<s>
Casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
is	O
any	O
of	O
three	O
arithmetical	O
procedures	O
:	O
</s>
<s>
A	O
sanity	B-Error_Name
test	I-Error_Name
in	O
which	O
the	O
above-mentioned	O
procedures	O
are	O
used	O
to	O
check	O
for	O
errors	O
in	O
arithmetical	O
calculations	O
.	O
</s>
<s>
More	O
generally	O
,	O
when	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
by	O
summing	O
digits	O
,	O
any	O
set	O
of	O
digits	O
which	O
add	O
up	O
to	O
9	O
,	O
or	O
a	O
multiple	O
of	O
9	O
,	O
can	O
be	O
ignored	O
.	O
</s>
<s>
To	O
check	O
the	O
result	O
of	O
an	O
arithmetical	O
calculation	O
by	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
,	O
each	O
number	O
in	O
the	O
calculation	O
is	O
replaced	O
by	O
its	O
digital	O
root	O
and	O
the	O
same	O
calculations	O
applied	O
to	O
these	O
digital	O
roots	O
.	O
</s>
<s>
While	O
extremely	O
useful	O
,	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
does	O
not	O
catch	O
all	O
errors	O
made	O
while	O
doing	O
calculations	O
.	O
</s>
<s>
In	O
particular	O
,	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
does	O
not	O
catch	O
transposition	O
errors	O
,	O
such	O
as	O
1324	O
instead	O
of	O
1234	O
.	O
</s>
<s>
A	O
form	O
of	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
known	O
to	O
ancient	O
Greek	O
mathematicians	O
was	O
described	O
by	O
the	O
Roman	O
bishop	O
Hippolytus	O
(	O
170	O
–	O
235	O
)	O
in	O
The	O
Refutation	O
of	O
all	O
Heresies	O
,	O
and	O
more	O
briefly	O
by	O
the	O
Syrian	O
Neoplatonist	O
philosopher	O
Iamblichus	O
(	O
c.245	O
–	O
c.325	O
)	O
in	O
his	O
commentary	O
on	O
the	O
Introduction	O
to	O
Arithmetic	O
of	O
Nicomachus	O
of	O
Gerasa.	O
,	O
.	O
</s>
<s>
The	O
earliest	O
known	O
surviving	O
work	O
which	O
describes	O
how	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
can	O
be	O
used	O
to	O
check	O
the	O
results	O
of	O
arithmetical	O
computations	O
is	O
the	O
Mahâsiddhânta	O
,	O
written	O
around	O
950	O
by	O
the	O
Indian	O
mathematician	O
and	O
astronomer	O
Aryabhata	O
II	O
(	O
c.920	O
–	O
c.1000	O
)	O
.	O
</s>
<s>
Writing	O
about	O
1020	O
,	O
the	O
Persian	O
polymath	O
,	O
Ibn	O
Sina	O
(	O
Avicenna	O
)	O
(	O
c.980	O
–	O
1037	O
)	O
,	O
also	O
gave	O
full	O
details	O
of	O
what	O
he	O
called	O
the	O
"	O
Hindu	O
method	O
"	O
of	O
checking	O
arithmetical	O
calculations	O
by	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
.	O
</s>
<s>
Fuller	O
explains	O
how	O
to	O
cast	O
out	O
nines	O
and	O
makes	O
other	O
claims	O
about	O
the	O
resulting	O
'	O
indigs	O
,	O
'	O
but	O
he	O
fails	O
to	O
note	O
that	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
can	O
result	O
in	O
false	O
positives	O
.	O
</s>
<s>
The	O
method	O
bears	O
striking	O
resemblance	O
to	O
standard	O
signal	O
processing	O
and	O
computational	O
error	B-Error_Name
detection	I-Error_Name
and	O
error	B-Error_Name
correction	I-Error_Name
methods	O
,	O
typically	O
using	O
similar	O
modular	O
arithmetic	O
in	O
checksums	B-Algorithm
and	O
simpler	O
check	B-Error_Name
digits	I-Error_Name
.	O
</s>
<s>
So	O
we	O
can	O
use	O
the	O
remainder	O
from	O
casting	B-Error_Name
out	I-Error_Name
nines	I-Error_Name
to	O
get	O
the	O
remainder	O
of	O
division	O
by	O
three	O
.	O
</s>
