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.[1][2][3]

أنظمة ترقيم حسب الثقافة
ترقيم هندي عربي
نظام ترقيم غربي
عربي شرقي
Khmer
عائلة هندية
براهمني
تايلندي
ترقيم غربي آسيا
صيني
Suzhou
Counting rods
منغولي 
ترقيم أبجدي
أبجدي
أرمني
سريالي
Ge'ez
عبري
اغريق (Ionian)
Āryabhaṭa
 
أنظمة أخرى
Attic
بابلي
مصري
Etruscan
مايا
Roman
Urnfield
List of numeral system topics
Positional systems by base
نظام عشري (10)
2, 4, 8, 16, 32, 64
1, 3, 6, 9, 12, 20, 24, 30, 36, 60, more…

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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 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 2 - 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) b

,

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

مراجع

  1. ^ Binary Coded Ternary and its Inverse, June 2016.
  2. ^ Impagliazzo, John; Proydakov, Eduard (2011-09-06). Perspectives on Soviet and Russian Computing: First IFIP WG 9.7 Conference, SoRuCom 2006, Petrozavodsk, Russia, July 3-7, 2006, Revised Selected Papers (in الإنجليزية). Springer. ISBN 9783642228162.
  3. ^ Brousentsov, N. P.; Maslov, S. P.; Ramil Alvarez, J.; Zhogolev, E.A. "Development of ternary computers at Moscow State University". Archived from the original on 13 يوليو 2018. Retrieved 20 January 2010. Unknown parameter |تاريخ الأرشيف= ignored (|archive-date= suggested) (help); Unknown parameter |مسار الأرشيف= ignored (|archive-url= suggested) (help); Check date values in: |تاريخ الأرشيف= (help)

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