<s>
In	O
mathematics	O
and	O
computer	B-General_Concept
science	I-General_Concept
,	O
the	O
floor	O
function	O
is	O
the	O
function	O
that	O
takes	O
as	O
input	O
a	O
real	O
number	O
,	O
and	O
gives	O
as	O
output	O
the	O
greatest	O
integer	O
less	O
than	O
or	O
equal	O
to	O
,	O
denoted	O
or	O
.	O
</s>
<s>
This	O
remained	O
the	O
standard	O
in	O
mathematics	O
until	O
Kenneth	O
E	O
.	O
Iverson	O
introduced	O
,	O
in	O
his	O
1962	O
book	O
A	B-Language
Programming	I-Language
Language	I-Language
,	O
the	O
names	O
"	O
floor	O
"	O
and	O
"	O
ceiling	O
"	O
and	O
the	O
corresponding	O
notations	O
and	O
.	O
</s>
<s>
In	O
the	O
LaTeX	B-Application
typesetting	I-Application
system	O
,	O
these	O
symbols	O
can	O
be	O
specified	O
with	O
the	O
and	O
commands	O
in	O
math	O
mode	O
,	O
and	O
extended	O
in	O
size	O
using	O
and	O
as	O
needed	O
.	O
</s>
<s>
This	O
is	O
truncation	B-Algorithm
to	O
zero	O
significant	O
digits	O
.	O
</s>
<s>
For	O
an	O
arbitrary	O
real	O
number	O
,	O
rounding	B-Algorithm
to	O
the	O
nearest	B-Algorithm
integer	I-Algorithm
with	O
tie	O
breaking	O
towards	O
positive	O
infinity	O
is	O
given	O
by	O
;	O
rounding	B-Algorithm
towards	O
negative	O
infinity	O
is	O
given	O
as	O
.	O
</s>
<s>
If	O
tie-breaking	O
is	O
away	O
from	O
0	O
,	O
then	O
the	O
rounding	B-Algorithm
function	I-Algorithm
is	O
(	O
see	O
sign	O
function	O
)	O
,	O
and	O
rounding	B-Algorithm
towards	O
even	O
can	O
be	O
expressed	O
with	O
the	O
more	O
cumbersome	O
,	O
which	O
is	O
the	O
above	O
expression	O
for	O
rounding	B-Algorithm
towards	O
positive	O
infinity	O
minus	O
an	O
integrality	O
indicator	O
for	O
.	O
</s>
<s>
In	O
most	O
programming	O
languages	O
,	O
the	O
simplest	O
method	O
to	O
convert	O
a	O
floating	O
point	O
number	O
to	O
an	O
integer	O
does	O
not	O
do	O
floor	O
or	O
ceiling	O
,	O
but	O
truncation	B-Algorithm
.	O
</s>
<s>
The	O
reason	O
for	O
this	O
is	O
historical	O
,	O
as	O
the	O
first	O
machines	O
used	O
ones	O
 '	O
complement	O
and	O
truncation	B-Algorithm
was	O
simpler	O
to	O
implement	O
(	O
floor	O
is	O
simpler	O
in	O
two	B-General_Concept
's	I-General_Concept
complement	I-General_Concept
)	O
.	O
</s>
<s>
FORTRAN	B-Application
was	O
defined	O
to	O
require	O
this	O
behavior	O
and	O
thus	O
almost	O
all	O
processors	O
implement	O
conversion	O
this	O
way	O
.	O
</s>
<s>
Many	O
programming	O
languages	O
(	O
including	O
C	B-Language
,	O
C++	B-Language
,	O
C#	B-Application
,	O
Java	B-Language
,	O
PHP	B-Application
,	O
R	B-Language
,	O
and	O
Python	B-Language
)	O
provide	O
standard	O
functions	O
for	O
floor	O
and	O
ceiling	O
,	O
usually	O
called	O
floor	O
and	O
ceil	B-General_Concept
,	O
or	O
less	O
commonly	O
ceiling	O
.	O
</s>
<s>
The	O
language	O
APL	B-Language
uses	O
⌊x	O
for	O
floor	O
.	O
</s>
<s>
The	O
J	B-Language
Programming	I-Language
Language	I-Language
,	O
a	O
follow-on	O
to	O
APL	B-Language
that	O
is	O
designed	O
to	O
use	O
standard	O
keyboard	O
symbols	O
,	O
uses	O
<	O
.	O
</s>
<s>
ALGOL	B-Language
usesentier	O
for	O
floor	O
.	O
</s>
<s>
In	O
Microsoft	B-Application
Excel	I-Application
the	O
floor	O
function	O
is	O
implemented	O
as	O
INT	O
(	O
which	O
rounds	O
down	O
rather	O
than	O
toward	O
zero	O
)	O
.	O
</s>
<s>
The	O
OpenDocument	B-Application
file	O
format	O
,	O
as	O
used	O
by	O
OpenOffice.org,	O
Libreoffice	B-Application
and	O
others	O
,	O
uses	O
the	O
same	O
function	O
names	O
;	O
INT	O
does	O
floor	O
and	O
FLOOR	O
has	O
a	O
third	O
argument	O
that	O
can	O
make	O
it	O
round	O
toward	O
zero	O
.	O
</s>
