<s>
Significant	B-Architecture
figures	I-Architecture
(	O
also	O
known	O
as	O
the	O
significant	B-Architecture
digits	I-Architecture
,	O
precision	B-General_Concept
or	O
resolution	O
)	O
of	O
a	O
number	O
in	O
positional	O
notation	O
are	O
digits	O
in	O
the	O
number	O
that	O
are	O
reliable	O
and	O
necessary	O
to	O
indicate	O
the	O
quantity	O
of	O
something	O
.	O
</s>
<s>
If	O
a	O
number	O
expressing	O
the	O
result	O
of	O
a	O
measurement	O
(	O
e.g.	O
,	O
length	O
,	O
pressure	O
,	O
volume	O
,	O
or	O
mass	O
)	O
has	O
more	O
digits	O
than	O
the	O
number	O
of	O
digits	O
allowed	O
by	O
the	O
measurement	O
resolution	O
,	O
then	O
only	O
as	O
many	O
digits	O
as	O
allowed	O
by	O
the	O
measurement	O
resolution	O
are	O
reliable	O
,	O
and	O
so	O
only	O
these	O
can	O
be	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
if	O
a	O
length	O
measurement	O
gives	O
114.8mm	O
while	O
the	O
smallest	O
interval	O
between	O
marks	O
on	O
the	O
ruler	O
used	O
in	O
the	O
measurement	O
is	O
1mm	O
,	O
then	O
the	O
first	O
three	O
digits	O
(	O
1	O
,	O
1	O
,	O
and	O
4	O
,	O
showing	O
114mm	O
)	O
are	O
certain	O
and	O
so	O
they	O
are	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
Digits	O
which	O
are	O
uncertain	O
but	O
reliable	O
are	O
also	O
considered	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
In	O
this	O
example	O
,	O
the	O
last	B-Architecture
digit	I-Architecture
(	O
8	O
,	O
which	O
adds	O
0.8mm	O
)	O
is	O
also	O
considered	O
a	O
significant	B-Architecture
figure	I-Architecture
even	O
though	O
there	O
is	O
uncertainty	O
in	O
it	O
.	O
</s>
<s>
And	O
so	O
all	O
three	O
are	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
The	O
following	O
digits	O
are	O
not	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
013kg	O
has	O
two	O
significant	B-Architecture
figures	I-Architecture
,	O
1	O
and	O
3	O
,	O
and	O
the	O
leading	O
zero	O
is	O
not	O
significant	O
since	O
it	O
is	O
not	O
necessary	O
to	O
indicate	O
the	O
mass	O
;	O
013kg	O
=	O
13kg	O
so	O
0	O
is	O
not	O
necessary	O
.	O
</s>
<s>
Spurious	O
digits	O
,	O
introduced	O
by	O
calculations	O
resulting	O
in	O
a	O
number	O
with	O
a	O
greater	O
precision	B-General_Concept
than	O
the	O
precision	B-General_Concept
of	O
the	O
used	O
data	O
in	O
the	O
calculations	O
,	O
or	O
in	O
a	O
measurement	O
reported	O
to	O
a	O
greater	O
precision	B-General_Concept
than	O
the	O
measurement	O
resolution	O
.	O
</s>
<s>
Of	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
a	O
number	O
,	O
the	O
most	O
significant	O
is	O
the	O
digit	O
with	O
the	O
highest	O
exponent	O
value	O
(	O
simply	O
the	O
left-most	O
significant	B-Architecture
figure	I-Architecture
)	O
,	O
and	O
the	O
least	O
significant	O
is	O
the	O
digit	O
with	O
the	O
lowest	O
exponent	O
value	O
(	O
simply	O
the	O
right-most	O
significant	B-Architecture
figure	I-Architecture
)	O
.	O
</s>
<s>
For	O
example	O
,	O
in	O
the	O
number	O
"	O
123	O
"	O
,	O
the	O
"	O
1	O
"	O
is	O
the	O
most	O
significant	B-Architecture
figure	I-Architecture
as	O
it	O
counts	O
hundreds	O
(	O
102	O
)	O
,	O
and	O
"	O
3	O
"	O
is	O
the	O
least	O
significant	B-Architecture
figure	I-Architecture
as	O
it	O
counts	O
ones	O
(	O
100	O
)	O
.	O
</s>
<s>
Significance	B-Algorithm
arithmetic	I-Algorithm
is	O
a	O
set	O
of	O
approximate	O
rules	O
for	O
roughly	O
maintaining	O
significance	O
throughout	O
a	O
computation	O
.	O
</s>
<s>
The	O
more	O
sophisticated	O
scientific	O
rules	O
are	O
known	O
as	O
propagation	B-General_Concept
of	I-General_Concept
uncertainty	I-General_Concept
.	O
</s>
<s>
Numbers	O
are	O
often	O
rounded	B-Algorithm
to	O
avoid	O
reporting	O
insignificant	O
figures	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
would	O
create	O
false	B-Algorithm
precision	I-Algorithm
to	O
express	O
a	O
measurement	O
as	O
12.34525kg	O
if	O
the	O
scale	O
was	O
only	O
measured	O
to	O
the	O
nearest	O
gram	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
significant	B-Architecture
figures	I-Architecture
are	O
the	O
first	O
5	O
digits	O
from	O
the	O
left-most	O
digit	O
(	O
1	O
,	O
2	O
,	O
3	O
,	O
4	O
,	O
and	O
5	O
)	O
,	O
and	O
the	O
number	O
needs	O
to	O
be	O
rounded	B-Algorithm
to	O
the	O
significant	B-Architecture
figures	I-Architecture
so	O
that	O
it	O
will	O
be	O
12.345kg	O
as	O
the	O
reliable	O
value	O
.	O
</s>
<s>
Numbers	O
can	O
also	O
be	O
rounded	B-Algorithm
merely	O
for	O
simplicity	O
rather	O
than	O
to	O
indicate	O
a	O
precision	B-General_Concept
of	O
measurement	O
,	O
for	O
example	O
,	O
in	O
order	O
to	O
make	O
the	O
numbers	O
faster	O
to	O
pronounce	O
in	O
news	O
broadcasts	O
.	O
</s>
<s>
Radix	O
10	O
(	O
base-10	O
,	O
decimal	B-Application
numbers	O
)	O
is	O
assumed	O
in	O
the	O
following	O
.	O
</s>
<s>
Note	O
that	O
identifying	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
a	O
number	O
requires	O
knowing	O
which	O
digits	O
are	O
reliable	O
(	O
e.g.	O
,	O
by	O
knowing	O
the	O
measurement	O
or	O
reporting	O
resolution	O
with	O
which	O
the	O
number	O
is	O
obtained	O
or	O
processed	O
)	O
since	O
only	O
reliable	O
digits	O
can	O
be	O
significant	O
;	O
e.g.	O
,	O
3	O
and	O
4	O
in	O
0.00234g	O
are	O
not	O
significant	O
if	O
the	O
measurable	O
smallest	O
weight	O
is	O
0.001g	O
.	O
</s>
<s>
91	O
has	O
two	O
significant	B-Architecture
figures	I-Architecture
(	O
9	O
and	O
1	O
)	O
if	O
they	O
are	O
measurement-allowed	O
digits	O
.	O
</s>
<s>
123.45	O
has	O
five	O
significant	B-Architecture
digits	I-Architecture
(	O
1	O
,	O
2	O
,	O
3	O
,	O
4	O
and	O
5	O
)	O
if	O
they	O
are	O
within	O
the	O
measurement	O
resolution	O
.	O
</s>
<s>
If	O
the	O
resolution	O
is	O
0.1	O
,	O
then	O
the	O
last	B-Architecture
digit	I-Architecture
5	O
is	O
not	O
significant	O
.	O
</s>
<s>
101.12003	O
consists	O
of	O
eight	O
significant	B-Architecture
figures	I-Architecture
if	O
the	O
resolution	O
is	O
to	O
0.00001	O
.	O
</s>
<s>
125.340006	O
has	O
seven	O
significant	B-Architecture
figures	I-Architecture
if	O
the	O
resolution	O
is	O
to	O
0.0001	O
:	O
1	O
,	O
2	O
,	O
5	O
,	O
3	O
,	O
4	O
,	O
0	O
,	O
and	O
0	O
.	O
</s>
<s>
0.00034	O
has	O
2	O
significant	B-Architecture
figures	I-Architecture
(	O
3	O
and	O
4	O
)	O
if	O
the	O
resolution	O
is	O
0.00001	O
.	O
</s>
<s>
Zeros	O
to	O
the	O
right	O
of	O
the	O
last	O
non-zero	O
digit	O
(	O
trailing	O
zeros	O
)	O
in	O
a	O
number	O
with	O
the	O
decimal	B-Application
point	O
are	O
significant	O
if	O
they	O
are	O
within	O
the	O
measurement	O
or	O
reporting	O
resolution	O
.	O
</s>
<s>
1.200	O
has	O
four	O
significant	B-Architecture
figures	I-Architecture
(	O
1	O
,	O
2	O
,	O
0	O
,	O
and	O
0	O
)	O
if	O
they	O
are	O
allowed	O
by	O
the	O
measurement	O
resolution	O
.	O
</s>
<s>
0.0980	O
has	O
three	O
significant	B-Architecture
digits	I-Architecture
(	O
9	O
,	O
8	O
,	O
and	O
the	O
last	O
zero	O
)	O
if	O
they	O
are	O
within	O
the	O
measurement	O
resolution	O
.	O
</s>
<s>
120.000	O
consists	O
of	O
six	O
significant	B-Architecture
figures	I-Architecture
(	O
1	O
,	O
2	O
,	O
and	O
the	O
four	O
subsequent	O
zeroes	O
)	O
if	O
,	O
as	O
before	O
,	O
they	O
are	O
within	O
the	O
measurement	O
resolution	O
.	O
</s>
<s>
45,600	O
has	O
3	O
,	O
4	O
or	O
5	O
significant	B-Architecture
figures	I-Architecture
depending	O
on	O
how	O
the	O
last	O
zeros	O
are	O
used	O
.	O
</s>
<s>
An	O
exact	O
number	O
has	O
an	O
infinite	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
If	O
the	O
number	O
of	O
apples	O
in	O
a	O
bag	O
is	O
4	O
(	O
exact	O
number	O
)	O
,	O
then	O
this	O
number	O
is	O
0000	O
...	O
(	O
with	O
infinite	O
trailing	O
zeros	O
to	O
the	O
right	O
of	O
the	O
decimal	B-Application
point	O
)	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
4	O
does	O
not	O
impact	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
or	O
digits	O
in	O
the	O
result	O
of	O
calculations	O
with	O
it	O
.	O
</s>
<s>
A	O
mathematical	O
or	O
physical	O
constant	O
has	O
significant	B-Architecture
figures	I-Architecture
to	O
its	O
known	O
digits	O
.	O
</s>
<s>
π	O
,	O
as	O
the	O
ratio	O
of	O
the	O
circumference	O
to	O
the	O
diameter	O
of	O
a	O
circle	O
,	O
is	O
14159265358979323	O
...	O
known	O
to	O
more	O
than	O
62	O
trillion	O
digits	O
calculated	O
as	O
of	O
19	O
August	O
2021	O
,	O
and	O
that	O
calculated	O
'	O
π	O
 '	O
approximation	O
has	O
that	O
many	O
significant	B-Architecture
digits	I-Architecture
,	O
while	O
in	O
practical	O
applications	O
far	O
fewer	O
are	O
used	O
(	O
and	O
π	O
itself	O
has	O
infinite	O
significant	B-Architecture
digits	I-Architecture
,	O
as	O
all	O
irrational	O
numbers	O
do	O
)	O
.	O
</s>
<s>
3	O
significant	O
decimal	B-Application
digits	O
,	O
with	O
7	O
correct	O
binary	O
digits	O
(	O
while	O
the	O
more	O
accurate	O
22/7	O
is	O
also	O
used	O
,	O
even	O
though	O
it	O
also	O
only	O
amounts	O
to	O
the	O
same	O
3	O
significant	O
correct	O
decimal	B-Application
digits	O
,	O
it	O
has	O
10	O
correct	O
binary	O
digits	O
)	O
,	O
which	O
is	O
a	O
good	O
enough	O
approximation	O
for	O
many	O
practical	O
uses	O
.	O
</s>
<s>
Most	O
calculators	O
,	O
and	O
computer	O
programs	O
,	O
can	O
handle	O
3.141592653589793	O
,	O
16	O
decimal	B-Application
digits	O
,	O
that	O
is	O
commonly	O
used	O
in	O
computers	O
and	O
used	O
by	O
NASA	O
for	O
"	O
JPL	O
's	O
highest	O
accuracy	O
calculations	O
,	O
which	O
are	O
for	O
interplanetary	O
navigation	O
"	O
.	O
</s>
<s>
The	O
significance	O
of	O
trailing	O
zeros	O
in	O
a	O
number	O
not	O
containing	O
a	O
decimal	B-Application
point	O
can	O
be	O
ambiguous	O
.	O
</s>
<s>
For	O
example	O
,	O
it	O
may	O
not	O
always	O
be	O
clear	O
if	O
the	O
number	O
1300	O
is	O
precise	O
to	O
the	O
nearest	O
unit	O
(	O
just	O
happens	O
coincidentally	O
to	O
be	O
an	O
exact	O
multiple	O
of	O
a	O
hundred	O
)	O
or	O
if	O
it	O
is	O
only	O
shown	O
to	O
the	O
nearest	O
hundreds	O
due	O
to	O
rounding	B-Algorithm
or	O
uncertainty	O
.	O
</s>
<s>
An	O
overline	O
,	O
sometimes	O
also	O
called	O
an	O
overbar	O
,	O
or	O
less	O
accurately	O
,	O
a	O
vinculum	O
,	O
may	O
be	O
placed	O
over	O
the	O
last	O
significant	B-Architecture
figure	I-Architecture
;	O
any	O
trailing	O
zeros	O
following	O
this	O
are	O
insignificant	O
.	O
</s>
<s>
For	O
example	O
,	O
130	O
has	O
three	O
significant	B-Architecture
figures	I-Architecture
(	O
and	O
hence	O
indicates	O
that	O
the	O
number	O
is	O
precise	O
to	O
the	O
nearest	O
ten	O
)	O
.	O
</s>
<s>
Less	O
often	O
,	O
using	O
a	O
closely	O
related	O
convention	O
,	O
the	O
last	O
significant	B-Architecture
figure	I-Architecture
of	O
a	O
number	O
may	O
be	O
underlined	O
;	O
for	O
example	O
,	O
"	O
1300	O
"	O
has	O
two	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
A	O
decimal	B-Application
point	O
may	O
be	O
placed	O
after	O
the	O
number	O
;	O
for	O
example	O
"	O
1300.	O
"	O
</s>
<s>
For	O
example	O
,	O
the	O
precision	B-General_Concept
of	O
measurement	O
specified	O
as	O
1300g	O
is	O
ambiguous	O
,	O
while	O
if	O
stated	O
as	O
1.30kg	O
it	O
is	O
not	O
.	O
</s>
<s>
Eliminate	O
ambiguous	O
or	O
non-significant	O
zeros	O
by	O
using	O
Scientific	O
Notation	O
:	O
For	O
example	O
,	O
1300	O
with	O
three	O
significant	B-Architecture
figures	I-Architecture
becomes	O
.	O
</s>
<s>
The	O
part	O
of	O
the	O
representation	O
that	O
contains	O
the	O
significant	B-Architecture
figures	I-Architecture
(	O
1.30	O
or	O
1.23	O
)	O
is	O
known	O
as	O
the	O
significand	B-Algorithm
or	O
mantissa	B-Algorithm
.	O
</s>
<s>
The	O
digits	O
in	O
the	O
base	O
and	O
exponent	O
(	O
or	O
)	O
are	O
considered	O
exact	O
numbers	O
so	O
for	O
these	O
digits	O
,	O
significant	B-Architecture
figures	I-Architecture
are	O
irrelevant	O
.	O
</s>
<s>
Explicitly	O
state	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
(	O
the	O
abbreviation	O
s.f.	O
</s>
<s>
State	O
the	O
expected	O
variability	O
(	O
precision	B-General_Concept
)	O
explicitly	O
with	O
a	O
plus	O
–	O
minus	O
sign	O
,	O
as	O
in	O
20000±1%	O
.	O
</s>
<s>
This	O
also	O
allows	O
specifying	O
a	O
range	O
of	O
precision	B-General_Concept
in-between	O
powers	O
of	O
ten	O
.	O
</s>
<s>
Rounding	B-Algorithm
to	O
significant	B-Architecture
figures	I-Architecture
is	O
a	O
more	O
general-purpose	O
technique	O
than	O
rounding	B-Algorithm
to	O
n	O
digits	O
,	O
since	O
it	O
handles	O
numbers	O
of	O
different	O
scales	O
in	O
a	O
uniform	O
way	O
.	O
</s>
<s>
The	O
former	O
might	O
be	O
in	O
error	O
by	O
hundreds	O
,	O
and	O
the	O
latter	O
might	O
be	O
in	O
error	O
by	O
hundreds	O
of	O
thousands	O
,	O
but	O
both	O
have	O
two	O
significant	B-Architecture
figures	I-Architecture
(	O
5	O
and	O
2	O
)	O
.	O
</s>
<s>
To	O
round	O
a	O
number	O
to	O
n	O
significant	B-Architecture
figures	I-Architecture
:	O
</s>
<s>
For	O
example	O
,	O
if	O
we	O
want	O
to	O
round	O
1.2459	O
to	O
3	O
significant	B-Architecture
figures	I-Architecture
,	O
then	O
this	O
step	O
results	O
in	O
1.25	O
.	O
</s>
<s>
If	O
the	O
n	O
+	O
1	O
digit	O
is	O
5	O
not	O
followed	O
by	O
other	O
digits	O
or	O
followed	O
by	O
only	O
zeros	O
,	O
then	O
rounding	B-Algorithm
requires	O
a	O
tie-breaking	O
rule	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
round	O
1.25	O
to	O
2	O
significant	B-Architecture
figures	I-Architecture
:	O
</s>
<s>
This	O
is	O
the	O
default	O
rounding	B-Algorithm
method	O
implied	O
in	O
many	O
disciplines	O
if	O
the	O
required	O
rounding	B-Algorithm
method	O
is	O
not	O
specified	O
.	O
</s>
<s>
With	O
this	O
method	O
,	O
1.25	O
is	O
rounded	B-Algorithm
down	O
to	O
1.2	O
.	O
</s>
<s>
If	O
this	O
method	O
applies	O
to	O
1.35	O
,	O
then	O
it	O
is	O
rounded	B-Algorithm
up	O
to	O
1.4	O
.	O
</s>
<s>
For	O
an	O
integer	O
in	O
rounding	B-Algorithm
,	O
replace	O
the	O
digits	O
after	O
the	O
n	O
digit	O
with	O
zeros	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
1254	O
is	O
rounded	B-Algorithm
to	O
2	O
significant	B-Architecture
figures	I-Architecture
,	O
then	O
5	O
and	O
4	O
are	O
replaced	O
to	O
0	O
so	O
that	O
it	O
will	O
be	O
1300	O
.	O
</s>
<s>
For	O
a	O
number	O
with	O
the	O
decimal	B-Application
point	O
in	O
rounding	B-Algorithm
,	O
remove	O
the	O
digits	O
after	O
the	O
n	O
digit	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
14.895	O
is	O
rounded	B-Algorithm
to	O
3	O
significant	B-Architecture
figures	I-Architecture
,	O
then	O
the	O
digits	O
after	O
8	O
are	O
removed	O
so	O
that	O
it	O
will	O
be	O
14.9	O
.	O
</s>
<s>
In	O
financial	O
calculations	O
,	O
a	O
number	O
is	O
often	O
rounded	B-Algorithm
to	O
a	O
given	O
number	O
of	O
places	O
.	O
</s>
<s>
For	O
example	O
,	O
to	O
two	O
places	O
after	O
the	O
decimal	B-Application
separator	O
for	O
many	O
world	O
currencies	O
.	O
</s>
<s>
This	O
is	O
done	O
because	O
greater	O
precision	B-General_Concept
is	O
immaterial	O
,	O
and	O
usually	O
it	O
is	O
not	O
possible	O
to	O
settle	O
a	O
debt	O
of	O
less	O
than	O
the	O
smallest	O
currency	O
unit	O
.	O
</s>
<s>
In	O
UK	O
personal	O
tax	O
returns	O
,	O
income	O
is	O
rounded	B-Algorithm
down	O
to	O
the	O
nearest	O
pound	O
,	O
whilst	O
tax	O
paid	O
is	O
calculated	O
to	O
the	O
nearest	O
penny	O
.	O
</s>
<s>
As	O
an	O
illustration	O
,	O
the	O
decimal	B-Application
quantity	O
12.345	O
can	O
be	O
expressed	O
with	O
various	O
numbers	O
of	O
significant	B-Architecture
figures	I-Architecture
or	O
decimal	B-Architecture
places	I-Architecture
.	O
</s>
<s>
If	O
insufficient	O
precision	B-General_Concept
is	O
available	O
then	O
the	O
number	O
is	O
rounded	B-Algorithm
in	O
some	O
manner	O
to	O
fit	O
the	O
available	O
precision	B-General_Concept
.	O
</s>
<s>
The	O
following	O
table	O
shows	O
the	O
results	O
for	O
various	O
total	O
precision	B-General_Concept
at	O
two	O
rounding	B-Algorithm
ways	O
(	O
N/A	O
stands	O
for	O
Not	O
Applicable	O
)	O
.	O
</s>
<s>
The	O
representation	O
of	O
a	O
non-zero	O
number	O
x	O
to	O
a	O
precision	B-General_Concept
of	O
p	O
significant	B-Architecture
digits	I-Architecture
has	O
a	O
numerical	O
value	O
that	O
is	O
given	O
by	O
the	O
formula	O
:	O
</s>
<s>
σx	O
has	O
only	O
one	O
or	O
two	O
significant	B-Architecture
figures	I-Architecture
as	O
more	O
precise	O
uncertainty	O
has	O
no	O
meaning	O
.	O
</s>
<s>
The	O
digit	O
positions	O
of	O
the	O
last	O
significant	B-Architecture
figures	I-Architecture
in	O
xbest	O
and	O
σx	O
are	O
the	O
same	O
,	O
otherwise	O
the	O
consistency	O
is	O
lost	O
.	O
</s>
<s>
1.79	O
±	O
0.9	O
(	O
incorrect	O
)	O
also	O
does	O
not	O
make	O
sense	O
since	O
the	O
rounding	B-Algorithm
guideline	O
for	O
addition	O
and	O
subtraction	O
below	O
tells	O
that	O
the	O
edges	O
of	O
the	O
true	O
value	O
range	O
are	O
2.7	O
and	O
0.9	O
,	O
that	O
are	O
less	O
accurate	O
than	O
the	O
best	O
estimate	O
.	O
</s>
<s>
In	O
chemistry	O
(	O
and	O
may	O
also	O
be	O
for	O
other	O
scientific	O
branches	O
)	O
,	O
uncertainty	O
may	O
be	O
implied	O
by	O
the	O
last	O
significant	B-Architecture
figure	I-Architecture
if	O
it	O
is	O
not	O
explicitly	O
expressed	O
.	O
</s>
<s>
The	O
implied	O
uncertainty	O
is	O
±	O
the	O
half	O
of	O
the	O
minimum	O
scale	O
at	O
the	O
last	O
significant	B-Architecture
figure	I-Architecture
position	O
.	O
</s>
<s>
As	O
there	O
are	O
rules	O
to	O
determine	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
directly	O
measured	O
quantities	O
,	O
there	O
are	O
also	O
guidelines	O
(	O
not	O
rules	O
)	O
to	O
determine	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
quantities	O
calculated	O
from	O
these	O
measured	O
quantities	O
.	O
</s>
<s>
Significant	B-Architecture
figures	I-Architecture
in	O
measured	O
quantities	O
are	O
most	O
important	O
in	O
the	O
determination	O
of	O
significant	B-Architecture
figures	I-Architecture
in	O
calculated	O
quantities	O
with	O
them	O
.	O
</s>
<s>
A	O
mathematical	O
or	O
physical	O
constant	O
(	O
e.g.	O
,	O
in	O
the	O
formula	O
for	O
the	O
area	O
of	O
a	O
circle	O
with	O
radius	O
as	O
)	O
has	O
no	O
effect	O
on	O
the	O
determination	O
of	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
result	O
of	O
a	O
calculation	O
with	O
it	O
if	O
its	O
known	O
digits	O
are	O
equal	O
to	O
or	O
more	O
than	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
measured	O
quantities	O
used	O
in	O
the	O
calculation	O
.	O
</s>
<s>
An	O
exact	O
number	O
such	O
as	O
in	O
the	O
formula	O
for	O
the	O
kinetic	O
energy	O
of	O
a	O
mass	O
with	O
velocity	O
as	O
has	O
no	O
bearing	O
on	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
calculated	O
kinetic	O
energy	O
since	O
its	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
is	O
infinite	O
(	O
500000	O
...	O
)	O
.	O
</s>
<s>
If	O
the	O
guidelines	O
give	O
the	O
implied	O
uncertainty	O
too	O
far	O
from	O
the	O
measured	O
ones	O
,	O
then	O
it	O
may	O
be	O
needed	O
to	O
decide	O
significant	B-Architecture
digits	I-Architecture
that	O
give	O
comparable	O
uncertainty	O
.	O
</s>
<s>
For	O
quantities	O
created	O
from	O
measured	O
quantities	O
via	O
multiplication	O
and	O
division	O
,	O
the	O
calculated	O
result	O
should	O
have	O
as	O
many	O
significant	B-Architecture
figures	I-Architecture
as	O
the	O
least	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
among	O
the	O
measured	O
quantities	O
used	O
in	O
the	O
calculation	O
.	O
</s>
<s>
with	O
one	O
,	O
two	O
,	O
and	O
one	O
significant	B-Architecture
figures	I-Architecture
respectively	O
.	O
</s>
<s>
For	O
the	O
first	O
example	O
,	O
the	O
first	O
multiplication	O
factor	O
has	O
four	O
significant	B-Architecture
figures	I-Architecture
and	O
the	O
second	O
has	O
one	O
significant	B-Architecture
figure	I-Architecture
.	O
</s>
<s>
The	O
factor	O
with	O
the	O
fewest	O
or	O
least	O
significant	B-Architecture
figures	I-Architecture
is	O
the	O
second	O
one	O
with	O
only	O
one	O
,	O
so	O
the	O
final	O
calculated	O
result	O
should	O
also	O
have	O
one	O
significant	B-Architecture
figure	I-Architecture
.	O
</s>
<s>
For	O
unit	O
conversion	O
,	O
the	O
implied	O
uncertainty	O
of	O
the	O
result	O
can	O
be	O
unsatisfactorily	O
higher	O
than	O
that	O
in	O
the	O
previous	O
unit	O
if	O
this	O
rounding	B-Algorithm
guideline	O
is	O
followed	O
;	O
For	O
example	O
,	O
8	O
inch	O
has	O
the	O
implied	O
uncertainty	O
of	O
±	O
0.5inch	O
=	O
±	O
1.27cm	O
.	O
</s>
<s>
If	O
it	O
is	O
converted	O
to	O
the	O
centimeter	O
scale	O
and	O
the	O
rounding	B-Algorithm
guideline	O
for	O
multiplication	O
and	O
division	O
is	O
followed	O
,	O
then	O
0.32cm	O
≈	O
20cm	O
with	O
the	O
implied	O
uncertainty	O
of	O
±5cm	O
.	O
</s>
<s>
If	O
this	O
implied	O
uncertainty	O
is	O
considered	O
as	O
too	O
overestimated	O
,	O
then	O
more	O
proper	O
significant	B-Architecture
digits	I-Architecture
in	O
the	O
unit	O
conversion	O
result	O
may	O
be	O
2.32cm	O
≈	O
20.cm	O
with	O
the	O
implied	O
uncertainty	O
of	O
±	O
0.5cm	O
.	O
</s>
<s>
Another	O
exception	O
of	O
applying	O
the	O
above	O
rounding	B-Algorithm
guideline	O
is	O
to	O
multiply	O
a	O
number	O
by	O
an	O
integer	O
,	O
such	O
as	O
1.234	O
×	O
9	O
.	O
</s>
<s>
If	O
the	O
above	O
guideline	O
is	O
followed	O
,	O
then	O
the	O
result	O
is	O
rounded	B-Algorithm
as	O
1.234	O
×	O
000	O
....	O
=	O
11.16	O
≈	O
11.11	O
.	O
</s>
<s>
However	O
,	O
this	O
multiplication	O
is	O
essentially	O
adding	O
1.234	O
to	O
itself	O
9	O
times	O
such	O
as	O
1.234	O
+	O
1.234	O
+	O
…	O
+	O
1.234	O
so	O
the	O
rounding	B-Algorithm
guideline	O
for	O
addition	O
and	O
subtraction	O
described	O
below	O
is	O
more	O
proper	O
rounding	B-Algorithm
approach	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
final	O
answer	O
is	O
1.234	O
+	O
1.234	O
+	O
…	O
+	O
1.234	O
=	O
11.10	O
=	O
11.106	O
(	O
one	O
significant	B-Architecture
digit	I-Architecture
increase	O
)	O
.	O
</s>
<s>
For	O
quantities	O
created	O
from	O
measured	O
quantities	O
via	O
addition	O
and	O
subtraction	O
,	O
the	O
last	O
significant	B-Architecture
figure	I-Architecture
position	O
(	O
e.g.	O
,	O
hundreds	O
,	O
tens	O
,	O
ones	O
,	O
tenths	O
,	O
hundredths	O
,	O
and	O
so	O
forth	O
)	O
in	O
the	O
calculated	O
result	O
should	O
be	O
the	O
same	O
as	O
the	O
leftmost	O
or	O
largest	O
digit	O
position	O
among	O
the	O
last	O
significant	B-Architecture
figures	I-Architecture
of	O
the	O
measured	O
quantities	O
in	O
the	O
calculation	O
.	O
</s>
<s>
with	O
the	O
last	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
ones	O
place	O
,	O
tenths	O
place	O
,	O
and	O
ones	O
place	O
respectively	O
.	O
</s>
<s>
For	O
the	O
first	O
example	O
,	O
the	O
first	O
term	O
has	O
its	O
last	O
significant	B-Architecture
figure	I-Architecture
in	O
the	O
thousandths	O
place	O
and	O
the	O
second	O
term	O
has	O
its	O
last	O
significant	B-Architecture
figure	I-Architecture
in	O
the	O
ones	O
place	O
.	O
</s>
<s>
The	O
leftmost	O
or	O
largest	O
digit	O
position	O
among	O
the	O
last	O
significant	B-Architecture
figures	I-Architecture
of	O
these	O
terms	O
is	O
the	O
ones	O
place	O
,	O
so	O
the	O
calculated	O
result	O
should	O
also	O
have	O
its	O
last	O
significant	B-Architecture
figure	I-Architecture
in	O
the	O
ones	O
place	O
.	O
</s>
<s>
The	O
rule	O
to	O
calculate	O
significant	B-Architecture
figures	I-Architecture
for	O
multiplication	O
and	O
division	O
are	O
not	O
the	O
same	O
as	O
the	O
rule	O
for	O
addition	O
and	O
subtraction	O
.	O
</s>
<s>
For	O
multiplication	O
and	O
division	O
,	O
only	O
the	O
total	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
in	O
each	O
of	O
the	O
factors	O
in	O
the	O
calculation	O
matters	O
;	O
the	O
digit	O
position	O
of	O
the	O
last	O
significant	B-Architecture
figure	I-Architecture
in	O
each	O
factor	O
is	O
irrelevant	O
.	O
</s>
<s>
For	O
addition	O
and	O
subtraction	O
,	O
only	O
the	O
digit	O
position	O
of	O
the	O
last	O
significant	B-Architecture
figure	I-Architecture
in	O
each	O
of	O
the	O
terms	O
in	O
the	O
calculation	O
matters	O
;	O
the	O
total	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
in	O
each	O
term	O
is	O
irrelevant	O
.	O
</s>
<s>
The	O
base-10	O
logarithm	O
of	O
a	O
normalized	B-Algorithm
number	I-Algorithm
(	O
i.e.	O
,	O
a	O
×	O
10b	O
with	O
1	O
≤	O
a	O
<	O
10	O
and	O
b	O
as	O
an	O
integer	O
)	O
,	O
is	O
rounded	B-Algorithm
such	O
that	O
its	O
decimal	B-Application
part	O
(	O
called	O
mantissa	B-Algorithm
)	O
has	O
as	O
many	O
significant	B-Architecture
figures	I-Architecture
as	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
normalized	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
log10( 	O
3.000	O
×	O
104	O
)	O
=	O
log10(104 )	O
+	O
log10( 	O
3.000	O
)	O
=	O
000000	O
...	O
(	O
exact	O
number	O
so	O
infinite	O
significant	B-Architecture
digits	I-Architecture
)	O
+	O
477212547	O
...	O
=	O
4.477212547	O
≈	O
4.4771	O
.	O
</s>
<s>
When	O
taking	O
the	O
antilogarithm	O
of	O
a	O
normalized	B-Algorithm
number	I-Algorithm
,	O
the	O
result	O
is	O
rounded	B-Algorithm
to	O
have	O
as	O
many	O
significant	B-Architecture
figures	I-Architecture
as	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
decimal	B-Application
part	O
of	O
the	O
number	O
to	O
be	O
antiloged	O
.	O
</s>
<s>
where	O
is	O
the	O
condition	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
See	O
the	O
significance	B-Algorithm
arithmetic	I-Algorithm
article	O
to	O
find	O
its	O
derivation	O
.	O
</s>
<s>
When	O
performing	O
multiple	O
stage	O
calculations	O
,	O
do	O
not	O
round	O
intermediate	O
stage	O
calculation	O
results	O
;	O
keep	O
as	O
many	O
digits	O
as	O
is	O
practical	O
(	O
at	O
least	O
one	O
more	O
digit	O
than	O
the	O
rounding	B-Algorithm
rule	O
allows	O
per	O
stage	O
)	O
until	O
the	O
end	O
of	O
all	O
the	O
calculations	O
to	O
avoid	O
cumulative	O
rounding	B-Algorithm
errors	I-Algorithm
while	O
tracking	O
or	O
recording	O
the	O
significant	B-Architecture
figures	I-Architecture
in	O
each	O
intermediate	O
result	O
.	O
</s>
<s>
Then	O
,	O
round	O
the	O
final	O
result	O
,	O
for	O
example	O
,	O
to	O
the	O
fewest	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
(	O
for	O
multiplication	O
or	O
division	O
)	O
or	O
leftmost	O
last	O
significant	B-Architecture
digit	I-Architecture
position	O
(	O
for	O
addition	O
or	O
subtraction	O
)	O
among	O
the	O
inputs	O
in	O
the	O
final	O
calculation	O
.	O
</s>
<s>
However	O
assuming	O
a	O
normal	O
good	O
quality	O
ruler	O
,	O
it	O
should	O
be	O
possible	O
to	O
estimate	O
tenths	O
between	O
the	O
nearest	O
two	O
marks	O
to	O
achieve	O
an	O
extra	O
decimal	B-Architecture
place	I-Architecture
of	O
accuracy	O
.	O
</s>
<s>
When	O
estimating	O
the	O
proportion	O
of	O
individuals	O
carrying	O
some	O
particular	O
characteristic	O
in	O
a	O
population	O
,	O
from	O
a	O
random	O
sample	O
of	O
that	O
population	O
,	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
should	O
not	O
exceed	O
the	O
maximum	O
precision	B-General_Concept
allowed	O
by	O
that	O
sample	O
size	O
.	O
</s>
<s>
Traditionally	O
,	O
in	O
various	O
technical	O
fields	O
,	O
"	O
accuracy	O
"	O
refers	O
to	O
the	O
closeness	O
of	O
a	O
given	O
measurement	O
to	O
its	O
true	O
value	O
;	O
"	O
precision	B-General_Concept
"	O
refers	O
to	O
the	O
stability	O
of	O
that	O
measurement	O
when	O
repeated	O
many	O
times	O
.	O
</s>
<s>
Hoping	O
to	O
reflect	O
the	O
way	O
in	O
which	O
the	O
term	O
"	O
accuracy	O
"	O
is	O
actually	O
used	O
in	O
the	O
scientific	O
community	O
,	O
there	O
is	O
a	O
recent	O
standard	O
,	O
ISO	O
5725	O
,	O
which	O
keeps	O
the	O
same	O
definition	O
of	O
precision	B-General_Concept
but	O
defines	O
the	O
term	O
"	O
trueness	O
"	O
as	O
the	O
closeness	O
of	O
a	O
given	O
measurement	O
to	O
its	O
true	O
value	O
and	O
uses	O
the	O
term	O
"	O
accuracy	O
"	O
as	O
the	O
combination	O
of	O
trueness	O
and	O
precision	B-General_Concept
.	O
</s>
<s>
(	O
See	O
the	O
accuracy	O
and	O
precision	B-General_Concept
article	O
for	O
a	O
full	O
discussion	O
.	O
)	O
</s>
<s>
In	O
either	O
case	O
,	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
roughly	O
corresponds	O
to	O
precision	B-General_Concept
,	O
not	O
to	O
accuracy	O
or	O
the	O
newer	O
concept	O
of	O
trueness	O
.	O
</s>
<s>
Computer	O
representations	O
of	O
floating-point	O
numbers	O
use	O
a	O
form	O
of	O
rounding	B-Algorithm
to	O
significant	B-Architecture
figures	I-Architecture
(	O
while	O
usually	O
not	O
keeping	O
track	O
of	O
how	O
many	O
)	O
,	O
in	O
general	O
with	O
binary	O
numbers	O
.	O
</s>
<s>
The	O
number	O
of	O
correct	O
significant	B-Architecture
figures	I-Architecture
is	O
closely	O
related	O
to	O
the	O
notion	O
of	O
relative	B-Algorithm
error	I-Algorithm
(	O
which	O
has	O
the	O
advantage	O
of	O
being	O
a	O
more	O
accurate	O
measure	O
of	O
precision	B-General_Concept
,	O
and	O
is	O
independent	O
of	O
the	O
radix	O
,	O
also	O
known	O
as	O
the	O
base	O
,	O
of	O
the	O
number	O
system	O
used	O
)	O
.	O
</s>
