999’ 
iooiooi ©f. 
10101 ©r, 
- • 
IOOOOOO & c , y 
X 3 
IOOOOOOOOO ©c. 
X 3 
x 23 
X 3 
i . : tooioo i © 
999 looooooooo © 
— 333 ^ — 
1000 ©*. 
— - 3030 3 & e . 
IOOOOOO ©c. 
232323 fife. 
IOOOOOO ©V. 
3003003 ©V. 
IOOOOOOOOO ©0. 
\ 2 ? , 23 023023 © c . — 
J IOOOOOOOOO ©<■. 
X 7 8 5 = 
785785785 w_ 
IOOOOOOOOO ©C. 
Hence 
