Duodecimal (Base 12) to Ternary (Base 3) Converter
Convert duodecimal numbers to ternary.
193 (base 12) = 100110 (base 3)
Digits in base 12: 0 to 9 and A to B. Digits in base 3: 0 to 2.
How to convert Duodecimal to Ternary
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 12 the place values, from the right, are 120 = 1, 121 = 12, 122 = 144 and so on.
Most pairs have no direct trick, so go through decimal, the base we all calculate in. First turn the base 12 number into decimal, then divide by 3 and read the remainders from the bottom up. Two short steps, and you can check the number in the middle.
The method
decimal value = each digit × its place value, all added upplace values in base 12: 1, 12, 144, 1728, 20736, and so ondivide by 3, keep the remainder, repeat until the quotient is 0the answer is the remainders read from last to firstdigits in base 12: 0 to 9 and A to Bdigits in base 3: 0 to 2
Worked example 1: 39 from duodecimal to ternary
Step 1: base 12 to decimal39 (base 12) = 3 × 121 + 9 × 120= 36 + 9= 45Step 2: decimal to base 345 ÷ 3 = 15 remainder 015 ÷ 3 = 5 remainder 05 ÷ 3 = 1 remainder 21 ÷ 3 = 0 remainder 1Read the remainders from the bottom up: 120039 (base 12) = 1200 (base 3)
Letters stand for values from 10 up: A = 10, B = 11, C = 12 and so on. Remainders come out last digit first, which is why you read them from the bottom up. Digits that are 0 add nothing to the sum, so they are left out of the expansion.
Worked example 2: 193 from duodecimal to ternary
Step 1: base 12 to decimal193 (base 12) = 1 × 122 + 9 × 121 + 3 × 120= 144 + 108 + 3= 255Step 2: decimal to base 3255 ÷ 3 = 85 remainder 085 ÷ 3 = 28 remainder 128 ÷ 3 = 9 remainder 19 ÷ 3 = 3 remainder 03 ÷ 3 = 1 remainder 01 ÷ 3 = 0 remainder 1Read the remainders from the bottom up: 100110193 (base 12) = 100110 (base 3)
Worked example 3: 6B4 from duodecimal to ternary
Step 1: base 12 to decimal6B4 (base 12) = 6 × 122 + 11 × 121 + 4 × 120= 864 + 132 + 4= 1,000Step 2: decimal to base 31,000 ÷ 3 = 333 remainder 1333 ÷ 3 = 111 remainder 0111 ÷ 3 = 37 remainder 037 ÷ 3 = 12 remainder 112 ÷ 3 = 4 remainder 04 ÷ 3 = 1 remainder 11 ÷ 3 = 0 remainder 1Read the remainders from the bottom up: 11010016B4 (base 12) = 1101001 (base 3)
Need the other direction? Open the Ternary to Duodecimal converter.
Duodecimal to Ternary examples
The decimal number 255 is written 193 in duodecimal and 100110 in ternary.
| Duodecimal | Ternary | Decimal |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 7 | 21 | 7 |
| 8 | 22 | 8 |
| A | 101 | 10 |
| 13 | 120 | 15 |
| 14 | 121 | 16 |
| 28 | 1012 | 32 |
| 54 | 2101 | 64 |
| 84 | 10201 | 100 |
| 193 | 100110 | 255 |
| 6B4 | 1101001 | 1000 |
Where Duodecimal to Ternary comes up
Duodecimal (base 12) uses 0 to 9 plus A for ten and B for eleven. Twelve divides evenly by 2, 3, 4 and 6. That is why dozens, the 12-hour clock, twelve inches to a foot and twelve months feel natural. A few mathematicians even argue it is a better base than ten.
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.
Going to a smaller base spreads the same value over more digits, so the result is longer. The number 255 needs 3 digits in duodecimal and 6 in ternary.
Duodecimal to Ternary FAQ
How do I convert duodecimal to ternary?
Convert to decimal first by adding up each digit times its place value, then divide that decimal number by 3 repeatedly and read the remainders from last to first.
Which digits does duodecimal use?
Base 12 uses the digits 0 to 9 and A to B. For example, C is not a valid digit in base 12. Letters are not case sensitive on this converter, and results are shown in capitals.
Will the ternary number be longer or shorter?
Longer. Going to a smaller base spreads the same value over more digits, so the result is longer. Roughly 2.26 ternary digits for every duodecimal digit. The number 255 needs 3 digits in duodecimal and 6 in ternary.
What is 193 in ternary?
193 in duodecimal is 100110 in ternary. 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.