DEC
229
DEC
**act p 0 J. 1 -—
^ t J on of the unit, than when fewer figures are used ;^ct ion 8 ’ Wort h the trouble, the numerator of a decimal7 aug (r . lria ? be found to any degree of exactness at pleasure,®ith er ,.® n t ln g the number of figures in the denominator,'%,} the division terminate, or till as many figures belllte tidoo ' Vl ^ render the operation sufiiciently exact for the
. A SJ ^rpose.
! n g awav^ fraction may be sufiiciently denoted, by throw-,llst ead f . denominator, and using any character or mark“He, t Wo ll > since the denominator is always 1, followed byJ 'at j s ’ nree, or a series of ciphers, which is the only thingy th e n la °* e ’ to ascertain this point, the number expressedufys Aerator, is always less than the denominator, and>f Ore C0 "^ ts of as many figures as there are ciphers ;
. cittial f a P 0 >nt be placed before the numerator of afi a deci r J a , Ct ' on > it will show that the number following it
A , j 1 (till auww lllelb UlC UULUVU IVllUMlUg XV
jl^re, a _ fraction, and by reckoning a cipher for everyi e *pre SU PP 0S ' n g unity placed before them, the numbera ^ v 'ded 8Se< ^ w ’h show how many decimal parts the unitp arta ln to, and the figures themselves that portion of
Thus — is represented by .6
! ‘hi
! \
,Iis £fiad
’ •« itTiJ 0 * Sa y’ n 8 66 hundredths, 785 thousandths, &c.,, „> 7 tenth Secon d> 6 tenths and 6 hundredths ; as in the°Urth of 5 ® hundredths, and 5 thousandths ; and as in
hi. * 0 ■» ii-Ji.1.- -- r\
for
and
Qft. "1 I.Q O I -—*V4A V/kAVUD) ***»'» V J — ——
**Pies tho , n< fredths and 5 thousandths, as the cipher, 0,Place of tenths;
6 _ 66Too “ Too8 5 _ 785
100 + 1000 — 1000
ol 8 5 085
a Jh e p Q . 100 T 1000 1000
it to v ' s not only useful in marking the following9t e 8e Parati n 6 a decimal fraction, but is likewise necessary^OhcernJI * ' e ^ ec ' m al parts from integers, when both
f|v
‘Wt' 0 " 8 deer ^ aS . ^ een sa 'd> it is observable that decimal’I'h ^ an d, th 6 * 8 - 6 " n tl 16 same order from unity towards thetoip a s, i n ’ g a integers increase towards the right.
Ces botl 1® tl' e place where the numbering
l ke a 0 f,, 1 f° r integers and for decimals ; going over^dtj^aiberi 6 ln * e S ers , w e have units, tens, hundreds, 348;Hii„,’ thmic ^the decimals, we have units, tenths, hun-
< Sth"r th8 >
tti '•utils s t en thousandths, which is 5 tenths,
°f tho thousandths, and 3 ten thousandths ; in this
%
si O' the /. uuuls , ana o ten tnousanatns ; in mis
a head racti °ns, the unit’s place was not reckoned,
^ tvin^ii'g no COUnte d into the whole numbers.
P , 'ocee ( i'' V ’ tf at the notation is completely understood,
.
^ °ceed tu <= notation is completely uuut
!f. e duce a to the reduction of decimal fractions.
<^d°d r ° f
the f ^ ai - f ract i° n to a decimal: set down the. "u, # - as ro an ract ion, with a point upon the right sideInsist o>y be t) C1 Ph ers in succession, towards the right-t 0t '*°at!i One si na .i 10ll 8ht necessary; then, if the denominator’C^'Cal hrie ^ f ^gure, or of two, not exceeding 12, draw ai>cSuZ°r, th . e row of figures so set down, and8 Vj a.^hondr, 11 .*he left side of the left-hand figure; setl ^isio n J) u P on the left of this line, proceed as8 fignj h ac ing a point under the other, then if thetowards the right begin under the first
3 N
figure after the point, these figures will be the decimal, butif not, the number corresponding to the upper row must bemade out, by adding a cipher to the left-hand.
Example I. Required the decimal of4) 1.00
. 25 the decimal required.Example II. Required the decimal of A.
2 ) 1.0
. 5 the decimal required.Example III. Required the decimal of f.
4) 3.00
.75 the decimal required.
The reader who wishes to employ decimals in his calcula-tions, should have the decimals .25, .5, .75, of 1, J, fixedon his memory.
Example IV. Required the decimal of 1 inch in terms of afoot. Here 1 inch is the twelfth part of a foot, therefore thevulgar fraction is fa.
12 ) 1.00000
.08333
Example V. Required the decimal of 2 inches. Now2 inches is fa, or £ ; therefore
12) 2.00000 or 6) 1.00000
.16666 .16666
Example VI. What is the decimal of 3 inches ? 3 inchesis equal to -fa or therefore the decimal will be .25,
as above.
Example VTI. Required the decimal of 4 inches. 4 inchesis equal to A — J, therefore,
12) 4.00000 or 3) 1.00000
.33333 .33333
In this manner, the decimals for every number of inchesunder twelve are to be found, as the following table shows:The decimal of 1 inch = .08333
2 = .16666
3 = .25
4 = .33333
5 = .41666
6 = .5
7 = .58333
8 = .66666
9 = .75
10 = .83333
11 = .91666
In most practical cases, three figures of decimals will befound sufficient.
When inches, seconds, thirds, &c. are to be reduced to adecimal, the best method is to reduce the feet, inches, &c. tothe last denomination, then divide as often by 12 in successionas there are denominations, and the last quotient will be thedecimal required.
Example I. Required the decimal of 9 firsts, or inches,and 6 seconds.
Now here are two denominations, therefore
9 6
12
12 ) 114 seconds in the whole.
12) 9.5
.79166 the decimal required.