Ternary numeral system

Ternary or trinary is the base-3 numeral system. Analogous to a "bit", a ternary digit is known as a trit (trinary digit). One trit contains about 1.58496 ( $\log _{2}3$ ) bits 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

Sum of the digits in trinary as opposed to binary

Whereas in binary, where the sum of all previous digit values before 2ⁿ can be found using the formula 2ⁿ-1, in trinary the following formula can be used: (3ⁿ-1)/2.

An example is where in binary the fourth digit has a value of 8, the sum of all the binary numbers before 8 can be found out using the above formula as 2³-1, which is 7. In trinary the fourth digit has a value of 27 and the sum of all previous trinary numbers can be found out using the above formula, as (3³-1)/2, which is 13.

The formula is 3ⁿ because we are counting to base 3 and we divide by 2 now because the maximum value of each digit is 2. The general formula for the nth digit it a base-N number is:

Nⁿ

and the sum of previous digits:

(Nⁿ-1)/(N-1)

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 three outs, each out is considered 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 3⅔. 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. Additionally, it turns out that the ternary representation is useful for defining the Cantor Set and related point sets, because one can define a unique representation for every point. (ie, you do not have non-unique representations for the same number such as .999.. and 1.00...)

External links

Third Base
Ternary Arithmetic
The ternary calculating machine of Thomas Fowler
Ternary Base Conversion includes fractional part, from Maths Is Fun
Gideon Frieder's replacement ternary numeral system
Binary/Ternary/Nonary/Decimal/Hexadecimal Converter