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# JB/135/080/003

Completed

To add decimals wherein
are single reptends—

Make every line end at
the same place filling up
the vacancies by the repeating
digits and annexing
a cypher or cyphers
to the finite terms then
add as before only increase
the sum of the right hand
row with as many units
as it contains nines &
the figure in the Sum
under that place will be
a repetend—

To subtract finite decimals.

Having first set down the
greater the two numbers
given whether it be a whole
number mixed number or
decimal set down the less
under it according to the
directions given in addition
then subtract as in whole
numbers imagining all
the vacant places filled
with cyphers.—

To subtract decimals that
have repetends.—

Make the repetends similar
and conterminous &
subtract as in the last
case observing only if the
repetend of the number to
be subtracted be greater
than the repetend of the
number it is to be taken
from then the right hand
figure of the remainder
must be less by unity
than it would be if the
expressions were finite &
the repetend in the remainder
will consist of as many
places as there are in
the other two numbers.—

---page break---

When both factors are
finite decimals whether
they are single or joined
with integers,

Multiply them as
if they were all whole numbers
and from the product
towards the right hand cut
off so many places for
decimal parts in the
product as there were
in both the multiplier &
multiplicand counted together
but if it so happen
that there are not so many
places in the product
supply the defect by prefixing
cyphers.—

Two decimal fractions
being given to reserve in
their product any assigned
number of places.—

Sets the Units place
of the multiplier directly
under that figure of the
decimal part of the multiplicand
whose place
you would reserve in the
product and insert the
order of all its other places
that is the write the decimals
on the left hand
and the integers if any
on the right.—

Then in multiplying
always begin at that
figure of the multiplicand
which stands over the
figure wherewith you are
then multiplying setting
down the first figure of
each particular product
directly underneath one
another due regard being
had to the increase which

---page break---
<p>which would arise out
of the two next figures
to the right hand of that
figure in the multiplicand
which you then begin with
carrying one from 5 to
15 two from 15 to 25 three
from 25 to 35 four from
35 to 45 &c and the Sum
of these lines will give
the product.—

In any of the following
cases in division if the
dividend be greater than
the divisor the quotient
will be either a whole or
a mixt number but
when the dividend is less
than the divisor the quotient
must necessarily be
a fraction for a less number
is contained in a greater
once at the least but
the greater is not contained
once in the less.—

When the divisor &
dividend are both finite
decimals

Divide as in whole
numbers and from the
right hand of the quotient
points off for decimals so
many places as the decimal
places in the dividend
exceed those in the
divisor and those to the
left if any are integers
but if the places of the
quotient are not so many
as this rule requires
supply the defect by prefixing
cyphers to the quotient
but if the decimal places
in the divisor be more
than those in the dividen annex

---page break---
annex cyphers to the dividend
to make them equal and
the quotient will be integers
until all those cyphers
are used.—

To contract the work of
division when the divisor
consists of many decimal
places.—

Having determined the
value of the quotient figures
let each remainder be a
now a dividend and for
every such dividend point
off one figure from the
right of the divisor observing
at each multiplication
to have regard to increase
of the figures so cut
off as in contracted
multiplication.—

If any whole mixed
or decimal number is
given to be divided by
10, 100, 1000 &c only remove
the separating point towards
the left hand so many
places as there are cyphers
in the divisor also in
multiplication the separating
point is moved to the
ri
ght hand so many
places are there cyphers
in the multiplier.—

If the dividend be a
repetend.—

If it be a single
repetend being down the
circulating figure until
the quotient either repeats
or is as exact as required
but if the repetend in the
dividend be a compound
one then being down the
circulating figures in the
same order they stand in
and when you have got thro'

---page break---
<p>Decimal Fractions.

through them all being
down the first figure in
the repetend over again
and so proceed until your
quotient either repeats or
becomes as exact as is
necessary.—

A series of 9s infinitely
continued is equal to unity
or one in the next left
hand place.—

Any single repetend
divided by 10 and the
quotient subtracted from
the said repetend the
remainder will be the
same number complete
or teminate.—

Hence it follows that
if a compound repetend
be divided by an unit
with so many cyphers
annexed as are equal to the
places of the repetend and
the quotient subtracted
from the said repetend the
remainder will be the same
number complete or terminate
that constituted the
repetend.—

To perform the work
of multiplication by
division or of division
by multiplication.—

Divide an Unit with
cyphers annexed by the
given multiplier or divisor
the quotient will be
the divisor or multiplier
sought.—

Identifier: | JB/135/080/003
"JB/" can not be assigned to a declared number type with value 135.

135

080

decimal fractions

003

private material

2

recto

sir samuel bentham

1798

1798

46198