Ternary numeral system

Ternary or trinary is the base-3 numeral system. Ternary digits are known as trits (trinary digit), with a name analogous to "bit". One trit contains about 1.58596 ($\log _{2}3$) bit of information. Although ternary most often refers to a system in which the three digits, 0, 1, and 2, are all nonnegative integers, the adjective also lends its name to the balanced ternary system, used in comparison logic and ternary computers.

Comparison to other radixes

Compared to decimal and binary

Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 corresponds to binary 101101101 (9 digits) and to ternary 111112 (6 digits). However, they are still far less compact than the corresponding representations in bases such as decimal — see below for a compact way to codify ternary using nonary and septemvigesimal.

Numbers one to twenty-seven in standard ternary

Ternary	1	2	10	11	12	20	21	22	100
Binary	1	10	11	100	101	110	111	1000	1001
Decimal	1	2	3	4	5	6	7	8	9
Ternary	101	102	110	111	112	120	121	122	200
Binary	1010	1011	1100	1101	1110	1111	10000	10001	10010
Decimal	10	11	12	13	14	15	16	17	18
Ternary	201	202	210	211	212	220	221	222	1000
Binary	10011	10100	10101	10110	10111	11000	11001	11010	11011
Decimal	19	20	21	22	23	24	25	26	27

Powers of three in ternary

Ternary	1	10	100	1 000	10 000
Binary	1	11	1001	1 1011	101 0001
Decimal	1	3	9	27	81
Power	30	31	32	33	34
Ternary	100 000	1 000 000	10 000 000	100 000 000	1 000 000 000
Binary	1111 0011	10 1101 1001	1000 1000 1011	1 1001 1010 0001	100 1100 1110 0011
Decimal	243	729	2 187	6 561	19 683
Power	35	36	37	38	39

As for rational numbers, ternary offers a convenient way to represent one third (as opposed to its cumbersome representation as an infinite string of recurring digits in decimal); but a major drawback is that, in turn, ternary does not offer a finite representation for the most basic fraction: one half (and thus, neither for one quarter, one sixth, one eighth, one tenth, etc.), because 2 is not a prime factor of the base.

Fractions in ternary

Ternary	0.111111111111...	0.1	0.020202020202...	0.012101210121...	0.011111111111...	0.010212010212...
Binary	0.1	0.010101010101...	0.01	0.001100110011...	0.00101010101...	0.001001001001...
Decimal	0.5	0.333333333333...	0.25	0.2	0.166666666666...	0.142857142857...
Fraction	1/2	1/3	1/4	1/5	1/6	1/7
Ternary	0.010101010101...	0.01	0.002200220022...	0.002110021100...	0.002020202020...	0.002002002002...
Binary	0.001	0.000111000111...	0.000110011001...	0.000101110100...	0.000101010101...	0.000100111011...
Decimal	0.125	0.111111111111...	0.1	0.090909090909...	0.083333333333...	0.076923076923...
Fraction	1/8	1/9	1/10	1/11	1/12	1/13

Compact ternary representation: base 9 and 27

Nonary (base 9, each digit is two ternary digits) or septemvigesimal (base 27, each digit is three ternary digits) is often used, similar to how octal and hexadecimal systems are used in place of binary. Ternary also has a unit similar to a byte, the tryte, which is six ternary digits.

Practical usage

A base-three system is used in Islam to keep track of counting Tasbih to 99 or to 100 on a single hand for counting prayers (as alternative for the Misbaha). The benefit —apart from allowing a single hand to count up to 99 or to 100— is that counting doesn't distract the mind too much since the counter needs only to divide Tasbihs into groups of three.

A rare ternary point is used to denote fractional parts of an inning in baseball. Since each inning consists of 3 outs, each out is considered ${\begin{matrix}{\frac {1}{3}}\end{matrix}}$ (one third) of an inning and is denoted as .1. For example, if a player pitched all of the 4th, 5th and 6th innings, plus 2 outs of the 7th inning, his Innings pitched column for that game would be listed as 3.2, meaning $3{\begin{matrix}{\frac {2}{3}}\end{matrix}}$. (In this usage, only the fractional part of the number is written in ternary form.)

Ternary numbers can be used to convey self-similar structures like a Sierpinski Triangle or a Cantor set conveniently.

Ternary encoding can also be applied in most Relational Databases, with NULL acting as the third trit. By manually encoding datatypes it is possible to effectively reduce the storage requirements by a third.

External links

• Third Base (PDF)
• Ternary Arithmetic
• The ternary calculating machine of Thomas Fowler
• Ternary Base Conversion includes fractional part, from Math Is Fun
• Gideon Frieder's replacement ternary numeral system