Ternary (Base 3) to Decimal (Base 10) Converter
Convert ternary numbers to decimal.
100110 (base 3) = 255 (base 10)
Digits in base 3: 0 to 2. Digits in base 10: 0 to 9.
How to convert Ternary to Decimal
Every positional number system works the same way: a digit is worth its own value times the base raised to the power of its position. In base 3 the place values, from the right, are 30 = 1, 31 = 3, 32 = 9 and so on.
To get from base 3 to decimal, multiply each digit by 3 raised to the power of its position, counting from 0 on the right, and add the results.
The method
decimal value = each digit × its place value, all added upplace values in base 3: 1, 3, 9, 27, 81, and so ondigits in base 3: 0 to 2digits in base 10: 0 to 9
Worked example 1: 1200 from ternary to decimal
1200 (base 3) = 1 × 33 + 2 × 32= 27 + 18= 451200 (base 3) = 45 (base 10)
Digits that are 0 add nothing to the sum, so they are left out of the expansion.
Worked example 2: 100110 from ternary to decimal
100110 (base 3) = 1 × 35 + 1 × 32 + 1 × 31= 243 + 9 + 3= 255100110 (base 3) = 255 (base 10)
Worked example 3: 1101001 from ternary to decimal
1101001 (base 3) = 1 × 36 + 1 × 35 + 1 × 33 + 1 × 30= 729 + 243 + 27 + 1= 1,0001101001 (base 3) = 1000 (base 10)
Need the other direction? Open the Decimal to Ternary converter.
Ternary to Decimal examples
The decimal number 255 is written 100110 in ternary and 255 in decimal.
| Ternary | Decimal |
|---|---|
| 1 | 1 |
| 2 | 2 |
| 21 | 7 |
| 22 | 8 |
| 101 | 10 |
| 120 | 15 |
| 121 | 16 |
| 1012 | 32 |
| 2101 | 64 |
| 10201 | 100 |
| 100110 | 255 |
| 1101001 | 1000 |
Where Ternary to Decimal comes up
Ternary uses the digits 0, 1 and 2, and each ternary digit (a trit) carries about 1.58 bits of information. Ternary is rare in real hardware, though the Soviet Setun computer of 1958 used a balanced ternary design. Today it shows up mostly in computer science courses and puzzles.
Decimal is the base 10 system everyone learns first, with the digits 0 to 9. It is the form people read: prices, measurements, ages and IDs. It is also the base we calculate in by hand, which is why the worked steps on this page pass through decimal.
Going to a larger base packs more value into each digit, so the result is shorter. The number 255 needs 6 digits in ternary and 3 in decimal.
Ternary to Decimal FAQ
How do I convert ternary to decimal?
Multiply each base 3 digit by 3 raised to the power of its position, counting from 0 on the right, then add everything up.
Which digits does ternary use?
Base 3 uses the digits 0 to 2. For example, 3 is not a valid digit in base 3.
Will the decimal number be longer or shorter?
Shorter. Going to a larger base packs more value into each digit, so the result is shorter. Roughly 0.48 decimal digits for every ternary digit. The number 255 needs 6 digits in ternary and 3 in decimal.
What is 100110 in decimal?
100110 in ternary is 255 in decimal. Both are the decimal number 255.
Can I convert negative numbers or fractions?
The converter handles whole numbers, and a leading minus sign works for negatives. Fractions use a different method, where you repeatedly multiply the fractional part by the new base, and they are not supported here.