نظام عد ثلاثي

نظام العد الثلاثي (بالإنجليزية: Ternary numeral system) هو نظام عد ذو رقم أساس 3، ويسمى هذا النظام عد ثلاثي فالرقم 3 أو -3 في النظام العشري فما فوق يساوي في النظام الثلاثي 10 أو -10 أما 4 فيساوي 11 أما 6 فيساوي 20 وهكذا.

 Comparison to other bases

A ternary multiplication table

×	1	2	10	11	12	20	21	22	100
1	1	2	10	11	12	20	21	22	100
2	2	11	20	22	101	110	112	121	200
10	10	20	100	110	120	200	210	220	1000
11	11	22	110	121	202	220	1001	1012	1100
12	12	101	120	202	221	1010	1022	1111	1200
20	20	110	200	220	1010	1100	1120	1210	2000
21	21	112	210	1001	1022	1120	1211	2002	2100
22	22	121	220	1012	1111	1210	2002	2101	2200
100	100	200	1000	1100	1200	2000	2100	2200	10000

Representations of integer numbers in ternary do not get uncomfortably lengthy as quickly as in binary. For example, decimal 365 corresponds to binary 101101101 (nine digits) and to ternary 111112 (six 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 from 1 to 27 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 3 in ternary

Ternary	1	10	100	1000	10000
Binary	1	11	1001	11011	1010001
Decimal	1	3	9	27	81
Power	3⁰	3¹	3²	3³	3⁴
Ternary	100000	1000000	10000000	100000000	1000000000
Binary	11110011	1011011001	100010001011	1100110100001	100110011100011
Decimal	243	729	2187	6561	19683
Power	3⁵	3⁶	3⁷	3⁸	3⁹

As for rational numbers, ternary offers a convenient way to represent 1÷3 (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 1÷2 (neither for 1÷4, 1÷8, etc.), because 2 is not a prime factor of the base; as with base 2, 1÷10 is not representable exactly (that would need e.g. base 10); nor is 1÷6.

Fractions in ternary

Fraction	1/2	1/3	1/4	1/5	1/6	1/7	1/8	1/9	1/10	1/11	1/12	1/13
Ternary	0.1	0.1	0.02	0.0121	0.01	0.010212	0.01	0.01	0.0022	0.00211	0.002	0.002
Binary	0.1	0.01	0.01	0.0011	0.001	0.001	0.001	0.000111	0.00011	0.0001011101	0.0001	0.000100111011
Decimal	0.5	0.3	0.25	0.2	0.16	0.142857	0.125	0.1	0.1	0.09	0.083	0.076923

 Sum of the digits in ternary as opposed to binary

The value of a binary number with n bits that are all 1 is 2ⁿ - 1.

Similarly, for a number N(b, d) with base b and d digits, all of which are the maximal digit value b - 1, we can write:

N(b, d) = (b - 1) bᵈ⁻¹ + (b - 1) bᵈ⁻² + … + (b - 1) b¹ + (b - 1) b⁰

,

N(b, d) = (b - 1) (bᵈ⁻¹ + bᵈ⁻² + … + b¹ + 1),

N(b, d) = (b - 1) M.

bM = bᵈ + bᵈ⁻¹ + … + b² + b¹, and

-M = -bᵈ⁻¹ - bᵈ⁻² - … - b¹ - 1, so

bM - M = bᵈ - 1, or

M = (bᵈ - 1)÷(b - 1).

Then

N(b, d) = (b - 1)M,

N(b, d) = (b - 1) (bᵈ - 1)÷(b  - 1), and

N(b, d) = bᵈ - 1.

For a three-digit ternary number, N(3, 3) = 3³ - 1 = 26 = 2 × 3² + 2 × 3¹ + 2 × 3⁰ = 18 + 6 + 2.

 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) can be used for compact representation of ternary, similar to how octal and hexadecimal systems are used in place of binary.

 انظر أيضاً

• نظام عد ثنائي

 وصلات خارجية

• 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
• Visualization of numeral systems