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

Updated on January 11, 2020

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".

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.

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 · 2,187 · 6,561

Positional

• 6,561 · 2,187 · 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 · 2,187 · 6,561

Positional

• 6,561 · 2,187 · 729 · 243 · 81 · 27 · 9 · 3 · 1

(convenience relist)

• What is 22 base three minus 12 base three?

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

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• shofahim

8 months ago

wow.what a narrate? the way of your teaching is extra-ordinary as well as awesome.i am from Bangladesh. you are really outstading guy......

• Robert Sacchi

2 years ago

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

• amalloy

9 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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