<s>
Significance	B-Algorithm
arithmetic	I-Algorithm
is	O
a	O
set	O
of	O
rules	O
(	O
sometimes	O
called	O
significant	B-Architecture
figure	I-Architecture
rules	O
)	O
for	O
approximating	O
the	O
propagation	B-General_Concept
of	I-General_Concept
uncertainty	I-General_Concept
in	O
scientific	O
or	O
statistical	O
calculations	O
.	O
</s>
<s>
These	O
rules	O
can	O
be	O
used	O
to	O
find	O
the	O
appropriate	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
to	O
use	O
to	O
represent	O
the	O
result	O
of	O
a	O
calculation	O
.	O
</s>
<s>
If	O
a	O
calculation	O
is	O
done	O
without	O
analysis	O
of	O
the	O
uncertainty	O
involved	O
,	O
a	O
result	O
that	O
is	O
written	O
with	O
too	O
many	O
significant	B-Architecture
figures	I-Architecture
can	O
be	O
taken	O
to	O
imply	O
a	O
higher	O
precision	B-Architecture
than	O
is	O
known	O
,	O
and	O
a	O
result	O
that	O
is	O
written	O
with	O
too	O
few	O
significant	B-Architecture
figures	I-Architecture
results	O
in	O
an	O
avoidable	O
loss	O
of	O
precision	B-Architecture
.	O
</s>
<s>
Understanding	O
these	O
rules	O
requires	O
a	O
good	O
understanding	O
of	O
the	O
concept	O
of	O
significant	B-Architecture
and	I-Architecture
insignificant	I-Architecture
figures	I-Architecture
.	O
</s>
<s>
The	O
rules	O
of	O
significance	B-Algorithm
arithmetic	I-Algorithm
are	O
an	O
approximation	O
based	O
on	O
statistical	O
rules	O
for	O
dealing	O
with	O
probability	O
distributions	O
.	O
</s>
<s>
See	O
the	O
article	O
on	O
propagation	B-General_Concept
of	I-General_Concept
uncertainty	I-General_Concept
for	O
these	O
more	O
advanced	O
and	O
precise	O
rules	O
.	O
</s>
<s>
Significance	B-Algorithm
arithmetic	I-Algorithm
rules	O
rely	O
on	O
the	O
assumption	O
that	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
operands	O
gives	O
accurate	O
information	O
about	O
the	O
uncertainty	O
of	O
the	O
operands	O
and	O
hence	O
the	O
uncertainty	O
of	O
the	O
result	O
.	O
</s>
<s>
For	O
alternatives	O
see	O
Interval	B-Algorithm
arithmetic	I-Algorithm
and	O
Floating-point	O
error	O
mitigation	O
.	O
</s>
<s>
An	O
important	O
caveat	O
is	O
that	O
significant	B-Architecture
figures	I-Architecture
apply	O
only	O
to	O
measured	O
values	O
.	O
</s>
<s>
Values	O
known	O
to	O
be	O
exact	O
should	O
be	O
ignored	O
for	O
determining	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
that	O
belong	O
in	O
the	O
result	O
.	O
</s>
<s>
Physical	O
constants	O
such	O
as	O
the	O
gravitational	O
constant	O
,	O
however	O
,	O
have	O
a	O
limited	O
number	O
of	O
significant	B-Architecture
digits	I-Architecture
,	O
because	O
these	O
constants	O
are	O
known	O
to	O
us	O
only	O
by	O
measurement	O
.	O
</s>
<s>
When	O
multiplying	O
or	O
dividing	O
numbers	O
,	O
the	O
result	O
is	O
rounded	B-Algorithm
to	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
in	O
the	O
factor	O
with	O
the	O
least	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
Here	O
,	O
the	O
quantity	O
of	O
significant	B-Architecture
figures	I-Architecture
in	O
each	O
of	O
the	O
factors	O
is	O
important	O
—	O
not	O
the	O
position	O
of	O
the	O
significant	B-Architecture
figures	I-Architecture
.	O
</s>
<s>
For	O
instance	O
,	O
using	O
significance	B-Algorithm
arithmetic	I-Algorithm
rules	O
:	O
</s>
<s>
If	O
,	O
in	O
the	O
above	O
,	O
the	O
numbers	O
are	O
assumed	O
to	O
be	O
measurements	O
(	O
and	O
therefore	O
probably	O
inexact	O
)	O
then	O
"	O
8	O
"	O
above	O
represents	O
an	O
inexact	O
measurement	O
with	O
only	O
one	O
significant	B-Architecture
digit	I-Architecture
.	O
</s>
<s>
Therefore	O
,	O
the	O
result	O
of	O
is	O
rounded	B-Algorithm
to	O
a	O
result	O
with	O
only	O
one	O
significant	B-Architecture
digit	I-Architecture
,	O
i.e.	O
,	O
instead	O
of	O
the	O
unrounded	O
"	O
64	O
"	O
that	O
one	O
might	O
expect	O
.	O
</s>
<s>
In	O
many	O
cases	O
,	O
the	O
rounded	B-Algorithm
result	O
is	O
less	O
accurate	O
than	O
the	O
non-rounded	O
result	O
;	O
a	O
measurement	O
of	O
"	O
8	O
"	O
has	O
an	O
actual	O
underlying	O
quantity	O
between	O
7.5	O
and	O
8.5	O
.	O
</s>
<s>
When	O
adding	O
or	O
subtracting	O
using	O
significant	B-Architecture
figures	I-Architecture
rules	O
,	O
results	O
are	O
rounded	B-Algorithm
to	O
the	O
position	O
of	O
the	O
least	O
significant	B-Architecture
digit	I-Architecture
in	O
the	O
most	O
uncertain	O
of	O
the	O
numbers	O
being	O
added	O
(	O
or	O
subtracted	O
)	O
.	O
</s>
<s>
That	O
is	O
,	O
the	O
result	O
is	O
rounded	B-Algorithm
to	O
the	O
last	B-Architecture
digit	I-Architecture
that	O
is	O
significant	O
in	O
each	O
of	O
the	O
numbers	O
being	O
summed	O
.	O
</s>
<s>
Here	O
the	O
position	O
of	O
the	O
significant	B-Architecture
figures	I-Architecture
is	O
important	O
,	O
but	O
the	O
quantity	O
of	O
significant	B-Architecture
figures	I-Architecture
is	O
irrelevant	O
.	O
</s>
<s>
The	O
answer	O
cannot	O
have	O
any	O
significant	B-Architecture
figures	I-Architecture
past	O
the	O
ones	O
place	O
.	O
</s>
<s>
The	O
answer	O
should	O
not	O
have	O
significant	B-Architecture
digits	I-Architecture
past	O
the	O
hundreds	O
place	O
.	O
</s>
<s>
The	O
answer	O
will	O
have	O
no	O
significant	B-Architecture
figures	I-Architecture
past	O
the	O
tens	O
place	O
.	O
</s>
<s>
It	O
may	O
seem	O
counter-intuitive	O
,	O
but	O
giving	O
the	O
nature	O
of	O
significant	B-Architecture
digits	I-Architecture
dictating	O
precision	B-Architecture
,	O
we	O
can	O
see	O
how	O
this	O
follows	O
from	O
the	O
standard	O
rules	O
.	O
</s>
<s>
The	O
significance	O
of	O
the	O
output	O
depends	O
on	O
the	O
condition	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
In	O
general	O
,	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
of	O
the	O
output	O
is	O
equal	O
to	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
of	O
the	O
function	O
input	O
(	O
function	O
argument	O
)	O
minus	O
the	O
order	O
of	O
magnitude	O
of	O
the	O
condition	B-Algorithm
number	I-Algorithm
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
of	O
a	O
differentiable	O
function	O
at	O
a	O
point	O
is	O
see	O
Condition	B-Algorithm
number	I-Algorithm
:	O
One	O
variable	O
for	O
details	O
.	O
</s>
<s>
Note	O
that	O
if	O
a	O
function	O
has	O
a	O
zero	O
at	O
a	O
point	O
,	O
its	O
condition	B-Algorithm
number	I-Algorithm
at	O
the	O
point	O
is	O
infinite	O
,	O
as	O
infinitesimal	O
changes	O
in	O
the	O
input	O
can	O
change	O
the	O
output	O
from	O
zero	O
to	O
non-zero	O
,	O
yielding	O
a	O
ratio	O
with	O
zero	O
in	O
the	O
denominator	O
,	O
hence	O
an	O
infinite	O
relative	O
change	O
.	O
</s>
<s>
The	O
condition	B-Algorithm
number	I-Algorithm
of	O
the	O
most	O
used	O
functions	O
are	O
as	O
follows	O
;	O
these	O
can	O
be	O
used	O
to	O
compute	O
significant	B-Architecture
figures	I-Architecture
for	O
all	O
elementary	O
functions	O
:	O
</s>
<s>
The	O
fact	O
that	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
of	O
the	O
function	O
output	O
is	O
equal	O
to	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
of	O
the	O
function	O
input	O
(	O
function	O
argument	O
)	O
minus	O
the	O
base-10	O
logarithm	O
of	O
the	O
condition	B-Algorithm
number	I-Algorithm
(	O
which	O
is	O
approximately	O
the	O
order	O
of	O
magnitude/number	O
of	O
digits	O
of	O
the	O
condition	B-Algorithm
number	I-Algorithm
)	O
can	O
be	O
easily	O
derived	O
from	O
first	O
principles	O
:	O
let	O
and	O
be	O
the	O
true	O
values	O
and	O
let	O
and	O
be	O
approximate	O
values	O
with	O
errors	O
and	O
respectively	O
,	O
so	O
that	O
and	O
.	O
</s>
<s>
The	O
significant	B-Architecture
figures	I-Architecture
of	O
a	O
number	O
is	O
related	O
to	O
the	O
uncertain	O
error	O
of	O
the	O
number	O
by	O
where	O
"	O
significant	B-Architecture
figures	I-Architecture
of	O
x	O
"	O
here	O
means	O
the	O
number	O
of	O
significant	B-Architecture
figures	I-Architecture
of	O
x	O
.	O
</s>
<s>
Because	O
significance	B-Algorithm
arithmetic	I-Algorithm
involves	O
rounding	B-Algorithm
,	O
it	O
is	O
useful	O
to	O
understand	O
a	O
specific	O
rounding	B-Algorithm
rule	O
that	O
is	O
often	O
used	O
when	O
doing	O
scientific	O
calculations	O
:	O
the	O
round-to-even	O
rule	O
(	O
also	O
called	O
banker	O
's	O
rounding	B-Algorithm
)	O
.	O
</s>
<s>
This	O
rule	O
helps	O
to	O
eliminate	O
the	O
upwards	O
skewing	O
of	O
data	O
when	O
using	O
traditional	O
rounding	B-Algorithm
rules	O
.	O
</s>
<s>
Whereas	O
traditional	O
rounding	B-Algorithm
always	O
rounds	O
up	O
when	O
the	O
following	O
digit	O
is	O
5	O
,	O
bankers	O
sometimes	O
round	O
down	O
to	O
eliminate	O
this	O
upwards	O
bias	O
.	O
</s>
<s>
See	O
the	O
article	O
on	O
rounding	B-Algorithm
for	O
more	O
information	O
on	O
rounding	B-Algorithm
rules	O
and	O
a	O
detailed	O
explanation	O
of	O
the	O
round-to-even	O
rule	O
.	O
</s>
<s>
Significant	B-Architecture
figures	I-Architecture
are	O
used	O
extensively	O
in	O
high	O
school	O
and	O
undergraduate	O
courses	O
as	O
a	O
shorthand	O
for	O
the	O
precision	B-Architecture
with	O
which	O
a	O
measurement	O
is	O
known	O
.	O
</s>
<s>
However	O
,	O
significant	B-Architecture
figures	I-Architecture
are	O
not	O
a	O
perfect	O
representation	O
of	O
uncertainty	O
,	O
and	O
are	O
not	O
meant	O
to	O
be	O
.	O
</s>
<s>
Instead	O
,	O
they	O
are	O
a	O
useful	O
tool	O
for	O
avoiding	O
expressing	O
more	O
information	O
than	O
the	O
experimenter	O
actually	O
knows	O
,	O
and	O
for	O
avoiding	O
rounding	B-Algorithm
numbers	I-Algorithm
in	O
such	O
a	O
way	O
as	O
to	O
lose	O
precision	B-Architecture
.	O
</s>
<s>
For	O
example	O
,	O
here	O
are	O
some	O
important	O
differences	O
between	O
significant	B-Architecture
figure	I-Architecture
rules	O
and	O
uncertainty	O
:	O
</s>
<s>
Significance	O
is	O
not	O
the	O
same	O
as	O
significant	B-Architecture
digits	I-Architecture
.	O
</s>
<s>
Manual	O
,	O
algebraic	O
propagation	B-General_Concept
of	I-General_Concept
uncertainty	I-General_Concept
—	O
the	O
nominal	O
topic	O
of	O
this	O
article	O
—	O
is	O
possible	O
,	O
but	O
challenging	O
.	O
</s>
<s>
Alternative	O
methods	O
include	O
the	O
method	O
and	O
the	O
Monte	B-Algorithm
Carlo	I-Algorithm
method	I-Algorithm
.	O
</s>
<s>
Another	O
option	O
is	O
interval	B-Algorithm
arithmetic	I-Algorithm
,	O
which	O
can	O
provide	O
a	O
strict	O
upper	O
bound	O
on	O
the	O
uncertainty	O
,	O
but	O
generally	O
it	O
is	O
not	O
a	O
tight	O
upper	O
bound	O
(	O
i.e.	O
</s>
<s>
For	O
most	O
purposes	O
,	O
Monte	O
Carlo	O
is	O
more	O
useful	O
than	O
interval	B-Algorithm
arithmetic	I-Algorithm
.	O
</s>
<s>
Kahan	O
considers	O
significance	B-Algorithm
arithmetic	I-Algorithm
to	O
be	O
unreliable	O
as	O
a	O
form	O
of	O
automated	O
error	O
analysis	O
.	O
</s>
<s>
If	O
the	O
confidence	O
interval	O
is	O
not	O
specified	O
it	O
has	O
traditionally	O
been	O
assumed	O
to	O
be	O
95%	O
corresponding	O
to	O
two	O
standard	B-General_Concept
deviations	I-General_Concept
from	O
the	O
mean	O
.	O
</s>
<s>
Confidence	O
intervals	O
at	O
one	O
standard	B-General_Concept
deviation	I-General_Concept
(	O
68%	O
)	O
and	O
three	O
standard	B-General_Concept
deviations	I-General_Concept
(	O
99%	O
)	O
are	O
also	O
commonly	O
used	O
.	O
</s>
