How to Convert from Ternary to Decimal

Last Updated: December 3, 2021

Just like the binary numeral system has two possible values for each place-value, the ternary numeral system has three possible values for each place-value. Ternary (also called base 3 or trinary) isn't as common as binary, octal or decimal or hexadecimal numeral systems, although ternary is sometimes used in computer science. We'll show you how to convert ternary numbers to decimals in a few simple steps.

This article deals with standard (unbalanced) system which uses the values 0, 1 & 2), not the balanced system (which uses the values -1, 0 & 1).

Steps

1
Write down the ternary number you want to convert. Let's use the ternary number 202212 (commonly written as 202212₃, to indicate the base in which the number is written) to follow along.
2
Write down the powers of three from right to left. Start on the right with 3⁰, then 3¹, then 3², until you have as many powers of three as there are digits in your ternary number. For our number, 202212₃, you would go up to 3⁵, since it has 6 digits, so you would have 243, 81, 27, 9, 3, 1.
- If you have a ternary number with n digits, you have to go up to 3^n-1, and not 3ⁿ, since you start with 3⁰, and not 3¹.
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3
Connect each digit from the ternary number with its corresponding power of three. The right-most digit belongs with the right-most power of three (3⁰), the second digit on the right belongs to the second power of three of the right (3¹), and so on.
4
Multiply each digit with its corresponding power of three. In our example (202212₃), this goes as follows;
- 1₁₀ (or 3⁰) corresponds with 2, so 2*1=1
- 3₁₀ (or 3¹) corresponds with 1, so 1*3=3
- 9₁₀ (or 3²) corresponds with 2, so 2*9=18
- 27₁₀ (or 3³) corresponds with 2, so 2*27=54
- 81₁₀ (or 3⁴) corresponds with 0, so 0*81=0
- 243₁₀ (or 3⁵) corresponds with 2, so 2*243=486
5
Add up all of the numbers you got from the previous step. In our case that would be 1+3+18+54+0+486=563. So 563 is the decimal form of 202212₃.
6
Write the answer with the correct base subscripts. This is to prevent confusion as to what number is written in which base. In our example, it would look like this: 202212₃=563₁₀.
7
Use the same logic to convert a ternary number with a decimal point to decimal form. You can use this method even when you want to covert a ternary number such as 1.1 to decimal. The "1" to the left of the decimal point is equal to 3⁰, or 1. The 1 to the right of the decimal is equal to 3^-1, or .333... (1/3). Add up 1 and .333 (1/3) and you get 1.333.. (4/3).