<s>
The	O
significand	B-Algorithm
(	O
also	O
mantissa	B-Algorithm
or	O
coefficient	O
,	O
sometimes	O
also	O
argument	O
,	O
or	O
ambiguously	O
fraction	O
or	O
characteristic	O
)	O
is	O
part	O
of	O
a	O
number	O
in	O
scientific	O
notation	O
or	O
in	O
floating-point	B-Algorithm
representation	I-Algorithm
,	O
consisting	O
of	O
its	O
significant	B-Architecture
digits	I-Architecture
.	O
</s>
<s>
Depending	O
on	O
the	O
interpretation	O
of	O
the	O
exponent	O
,	O
the	O
significand	B-Algorithm
may	O
represent	O
an	O
integer	O
or	O
a	O
fraction	O
.	O
</s>
<s>
The	O
number	O
123.45	O
can	O
be	O
represented	O
as	O
a	O
decimal	B-Application
floating-point	B-Algorithm
number	I-Algorithm
with	O
the	O
integer	O
12345	O
as	O
the	O
significand	B-Algorithm
and	O
a	O
10−2	O
power	O
term	O
,	O
also	O
called	O
characteristics	O
,	O
where	O
−2	O
is	O
the	O
exponent	O
(	O
and	O
10	O
is	O
the	O
base	O
)	O
.	O
</s>
<s>
Schmid	O
,	O
however	O
,	O
called	O
this	O
representation	O
with	O
a	O
significand	B-Algorithm
ranging	O
between	O
1.0	O
and	O
10	O
a	O
modified	B-Algorithm
normalized	I-Algorithm
form	I-Algorithm
.	O
</s>
<s>
For	O
base	O
2	O
,	O
this	O
1.xxxx	O
form	O
is	O
also	O
called	O
a	O
normalized	B-Algorithm
significand	I-Algorithm
.	O
</s>
<s>
Schmid	O
called	O
this	O
representation	O
with	O
a	O
significand	B-Algorithm
ranging	O
between	O
0.1	O
and	O
1.0	O
the	O
true	B-Algorithm
normalized	I-Algorithm
form	I-Algorithm
.	O
</s>
<s>
For	O
base	O
2	O
,	O
this	O
0.xxxx	O
form	O
is	O
also	O
called	O
a	O
normed	B-Algorithm
significand	I-Algorithm
.	O
</s>
<s>
For	O
a	O
normalized	B-Algorithm
number	I-Algorithm
,	O
the	O
most	O
significant	B-Architecture
digit	I-Architecture
is	O
always	O
non-zero	O
.	O
</s>
<s>
The	O
significand	B-Algorithm
is	O
characterized	O
by	O
its	O
width	O
in	O
(	O
binary	O
)	O
digits	O
,	O
and	O
depending	O
on	O
the	O
context	O
,	O
the	O
hidden	O
bit	O
may	O
or	O
may	O
not	O
be	O
counted	O
towards	O
the	O
width	O
of	O
the	O
significand	B-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
same	O
IEEE	O
754	O
double-precision	O
format	O
is	O
commonly	O
described	O
as	O
having	O
either	O
a	O
53-bit	O
significand	B-Algorithm
,	O
including	O
the	O
hidden	O
bit	O
,	O
or	O
a	O
52-bit	O
significand	B-Algorithm
,	O
excluding	O
the	O
hidden	O
bit	O
.	O
</s>
<s>
IEEE	O
754	O
defines	O
the	O
precision	B-Architecture
p	O
to	O
be	O
the	O
number	O
of	O
digits	O
in	O
the	O
significand	B-Algorithm
,	O
including	O
any	O
implicit	O
leading	O
bit	O
(	O
e.g.	O
,	O
p	O
=	O
53	O
for	O
the	O
double-precision	O
format	O
)	O
,	O
thus	O
in	O
a	O
way	O
independent	O
from	O
the	O
encoding	O
,	O
and	O
the	O
term	O
to	O
express	O
what	O
is	O
encoded	O
(	O
that	O
is	O
,	O
the	O
significand	B-Algorithm
without	O
its	O
leading	O
bit	O
)	O
is	O
trailing	O
significand	B-Algorithm
field	O
.	O
</s>
<s>
The	O
term	O
significand	B-Algorithm
was	O
introduced	O
by	O
George	O
Forsythe	O
and	O
Cleve	O
Moler	O
in	O
1967	O
and	O
is	O
the	O
word	O
used	O
in	O
the	O
IEEE	O
standard	O
.	O
</s>
<s>
However	O
,	O
in	O
1946	O
Arthur	O
Burks	O
used	O
the	O
terms	O
mantissa	B-Algorithm
and	O
characteristic	O
to	O
describe	O
the	O
two	O
parts	O
of	O
a	O
floating-point	B-Algorithm
number	I-Algorithm
(	O
Burks	O
et	O
al	O
.	O
)	O
</s>
<s>
Mantissa	B-Algorithm
and	O
characteristic	O
have	O
long	O
described	O
the	O
two	O
parts	O
of	O
the	O
logarithm	O
found	O
on	O
tables	O
of	O
common	O
logarithms	O
.	O
</s>
<s>
While	O
the	O
two	O
meanings	O
of	O
exponent	O
are	O
analogous	O
,	O
the	O
two	O
meanings	O
of	O
mantissa	B-Algorithm
are	O
not	O
equivalent	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
use	O
of	O
mantissa	B-Algorithm
for	O
significand	B-Algorithm
is	O
discouraged	O
by	O
some	O
including	O
the	O
creator	O
of	O
the	O
standard	O
,	O
William	O
Kahan	O
and	O
prominent	O
computer	O
programmer	O
and	O
author	O
of	O
The	B-General_Concept
Art	I-General_Concept
of	I-General_Concept
Computer	I-General_Concept
Programming	I-General_Concept
,	O
Donald	O
E	O
.	O
Knuth	O
.	O
</s>
<s>
The	O
confusion	O
is	O
because	O
scientific	O
notation	O
and	O
floating-point	B-Algorithm
representation	I-Algorithm
are	O
log-linear	O
,	O
not	O
logarithmic	O
.	O
</s>
<s>
To	O
multiply	O
two	O
numbers	O
,	O
given	O
their	O
logarithms	O
,	O
one	O
just	O
adds	O
the	O
characteristic	O
(	O
integer	O
part	O
)	O
and	O
the	O
mantissa	B-Algorithm
(	O
fractional	O
part	O
)	O
.	O
</s>
<s>
By	O
contrast	O
,	O
to	O
multiply	O
two	O
floating-point	B-Algorithm
numbers	I-Algorithm
,	O
one	O
adds	O
the	O
exponent	O
(	O
which	O
is	O
logarithmic	O
)	O
and	O
multiplies	O
the	O
significand	B-Algorithm
(	O
which	O
is	O
linear	O
)	O
.	O
</s>
