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

Updated on May 26, 2018
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This author has written several base numbering system and math tutorials.

If you understand the everyday decimal (base 10) number system, then you already understand the ternary, base 3 counting and numbering system. You just don’t know you know yet. The complete lesson immediately follows the short semantics note about "ternary" versus "trinary".

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.

Semantics Note

Ternary is the primary descriptor used to identify base 3 (using the digits 0 1 2) in mathematics as relates to numbering systems.

Trinary is the primary descriptor used to identify base three as relates to logic (using the digits -1 0 +1); but the term has also been used in place of ternary. This page does not address the logic definition of trinary.

This page is about and explains the base 3 number system; usually called ternary, but sometimes referred to as trinary.

Complete Lesson and Examples

Quick review of base 10 structure...

Base 10, Decimal Orders of Magnitude

1 · 10 · 100 · 1,000 · 10,000 · 100,000

Positional

100,000 · 10,000 · 1,000 · 100 · 10 · 1

We use the base 10 numbering/counting system 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 10.
  • The next order number represents itself times 10 x 10, or itself times 100.
  • The next order of magnitude would be 10 x 10 x 10, or 1000.

And so on.

A base 10 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.

The ternary or base 3 numbering system...

...uses the same structure, the only difference being the orders of magnitude. Base 3 or ternary has three numbers: 0, 1, and 2.

The orders of magnitude are times three.

  • The lowest order number represents itself times one.
  • The next order number represents itself times 3.
  • The next order number represents itself times 3 x 3, or itself times 9.
  • The next order of magnitude would be 3 x 3 x 3, or itself times 27.
  • The next order of magnitude would be 3 x 3 x 3 x 3, or itself times 81.

And so on.

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

A basic, first example of a ternary number would be the base 3 number 11111. This would mean there is:

  • one 1,
  • one 3,
  • one 9,
  • one 27,
  • and one 81.

Which represents 1 + 3 + 9 + 27 + 81 for a total of 121 in Base 10 decimal.

Another base 3 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 in base 10 decimal.

Another base 3 example would be the number 2101. This number means there are:

  • One 1,
  • No 3's,
  • One 9,
  • And two 27’s.

Which represents 1 + 0 + 9 + 54 for a total of 64 in base 10 decimal.

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)


Questions & Answers

  • What is 22 base three minus 12 base three?

    The answer is 10 in base 3 and 3 in base 10.

Comments

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    • Robert Sacchi profile image

      Robert Sacchi 

      10 months ago

      This same principal can be used for any base system, yes?

    • amalloy profile image

      amalloy 

      7 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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