Base-3 to Binary Converter
Convert base-3 (ternary) integers to binary instantly. Enter a ternary value such as 221 to see its decimal value, positional base-3 calculation, exact binary representation, and conversion steps.
What Is Base-3 to Binary Conversion?
Base-3 to binary conversion changes a number written in the ternary numeral
system into an equivalent base-2 number. Ternary uses three digits:
0, 1, and 2.
How to Convert Base-3 to Binary
1. Validate the digits
A base-3 number may contain only 0, 1, and 2.
2. Identify place values
Starting from the right, positions represent powers of 3.
3. Multiply each digit
Multiply each ternary digit by its corresponding power of 3.
4. Add the values
The sum gives the ordinary numerical value in decimal.
5. Convert that value to binary
Express the exact integer using powers of 2.
6. Read the binary result
The resulting 0-and-1 sequence is the equivalent base-2 number.
Base-3 Place Values
| Power | Value |
|---|---|
| 3⁰ | 1 |
| 3¹ | 3 |
| 3² | 9 |
| 3³ | 27 |
| 3⁴ | 81 |
| 3⁵ | 243 |
| 3⁶ | 729 |
Example: Convert Base-3 12 to Binary
Example: Convert Base-3 101 to Binary
Base-3 to Binary Conversion Table
| Base-3 | Decimal | Binary |
|---|---|---|
0 | 0 | 0 |
1 | 1 | 1 |
2 | 2 | 10 |
10 | 3 | 11 |
11 | 4 | 100 |
12 | 5 | 101 |
20 | 6 | 110 |
21 | 7 | 111 |
22 | 8 | 1000 |
100 | 9 | 1001 |
101 | 10 | 1010 |
221 | 25 | 11001 |
Why Can’t Base-3 Be Grouped Directly Into Binary Bits?
Direct fixed-size grouping works when one base is an exact power of another.
For example, octal digits map neatly to groups of three binary bits because
8 = 2³.
Base 3 is not a power of 2, so no fixed number of binary bits corresponds exactly to one ternary digit. The numeric value must instead be converted.
Base-3 vs Binary
| Feature | Base-3 | Binary |
|---|---|---|
| Base | 3 | 2 |
| Allowed digits | 0, 1, 2 | 0, 1 |
| Place values | Powers of 3 | Powers of 2 |
| Decimal 10 | 101 |
1010 |
| Decimal 25 | 221 |
11001 |
Do Leading Zeros Matter?
No. Leading zeros do not change a ternary integer’s numerical value.
Common Base-3 to Binary Mistakes
Using digits 3–9
Ternary permits only digits 0, 1, and 2.
Using powers of 2 too early
The ternary input must first be interpreted using powers of 3.
Grouping ternary digits
There is no direct fixed-size ternary-to-binary grouping rule.
Reading 101₃ as binary 101₂
The same digit string represents different values in different bases.
Related BinaryCon Tools
Base-3 to Binary Converter FAQs
How do you convert base 3 to binary?
What is base-3 2 in binary?
2 equals decimal 2, which is binary 10.
What is base-3 12 in binary?
12₃ = 1×3 + 2 = 5, so the binary result is 101.
What is base-3 101 in binary?
101₃ = 9 + 1 = 10, so it equals binary 1010.
What is base-3 221 in binary?
221₃ = 25₁₀, which equals binary 11001.
What digits are valid in base 3?
0, 1, and 2.