2001
November–December
1
C
OMPUTING
S
CIENCE
T
HIRD
B
ASE
Brian Hayes
A reprint from
American Scientist
the magazine of Sigma Xi, the Scientific Research Society
Volume 89, Number 6
November–December,
2001
pages 490–494
This reprint is provided for personal and noncommercial use. For any other use, please send a
request to Permissions,
American Scientist
, P.O. Box 13975, Research Triangle Park, NC, 27709, U.S.A.,
or by electronic mail to [email protected]
© 2001 Brian Hayes.
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P
eople count by tens and machines count by
twos—that pretty much sums up the way
we do arithmetic on this planet. But there
are countless other ways to count. Here I want to
offer three cheers for base 3, the ternary system.
The numerals in this sequence—beginning 0, 1, 2,
10, 11, 12, 20, 21, 22, 100, 101—are not as widely
known or widely used as their decimal and bina
ry cousins, but they have charms all their own.
They are the Goldilocks choice among number
ing systems: When base 2 is too small and base
10 is too big, base 3 is just right.
Cheaper by the Threesome
Under the skin, numbering systems are all alike.
Numerals in various bases may well look differ
ent, but the numbers they represent are the same.
In decimal notation, the numeral 19 is shorthand
for this expression:
1
×
10
1
+ 9
×
10
0
.
Likewise the binary numeral 10011 is understood
to mean:
1
×
2
4
+ 0
×
2
3
+ 0
×
2
2
+ 1
×
2
1
+ 1
×
2
0
,
which adds up to the same value. So does the
ternary version, 201:
2
×
3
2
+ 0
×
3
1
+ 1
×
3
0
.
The general formula for a numeral in any po
sitional notation goes something like this:
… d
3
r
3
+
d
2
r
2
+
d
1
r
1
+
d
0
r
0
.…
Here
r
is the base, or
radix
, and the coefficients
d
i
are the digits of the number. Usually,
r
is a posi
tive integer and the digits are integers in the range
from 0 to
r
–1, but neither of these restrictions is
strictly necessary. (You can build perfectly good
numbers on a negative or an irrational base, and
below we’ll meet numbers with negative digits.)
To say that all bases represent the same num
bers, however, is not to say that all numeric repre
sentations are equally good for all purposes. Base
10 is famously well suited to those of us who
count on our fingers. Base 2 dominates computing
technology because binary devices are simple and
reliable, with just two stable states—on or off, full
or empty. Computer circuitry also exploits a coin
cidence between binary arithmetic and binary log
ic: The same signal can represent either a numeric
value (1 or 0) or a logical value (
true
or
false
).
The cultural preference for base 10 and the en
gineering advantages of base 2 have nothing to
do with any intrinsic properties of the decimal and
binary numbering systems. Base 3, on the other
hand, does have a genuine mathematical distinc
tion in its favor. By one plausible measure, it is the
most efficient of all integer bases; it offers the most
economical way of representing numbers.
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 Binary numeral system, Decimal, American Scientist, Ternary logic, ternary representation, ternary digits

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