How to Do Ternary or Trinary, Base 3 Numbering System Conversions Lesson / Tutorial Examples

Latest update: November 24, 2024

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

How to Learn the Ternary Base 3 Numbering System

A complete lesson and examples.

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.

A 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 are:

• 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

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

(Convenience relist)

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

Other base numbering systems:  Try https://mathschool.etsy.com.

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