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# JB/135/080/002

Completed

Decimal Fractions

To reduce a Vulgar
Fraction into a decimal.

Annex Cyphers to the
numerator till it be
equal to or greater than
the denominator then
divide by the denominator
and the the quotient will
be the decimal sought.

If after you have
Cyphers annexed to the
numerator there is a
remainder annex Cyphers
thereto and continue your
division till it divide off
or arrive to what degree
of exactness you think
proper.—

Always observe to
set a point betwixt the
numerator and the Cyphers
annexed thereto and that
the quotient have as
many places as you annex
Cyphers to the numerator
and remainders; and if
it be deficient let the
want be supplied by prefixing
as many Cyphers
to the quotient as it falls
short.—

Those decimals
that are reduced from
such a Vulgar Fraction
whose numerator with
Cyphers annexed is an
aliquot part of or can
be measured by its denominator
are finite or
determinate decimals.—-
as No fraction will
produce a finite decimal
but such whose denominator
is 2 or 5 and
their multiples.—-

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But such as are produced
from a Vulgar fraction
whose numerator with
Cyphers annexed is no
aliquot part of or cannot
be measured by its denominator
will be indeterminate
or endless.—-

In circulating decimals
if one figure only
repeats it is called a
single repetend.—

To avoid the trouble
of writing down unnecessary
figures a single
repetend is denoted by
the repeating digit dashed.

Those decimals in
which two or more figures
circulate are called compound
repetends & the manner
of distinguishing
them is by dashing the
first and last figure of
the repetend by which
means we make one
place of the repetend
sufficient.—

To reduce Coin Weights
Measures &c into decimals.

Reduce the different species
into one viz the lowest
denomination they consist
of for a dividend then
reduce the integer into
the same denomination
for a divisor; the result
will be the decimal
required.—

Write the given denominations
or parts orderly
under each other the
inferior or least parts being
uppermost let these be the
dividends.—

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Against each part on
the left hand write the
number thereof contained
in one of its superior
let these be the divisors.

The begininning
with the upper one write
the quotient of each division
as decimal parts on the
right hand of the dividend
next below it, and let
this mixed number be
divided by its divisor &c
till all be finished and
the last quotient will be
the decimal sought. —

The decimal may
rule of practice namely
by considering the next
inferior denomination as
aliquot parts of the integer
and those still lower as
aliquot parts of the superior
ones or of each other
the Sum of all those
aliquot parts will be the
decimal required.—

To reduce any decimal
into the equivalent
known parts of Coin Weight
or Measure -

Multiply the given
number by the number
of Units contained in the
next inferior denomination
cutting off as many figures
from the product as the
given decimal consists of
then multiply the remaining
parts if any by the
next lower denomination
cutting off as before and
thus proceed till you have
or come to the lowest part and

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and the several figures
to the left hand of the
separating points will
be the several parts of t
he quantity required—

To reduce a decimal
into its least equivalent
Vulgar Fraction—
1st if the decimal be finite.

Under the given decimal
write an Unit with as many
Cyphers as the decimal
consists of Places then divide
both the numerator
and denominator by the
greatest common measure
which gives the least equivalent
Vulgar Fraction required

2d. If the given decimal be
a repetend—

The decimal is the
numerator of a Vulgar
Fraction whose denominator
consists of as many
nines as there are
recurring places in the
given decimal both
which divide by their
greatest common measure
as before and their
quotient will be the least
equivalent Vulgar Fraction.

3d. When the given decimal
is part final and part a
circulate.

To as many nines
as there are figures in
the repetend annex as
many cyphers as there
are finite places for a
denominator then multiply
the nines in the said
denominator by the finite
part & and to the product add the
repeating decimal for a nu

---page break---

numerator these divided
by their greatest common
measure, will give the
least equivalent fraction.

A general rule for reducing
decimal into Vulgar
Fractions—

Under the given decimal
set an Unit with as many
cyphers as there are
places in the given decimal
then set the finite
decimal as a numerator
even under the lowest figures
of the first numerator
with its proper denominator;
lastly subtract
the under numerator from
the upper one and the under
denominator from the
upper one the remainder
will be a Vulgar fraction
equivalent to the given decimal
which reduce to its
lowest terms.—

When decimal fractions
observe that the commas
be placed directly underneath
each other for the
primes seconds thirds &c
willl fall under those of
the same name and in
mixed number Units
will fall under Units
tens under tens &c.—

and from the Sum or difference
cut off so many
places for decimals as are
equal to the greatest number
of decimal places in
any of the given numbers.—

Identifier: | JB/135/080/002
"JB/" can not be assigned to a declared number type with value 135.

135

080

decimal fractions

002

private material

2

recto

sir samuel bentham

1798

1798

46198