How to Do Ternary or Trinary, Base 3 Number System Conversions - Includes Examples

Base 3 Conversion - Base 3 to Base 10 and Back - 0 1 2
Base 3 Conversion - Base 3 to Base 10 and Back - 0 1 2 | Source

How to Learn the Ternary Base 3 Numbering System

And a semantics note.

Ternary is the primary descriptor used to identify base 3 as relates to mathematics. (0 1 2)

Trinary is the primary descriptor used to identify base three as relates to logic; but the term has also been used in place of ternary. This article does not address the logic definition of trinary.

This article explains the base 3 number system; usually called ternary, sometimes called trinary.

Tutorial

Base 3 Conversion - Base 10 to Base 3 and Back - 0 1 2
Base 3 Conversion - Base 10 to Base 3 and Back - 0 1 2 | Source
Ternary aka Base 3 lesson. How to Learn and Do Ternary / Trinary / Base 3 Numerical System and Conversion
Ternary aka Base 3 lesson. How to Learn and Do Ternary / Trinary / Base 3 Numerical System and Conversion | Source

We use Base 10 in our day-to-day living. Base 10 has ten numbers (0-9) and orders of magnitude that are times ten. The lowest-order number represents itself times one. The next-order number represents itself times ten. The next order number represents itself times 10x10 or itself times 100. The next order of magnitude would be 10x10x10 or 1000. And so on.

An example would be the number 3528. This number means that there are:

Eight 1’s,

two 10’s,

five 100’s,

and three 1000's.

Which represents 8 + 20 + 500 + 3000; for a total of 3528.

Source

Orders of Magnitude in Base 3

  • 1 3 9 27 81 243 729 2187 6561

Positional

  • 6561 2187 729 243 81 27 9 3 1

Ternary ( base 3 ) uses the same structure...

...the only difference being the order of magnitude. Base 3 has three numbers (0 1 2) and orders of magnitude that are times three. The lowest-order number represents itself times one. The next-order number represents itself times three. The next order number represents itself times 3x3 or itself times 9. The next order of magnitude would be 3x3x3 or 27. And so on.

An example would be the number 1120. This number means that there are:

No 1’s,

two 3’s,

one 9,

and one 27.

Which represents 0 + 6 + 9 + 27; for a total of 42.

More Ternary ( Base 3 ) to Base 10 Conversion Examples

Column headings in the following table are simply a convenience relist of the relevant positional orders of magnitude as applies to each column. There is no significance attached as to where one column ends and the next one begins.

9 3 1
9 3 1
27 9 3 1
0=0
110=12
220=24
1=1
111=13
221=25
2=2
112=14
222=26
10=3
120=15
1000=27
11=4
121=16
1001=28
12=5
122=17
1002=29
20=6
200=18
1010=30
21=7
201=19
1011=31
22=8
202=20
1012=32
100=9
210=21
1020=33
101=10
211=22
1021=34
102=11
212=23
1022=35

Orders of Magnitude in Base 3

  • 1 3 9 27 81 243 729 2187 6561

Positional

  • 6561 2187 729 243 81 27 9 3 1

(convenience relist)


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Comments 1 comment

amalloy profile image

amalloy 5 years ago from Los Angeles

Cool. Have you thought about a writeup of "balanced ternary"? The base is still three, but the digits are +0- instead of 012. So to represent 15, you use +--0, that is, +27, -9, -3, +0.

This way you can represent negative numbers without needing any special notation like you would in regular ternary.

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