<s>
The	O
radix	B-Algorithm
economy	I-Algorithm
of	O
a	O
number	O
in	O
a	O
particular	O
base	O
(	O
or	O
radix	O
)	O
is	O
the	O
number	O
of	O
digits	O
needed	O
to	O
express	O
it	O
in	O
that	O
base	O
,	O
multiplied	O
by	O
the	O
base	O
(	O
the	O
number	O
of	O
possible	O
values	O
each	O
digit	O
could	O
have	O
)	O
.	O
</s>
<s>
Radix	B-Algorithm
economy	I-Algorithm
also	O
has	O
implications	O
for	O
organizational	O
structure	O
,	O
networking	O
,	O
and	O
other	O
fields	O
.	O
</s>
<s>
If	O
both	O
b	O
and	O
N	O
are	O
positive	O
integers	O
,	O
then	O
the	O
radix	B-Algorithm
economy	I-Algorithm
is	O
equal	O
to	O
the	O
number	O
of	O
digits	O
needed	O
to	O
express	O
the	O
number	O
N	O
in	O
base	O
b	O
,	O
multiplied	O
by	O
base	O
b	O
.	O
</s>
<s>
The	O
radix	B-Algorithm
economy	I-Algorithm
thus	O
measures	O
the	O
cost	O
of	O
storing	O
or	O
processing	O
the	O
number	O
N	O
in	O
base	O
b	O
if	O
the	O
cost	O
of	O
each	O
"	O
digit	O
"	O
is	O
proportional	O
to	O
b	O
.	O
</s>
<s>
A	O
base	O
with	O
a	O
lower	O
average	O
radix	B-Algorithm
economy	I-Algorithm
is	O
therefore	O
,	O
in	O
some	O
senses	O
,	O
more	O
efficient	O
than	O
a	O
base	O
with	O
a	O
higher	O
average	O
radix	B-Algorithm
economy	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
100	O
in	O
decimal	B-Application
has	O
three	O
digits	O
,	O
so	O
its	O
radix	B-Algorithm
economy	I-Algorithm
is	O
10×3	O
=	O
30	O
;	O
its	O
binary	O
representation	O
has	O
seven	O
digits	O
(	O
11001002	O
)	O
so	O
it	O
has	O
radix	B-Algorithm
economy	I-Algorithm
2×7	O
=	O
14	O
in	O
base	O
2	O
;	O
in	O
base	B-Algorithm
3	I-Algorithm
its	O
representation	O
has	O
five	O
digits	O
(	O
102013	O
)	O
with	O
a	O
radix	B-Algorithm
economy	I-Algorithm
of	O
3×5	O
=	O
15	O
;	O
in	O
base	O
36	O
(	O
2S36	O
)	O
its	O
radix	B-Algorithm
economy	I-Algorithm
is	O
36×2	O
=	O
72	O
.	O
</s>
<s>
If	O
the	O
number	O
is	O
imagined	O
to	O
be	O
represented	O
by	O
a	O
combination	O
lock	O
or	O
a	O
tally	O
counter	O
,	O
in	O
which	O
each	O
wheel	O
has	O
b	O
digit	O
faces	O
,	O
from	O
and	O
having	O
wheels	O
,	O
then	O
the	O
radix	B-Algorithm
economy	I-Algorithm
is	O
the	O
total	O
number	O
of	O
digit	O
faces	O
needed	O
to	O
inclusively	O
represent	O
any	O
integer	O
from	O
0	O
to	O
N	O
.	O
</s>
<s>
The	O
radix	B-Algorithm
economy	I-Algorithm
for	O
large	O
N	O
can	O
be	O
approximated	O
as	O
follows	O
:	O
</s>
<s>
The	O
asymptotically	O
best	O
radix	B-Algorithm
economy	I-Algorithm
is	O
obtained	O
for	O
base	B-Algorithm
3	I-Algorithm
,	O
since	O
attains	O
a	O
minimum	O
for	O
in	O
the	O
positive	O
integers	O
:	O
</s>
<s>
For	O
base	O
10	B-Application
,	O
we	O
have	O
:	O
</s>
<s>
Here	O
is	O
a	O
proof	O
that	O
base	O
e	O
is	O
the	O
real-valued	O
base	O
with	O
the	O
lowest	O
average	O
radix	B-Algorithm
economy	I-Algorithm
:	O
</s>
<s>
Then	O
for	O
a	O
constant	O
N	O
,	O
will	O
have	O
a	O
minimum	O
at	O
e	O
for	O
the	O
same	O
reason	O
f(x )	O
does	O
,	O
meaning	O
e	O
is	O
therefore	O
the	O
base	O
with	O
the	O
lowest	O
average	O
radix	B-Algorithm
economy	I-Algorithm
.	O
</s>
<s>
Since	O
2	O
/	O
ln(2 )	O
≈	O
2.89	O
and	O
3	B-Algorithm
/	O
ln(3 )	O
≈	O
2.73	O
,	O
it	O
follows	O
that	O
3	B-Algorithm
is	O
the	O
integer	O
base	O
with	O
the	O
lowest	O
average	O
radix	B-Algorithm
economy	I-Algorithm
.	O
</s>
<s>
The	O
radix	B-Algorithm
economy	I-Algorithm
of	O
bases	O
b1	O
and	O
b2	O
may	O
be	O
compared	O
for	O
a	O
large	O
value	O
of	O
N	O
:	O
</s>
<s>
The	O
average	O
radix	B-Algorithm
economies	I-Algorithm
of	O
various	O
bases	O
up	O
to	O
several	O
arbitrary	O
numbers	O
(	O
avoiding	O
proximity	O
to	O
powers	O
of	O
2	O
through	O
12	O
and	O
e	O
)	O
are	O
given	O
in	O
the	O
table	O
below	O
.	O
</s>
<s>
Also	O
shown	O
are	O
the	O
radix	B-Algorithm
economies	I-Algorithm
relative	O
to	O
that	O
of	O
e	O
.	O
Note	O
that	O
the	O
radix	B-Algorithm
economy	I-Algorithm
of	O
any	O
number	O
in	O
base	O
1	O
is	O
that	O
number	O
,	O
making	O
it	O
the	O
most	O
economical	O
for	O
the	O
first	O
few	O
integers	O
,	O
but	O
as	O
N	O
climbs	O
to	O
infinity	O
so	O
does	O
its	O
relative	O
economy	O
.	O
</s>
<s>
One	O
result	O
of	O
the	O
relative	O
economy	O
of	O
base	B-Algorithm
3	I-Algorithm
is	O
that	O
ternary	B-Data_Structure
search	I-Data_Structure
trees	I-Data_Structure
offer	O
an	O
efficient	O
strategy	O
for	O
retrieving	O
elements	O
of	O
a	O
database	O
.	O
</s>
<s>
Each	O
digit	O
of	O
a	O
number	O
would	O
be	O
stored	O
as	O
the	O
state	O
of	O
a	O
ring	B-General_Concept
counter	I-General_Concept
composed	O
of	O
several	O
triodes	O
.	O
</s>
<s>
(	O
Larger	O
radices	O
required	O
2r	O
triodes	O
arranged	O
as	O
r	O
flip-flops	B-General_Concept
,	O
as	O
in	O
ENIAC	B-Device
's	O
decimal	B-Application
counters	O
.	O
)	O
</s>
<s>
A	O
smaller	O
percentage	O
gain	O
is	O
realized	O
in	O
going	O
from	O
a	O
radix	O
3	B-Algorithm
to	O
a	O
radix	O
4	O
system.	O
"	O
</s>
