The base-3 method of counting in which only the digits 0, 1, and 2 are used. Ternary numbers arise in a number of problems in mathematics,
including some problems of weighing. However, according
to Knuth (1998), "no substantial application of balanced ternary notation has
been made" (balanced ternary uses digits , 0, and 1 instead of 0, 1, and 2).
The illustration above shows a graphical representation of the numbers 0 to 25 in ternary, and the following table gives the ternary equivalents of the first few decimal numbers. The concatenation of the ternary digits of the consecutive numbers 0, 1, 2, 3, ... gives (0), (1), (2), (1, 0), (1, 1), (1, 2), (2, 0), ... (OEIS A054635).
| 1 | 1 | 11 | 102 | 21 | 210 |
| 2 | 2 | 12 | 110 | 22 | 211 |
| 3 | 10 | 13 | 111 | 23 | 212 |
| 4 | 11 | 14 | 112 | 24 | 220 |
| 5 | 12 | 15 | 120 | 25 | 221 |
| 6 | 20 | 16 | 121 | 26 | 222 |
| 7 | 21 | 17 | 122 | 27 | 1000 |
| 8 | 22 | 18 | 200 | 28 | 1001 |
| 9 | 100 | 19 | 201 | 29 | 1002 |
| 10 | 101 | 20 | 202 | 30 | 1010 |
Ternary digits have the following multiplication table.
| 0 | 1 | 2 | |
| 0 | 0 | 0 | 0 |
| 1 | 0 | 1 | 2 |
| 2 | 0 | 2 | 11 |
A ternary representation can be used to uniquely identify totalistic cellular automaton rules, where the three colors (white, gray, and black) correspond
to the three numbers 0, 1 and 2 (Wolfram 2002, pp. 60-70
and 886). For
example, the ternary digits ,
lead to the code 600 totalistic cellular
automaton.
Every even number represented in ternary has an even number (possibly 0) of 1s. This is true since a
number is congruent mod
to the sum of its base-
digits. In the case
, there is only one digit (1) which is not a multiple of
, so all we have to do is "cast
out twos" and count the number of 1s in the base-3 representation.
The following table gives
for
, 2, ... in ternary.
|
(1)
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|
(2)
| |||
|
(3)
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|
(4)
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|
(5)
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|
(6)
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|
(7)
|
N. J. A. Sloane conjectured that for any integer ,
always has a 0 in its ternary expansion (Sloane 1973; Vardi 1991, p. 28). Known
values of
such that
lacks
a 0 are 1, 2, 3, 4, 15 (OEIS A102483), with
no others up to
(E. W. Weisstein, Apr. 8, 2006). The positions (counting from the
least significant ternary digits) of the first 0 digit in
,
,
..., are 0, 0, 0, 0, 3, 2, 2, 4, 4, 5, 4, 2, 2, 4, 0, 3, 4, (OEIS A117971).
Similarly,
always has a 1 in its ternary expansion except for
, 1, 3, and 9, with no others up to
(E. W. Weisstein, Apr. 8, 2006).
Erdős and Graham (1980) conjectured that no power of 2, , for
is a sum of distinct powers of
3. This is equivalent to the requirement that the ternary expansion of
always contains a 2 for
. The fact that the only values not having a two are
and 8 has been verified by Vardi (1991)
up to
. The positions
(counting from the least significant ternary digits) of the first 2 digit in
,
, ..., are 1, 0, 1, 2, 1, 4, 1, 0, 1, 2, 1, 3, 1, 3,
... (OEIS A117970).