<s>
In	O
arithmetic	O
,	O
long	B-Algorithm
division	I-Algorithm
is	O
a	O
standard	O
division	B-Algorithm
algorithm	I-Algorithm
suitable	O
for	O
dividing	O
multi-digit	O
Hindu-Arabic	O
numerals	O
(	O
Positional	O
notation	O
)	O
that	O
is	O
simple	O
enough	O
to	O
perform	O
by	O
hand	O
.	O
</s>
<s>
The	O
abbreviated	O
form	O
of	O
long	B-Algorithm
division	I-Algorithm
is	O
called	O
short	O
division	O
,	O
which	O
is	O
almost	O
always	O
used	O
instead	O
of	O
long	B-Algorithm
division	I-Algorithm
when	O
the	O
divisor	O
has	O
only	O
one	O
digit	O
.	O
</s>
<s>
Chunking	O
(	O
also	O
known	O
as	O
the	O
partial	O
quotients	O
method	O
or	O
the	O
hangman	O
method	O
)	O
is	O
a	O
less	O
mechanical	O
form	O
of	O
long	B-Algorithm
division	I-Algorithm
prominent	O
in	O
the	O
UK	O
which	O
contributes	O
to	O
a	O
more	O
holistic	O
understanding	O
of	O
the	O
division	O
process	O
.	O
</s>
<s>
(	O
Internally	O
,	O
those	O
devices	O
use	O
one	O
of	O
a	O
variety	O
of	O
division	B-Algorithm
algorithms	I-Algorithm
,	O
the	O
faster	O
ones	O
among	O
which	O
rely	O
on	O
approximations	O
and	O
multiplications	O
to	O
achieve	O
the	O
tasks	O
.	O
)	O
</s>
<s>
In	O
the	O
United	O
States	O
,	O
long	B-Algorithm
division	I-Algorithm
has	O
been	O
especially	O
targeted	O
for	O
de-emphasis	O
,	O
or	O
even	O
elimination	O
from	O
the	O
school	O
curriculum	O
,	O
by	O
reform	O
mathematics	O
,	O
though	O
traditionally	O
introduced	O
in	O
the	O
4th	O
or	O
5th	O
grades	O
.	O
</s>
<s>
In	O
English-speaking	O
countries	O
,	O
long	B-Algorithm
division	I-Algorithm
does	O
not	O
use	O
the	O
division	O
slash	O
or	O
division	O
sign	O
symbols	O
but	O
instead	O
constructs	O
a	O
tableau	O
.	O
</s>
<s>
The	O
combination	O
of	O
these	O
two	O
symbols	O
is	O
sometimes	O
known	O
as	O
a	O
long	B-Algorithm
division	I-Algorithm
symbol	O
or	O
division	O
bracket	O
.	O
</s>
<s>
We	O
could	O
extend	O
the	O
dividend	O
by	O
writing	O
it	O
as	O
,	O
say	O
,	O
000	O
...	O
and	O
continue	O
the	O
process	O
(	O
using	O
a	O
decimal	B-Application
point	O
in	O
the	O
quotient	O
directly	O
above	O
the	O
decimal	B-Application
point	O
in	O
the	O
dividend	O
)	O
,	O
in	O
order	O
to	O
get	O
a	O
decimal	B-Application
answer	O
,	O
as	O
in	O
the	O
following	O
example	O
.	O
</s>
<s>
In	O
this	O
example	O
,	O
the	O
decimal	B-Application
part	O
of	O
the	O
result	O
is	O
calculated	O
by	O
continuing	O
the	O
process	O
beyond	O
the	O
units	O
digit	O
,	O
"	O
bringing	O
down	O
"	O
zeros	O
as	O
being	O
the	O
decimal	B-Application
part	O
of	O
the	O
dividend	O
.	O
</s>
<s>
Find	O
the	O
location	O
of	O
all	O
decimal	B-Application
points	O
in	O
the	O
dividend	O
n	O
and	O
divisor	O
m	O
.	O
</s>
<s>
If	O
necessary	O
,	O
simplify	O
the	O
long	B-Algorithm
division	I-Algorithm
problem	O
by	O
moving	O
the	O
decimals	B-Application
of	O
the	O
divisor	O
and	O
dividend	O
by	O
the	O
same	O
number	O
of	O
decimal	B-Application
places	O
,	O
to	O
the	O
right	O
(	O
or	O
to	O
the	O
left	O
)	O
,	O
so	O
that	O
the	O
decimal	B-Application
of	O
the	O
divisor	O
is	O
to	O
the	O
right	O
of	O
the	O
last	O
digit	O
.	O
</s>
<s>
When	O
doing	O
long	B-Algorithm
division	I-Algorithm
,	O
keep	O
the	O
numbers	O
lined	O
up	O
straight	O
from	O
top	O
to	O
bottom	O
under	O
the	O
tableau	O
.	O
</s>
<s>
This	O
lets	O
us	O
maintain	O
an	O
invariant	B-Application
relation	I-Application
at	O
every	O
step	O
:	O
</s>
<s>
For	O
non-decimal	O
currencies	O
(	O
such	O
as	O
the	O
British	O
£sd	O
system	O
before	O
1971	O
)	O
and	O
measures	O
(	O
such	O
as	O
avoirdupois	O
)	O
mixed	O
mode	O
division	O
must	O
be	O
used	O
.	O
</s>
<s>
Long	B-Algorithm
division	I-Algorithm
of	O
23,480	O
/	O
37	O
now	O
proceeds	O
as	O
normal	O
yielding	O
634	O
with	O
remainder	O
22	O
.	O
</s>
<s>
Long	B-Algorithm
division	I-Algorithm
of	O
the	O
feet	O
gives	O
1	O
remainder	O
29	O
which	O
is	O
then	O
multiplied	O
by	O
twelve	O
to	O
get	O
348	O
inches	O
.	O
</s>
<s>
Long	B-Algorithm
division	I-Algorithm
continues	O
with	O
the	O
final	O
remainder	O
of	O
15	O
inches	O
being	O
shown	O
on	O
the	O
result	O
line	O
.	O
</s>
<s>
When	O
the	O
quotient	O
is	O
not	O
an	O
integer	O
and	O
the	O
division	O
process	O
is	O
extended	O
beyond	O
the	O
decimal	B-Application
point	O
,	O
one	O
of	O
two	O
things	O
can	O
happen	O
:	O
</s>
<s>
A	O
remainder	O
could	O
be	O
reached	O
that	O
is	O
identical	O
to	O
a	O
previous	O
remainder	O
that	O
occurred	O
after	O
the	O
decimal	B-Application
points	O
were	O
written	O
.	O
</s>
<s>
So	O
a	O
bar	O
is	O
drawn	O
over	O
the	O
repeating	O
sequence	O
to	O
indicate	O
that	O
it	O
repeats	O
forever	O
(	O
i.e.	O
,	O
every	O
rational	O
number	O
is	O
either	O
a	O
terminating	O
or	O
repeating	O
decimal	B-Application
)	O
.	O
</s>
<s>
Decimal	B-Application
numbers	O
are	O
not	O
divided	O
directly	O
,	O
the	O
dividend	O
and	O
divisor	O
are	O
multiplied	O
by	O
a	O
power	O
of	O
ten	O
so	O
that	O
the	O
division	O
involves	O
two	O
whole	O
numbers	O
.	O
</s>
<s>
Therefore	O
,	O
if	O
one	O
were	O
dividing	O
12	O
,	O
7	O
by	O
0	O
,	O
4	O
(	O
commas	O
being	O
used	O
instead	O
of	O
decimal	B-Application
points	O
)	O
,	O
the	O
dividend	O
and	O
divisor	O
would	O
first	O
be	O
changed	O
to	O
127	O
and	O
4	O
,	O
and	O
then	O
the	O
division	O
would	O
proceed	O
as	O
above	O
.	O
</s>
<s>
In	O
Austria	O
,	O
Germany	O
and	O
Switzerland	B-Protocol
,	O
the	O
notational	O
form	O
of	O
a	O
normal	O
equation	O
is	O
used	O
.	O
</s>
<s>
In	O
these	O
regions	O
the	O
decimal	B-Application
separator	O
is	O
written	O
as	O
a	O
comma	O
.	O
</s>
<s>
Otherwise	O
,	O
we	O
iterate	B-Algorithm
from	O
,	O
before	O
stopping	O
.	O
</s>
<s>
For	O
each	O
iteration	B-Algorithm
,	O
let	O
be	O
the	O
quotient	O
extracted	O
so	O
far	O
,	O
be	O
the	O
intermediate	O
dividend	O
,	O
be	O
the	O
intermediate	O
remainder	O
,	O
be	O
the	O
next	O
digit	O
of	O
the	O
original	O
dividend	O
,	O
and	O
be	O
the	O
next	O
digit	O
of	O
the	O
quotient	O
.	O
</s>
<s>
With	O
every	O
iteration	B-Algorithm
,	O
the	O
three	O
equations	O
are	O
true	O
:	O
</s>
<s>
In	O
base	B-Application
10	I-Application
,	O
using	O
the	O
example	O
above	O
with	O
and	O
,	O
the	O
initial	O
values	O
and	O
.	O
</s>
<s>
If	O
one	O
does	O
n't	O
have	O
the	O
addition	O
,	O
subtraction	O
,	O
or	O
multiplication	O
tables	O
for	O
base	O
memorised	O
,	O
then	O
this	O
algorithm	O
still	O
works	O
if	O
the	O
numbers	O
are	O
converted	O
to	O
decimal	B-Application
and	O
at	O
the	O
end	O
are	O
converted	O
back	O
to	O
base	O
.	O
</s>
<s>
If	O
the	O
remainder	O
is	O
equal	O
to	O
zero	O
at	O
any	O
iteration	B-Algorithm
,	O
then	O
the	O
quotient	O
is	O
a	O
-adic	O
fraction	O
,	O
and	O
is	O
represented	O
as	O
a	O
finite	O
decimal	B-Application
expansion	O
in	O
base	O
positional	O
notation	O
.	O
</s>
<s>
Otherwise	O
,	O
it	O
is	O
still	O
a	O
rational	O
number	O
but	O
not	O
a	O
-adic	O
rational	O
,	O
and	O
is	O
instead	O
represented	O
as	O
an	O
infinite	O
repeating	O
decimal	B-Application
expansion	O
in	O
base	O
positional	O
notation	O
.	O
</s>
<s>
With	O
every	O
iteration	B-Algorithm
,	O
the	O
following	O
operations	O
are	O
done	O
:	O
</s>
<s>
On	O
each	O
iteration	B-Algorithm
,	O
the	O
most	O
time-consuming	O
task	O
is	O
to	O
select	O
.	O
</s>
<s>
The	O
remainder	O
of	O
the	O
algorithm	O
are	O
addition	O
and	O
the	O
digit-shifting	O
of	O
and	O
to	O
the	O
left	O
one	O
digit	O
,	O
and	O
so	O
takes	O
time	O
and	O
in	O
base	O
,	O
so	O
each	O
iteration	B-Algorithm
takes	O
,	O
or	O
just	O
.	O
</s>
<s>
Long	B-Algorithm
division	I-Algorithm
of	O
integers	O
can	O
easily	O
be	O
extended	O
to	O
include	O
non-integer	O
dividends	O
,	O
as	O
long	O
as	O
they	O
are	O
rational	O
.	O
</s>
<s>
This	O
is	O
because	O
every	O
rational	O
number	O
has	O
a	O
recurring	O
decimal	B-Application
expansion	O
.	O
</s>
<s>
The	O
procedure	O
can	O
also	O
be	O
extended	O
to	O
include	O
divisors	O
which	O
have	O
a	O
finite	O
or	O
terminating	O
decimal	B-Application
expansion	O
(	O
i.e.	O
</s>
<s>
decimal	B-Application
fractions	O
)	O
.	O
</s>
<s>
A	O
generalised	O
version	O
of	O
this	O
method	O
called	O
polynomial	O
long	B-Algorithm
division	I-Algorithm
is	O
also	O
used	O
for	O
dividing	O
polynomials	O
(	O
sometimes	O
using	O
a	O
shorthand	O
version	O
called	O
synthetic	O
division	O
)	O
.	O
</s>
