How to Convert from Ternary to Decimal
Write down the ternary number you want to convert., Write down the powers of three from right to left., Connect each digit from the ternary number with its corresponding power of three., Multiply each digit with its corresponding power of three...
Step-by-Step Guide
-
Step 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. -
Step 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 3n-1, and not 3n, since you start with 30, and not
31. , 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. , 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 , In our case that would be 1+3+18+54+0+486=563.
So 563 is the decimal form of
2022123. , This is to prevent confusion as to what number is written in which base.
In our example, it would look like this: 2022123=56310. , 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). -
Step 3: Connect each digit from the ternary number with its corresponding power of three.
-
Step 4: Multiply each digit with its corresponding power of three.
-
Step 5: Add up all of the numbers you got from the previous step.
-
Step 6: Write the answer with the correct base subscripts.
-
Step 7: Use the same logic to convert a ternary number with a decimal point to decimal form.
Detailed Guide
Let's use the ternary number 202212 (commonly written as 2022123, to indicate the base in which the number is written) to follow along.
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 3n-1, and not 3n, since you start with 30, and not
31. , 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. , 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 , In our case that would be 1+3+18+54+0+486=563.
So 563 is the decimal form of
2022123. , This is to prevent confusion as to what number is written in which base.
In our example, it would look like this: 2022123=56310. , 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).
About the Author
Christopher Martin
Experienced content creator specializing in crafts guides and tutorials.
Rate This Guide
How helpful was this guide? Click to rate: