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

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