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
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1Write down the ternary number you want to convert. Let's use the ternary number 202212 (commonly written as 202212 3 , to indicate the base in which the number is written) to follow along.
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2Write down the powers of three from right to left. Start on the right with 3 0 , then 3 1 , then 3 2 , until you have as many powers of three as there are digits in your ternary number. For our number, 202212 3 , you would go up to 3 5 , 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 3 n , since you start with 3 0 , and not 3 1 .
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3Connect each digit from the ternary number with its corresponding power of three. The right-most digit belongs with the right-most power of three (3 0 ), the second digit on the right belongs to the second power of three of the right (3 1 ), and so on.
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4Multiply each digit with its corresponding power of three. In our example (202212 3 ), this goes as follows;
- 1 10 (or 3 0 ) corresponds with 2, so 2*1=1
- 3 10 (or 3 1 ) corresponds with 1, so 1*3=3
- 9 10 (or 3 2 ) corresponds with 2, so 2*9=18
- 27 10 (or 3 3 ) corresponds with 2, so 2*27=54
- 81 10 (or 3 4 ) corresponds with 0, so 0*81=0
- 243 10 (or 3 5 ) corresponds with 2, so 2*243=486
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5Add 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 202212 3 .
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6Write 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: 202212 3 =563 10 .
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7Use 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 3 0 , 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).Advertisement
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