Ternary (Base 3) to Octal (Base 8) Converter
Convert ternary numbers to octal.
100110 (base 3) = 377 (base 8)
Digits in base 3: 0 to 2. Digits in base 8: 0 to 7.
How to convert Ternary to Octal
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.
Most pairs have no direct trick, so go through decimal, the base we all calculate in. First turn the base 3 number into decimal, then divide by 8 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 3: 1, 3, 9, 27, 81, and so ondivide by 8, keep the remainder, repeat until the quotient is 0the answer is the remainders read from last to firstdigits in base 3: 0 to 2digits in base 8: 0 to 7
Worked example 1: 1200 from ternary to octal
Step 1: base 3 to decimal1200 (base 3) = 1 × 33 + 2 × 32= 27 + 18= 45Step 2: decimal to base 845 ÷ 8 = 5 remainder 55 ÷ 8 = 0 remainder 5Read the remainders from the bottom up: 551200 (base 3) = 55 (base 8)
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: 100110 from ternary to octal
Step 1: base 3 to decimal100110 (base 3) = 1 × 35 + 1 × 32 + 1 × 31= 243 + 9 + 3= 255Step 2: decimal to base 8255 ÷ 8 = 31 remainder 731 ÷ 8 = 3 remainder 73 ÷ 8 = 0 remainder 3Read the remainders from the bottom up: 377100110 (base 3) = 377 (base 8)
Worked example 3: 1101001 from ternary to octal
Step 1: base 3 to decimal1101001 (base 3) = 1 × 36 + 1 × 35 + 1 × 33 + 1 × 30= 729 + 243 + 27 + 1= 1,000Step 2: decimal to base 81,000 ÷ 8 = 125 remainder 0125 ÷ 8 = 15 remainder 515 ÷ 8 = 1 remainder 71 ÷ 8 = 0 remainder 1Read the remainders from the bottom up: 17501101001 (base 3) = 1750 (base 8)
Need the other direction? Open the Octal to Ternary converter.
Ternary to Octal examples
The decimal number 255 is written 100110 in ternary and 377 in octal.
| Ternary | Octal | Decimal |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 2 | 2 |
| 21 | 7 | 7 |
| 22 | 10 | 8 |
| 101 | 12 | 10 |
| 120 | 17 | 15 |
| 121 | 20 | 16 |
| 1012 | 40 | 32 |
| 2101 | 100 | 64 |
| 10201 | 144 | 100 |
| 100110 | 377 | 255 |
| 1101001 | 1750 | 1000 |
Where Ternary to Octal 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.
Octal uses the digits 0 to 7, and each octal digit stands for exactly three bits. Unix file permissions such as chmod 755 are written in octal, and it also appears in older machine listings where word sizes divide by three.
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 octal.
Ternary to Octal FAQ
How do I convert ternary to octal?
Convert to decimal first by adding up each digit times its place value, then divide that decimal number by 8 repeatedly and read the remainders from last to first.
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 octal number be longer or shorter?
Shorter. Going to a larger base packs more value into each digit, so the result is shorter. Roughly 0.53 octal digits for every ternary digit. The number 255 needs 6 digits in ternary and 3 in octal.
What is 100110 in octal?
100110 in ternary is 377 in octal. 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.