Ternary (Base 3) to Binary (Base 2) Converter
Convert ternary numbers to binary.
100110 (base 3) = 11111111 (base 2)
Digits in base 3: 0 to 2. Digits in base 2: 0 and 1.
How to convert Ternary to Binary
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 2 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 2, 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 2: 0 and 1
Worked example 1: 1200 from ternary to binary
Step 1: base 3 to decimal1200 (base 3) = 1 × 33 + 2 × 32= 27 + 18= 45Step 2: decimal to base 245 ÷ 2 = 22 remainder 122 ÷ 2 = 11 remainder 011 ÷ 2 = 5 remainder 15 ÷ 2 = 2 remainder 12 ÷ 2 = 1 remainder 01 ÷ 2 = 0 remainder 1Read the remainders from the bottom up: 1011011200 (base 3) = 101101 (base 2)
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 binary
Step 1: base 3 to decimal100110 (base 3) = 1 × 35 + 1 × 32 + 1 × 31= 243 + 9 + 3= 255Step 2: decimal to base 2255 ÷ 2 = 127 remainder 1127 ÷ 2 = 63 remainder 163 ÷ 2 = 31 remainder 131 ÷ 2 = 15 remainder 115 ÷ 2 = 7 remainder 17 ÷ 2 = 3 remainder 13 ÷ 2 = 1 remainder 11 ÷ 2 = 0 remainder 1Read the remainders from the bottom up: 11111111100110 (base 3) = 11111111 (base 2)
Worked example 3: 1101001 from ternary to binary
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 21,000 ÷ 2 = 500 remainder 0500 ÷ 2 = 250 remainder 0250 ÷ 2 = 125 remainder 0125 ÷ 2 = 62 remainder 162 ÷ 2 = 31 remainder 031 ÷ 2 = 15 remainder 115 ÷ 2 = 7 remainder 17 ÷ 2 = 3 remainder 13 ÷ 2 = 1 remainder 11 ÷ 2 = 0 remainder 1Read the remainders from the bottom up: 11111010001101001 (base 3) = 1111101000 (base 2)
Need the other direction? Open the Binary to Ternary converter.
Ternary to Binary examples
The decimal number 255 is written 100110 in ternary and 11111111 in binary.
| Ternary | Binary | Decimal |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 10 | 2 |
| 21 | 111 | 7 |
| 22 | 1000 | 8 |
| 101 | 1010 | 10 |
| 120 | 1111 | 15 |
| 121 | 10000 | 16 |
| 1012 | 100000 | 32 |
| 2101 | 1000000 | 64 |
| 10201 | 1100100 | 100 |
| 100110 | 11111111 | 255 |
| 1101001 | 1111101000 | 1000 |
Where Ternary to Binary 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.
Binary uses only 0 and 1, because a digital circuit has two stable states: off and on. Every number a computer stores is binary underneath. You meet it when you work with bit flags, bit masks, subnet masks and low-level debugging.
Going to a smaller base spreads the same value over more digits, so the result is longer. The number 255 needs 6 digits in ternary and 8 in binary.
Ternary to Binary FAQ
How do I convert ternary to binary?
Convert to decimal first by adding up each digit times its place value, then divide that decimal number by 2 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 binary number be longer or shorter?
Longer. Going to a smaller base spreads the same value over more digits, so the result is longer. Roughly 1.58 binary digits for every ternary digit. The number 255 needs 6 digits in ternary and 8 in binary.
What is 100110 in binary?
100110 in ternary is 11111111 in binary. 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.