Ternary (Base 3) — Definition, Formula & Examples

Ternary (Base 3) is a number system that uses only three digits — 0, 1, and 2 — where each place value represents a power of 3 instead of a power of 10.

The ternary numeral system is a positional notation with radix 3, in which a number is expressed as a finite sum $\sum_{k=0}^{n} d_k \cdot 3^k$ , where each digit $d_k \in \{0, 1, 2\}$ .

Key Formula

N = d_n \cdot 3^n + d_{n-1} \cdot 3^{n-1} + \cdots + d_1 \cdot 3^1 + d_0 \cdot 3^0

Where:

$N$ = The value of the number in decimal
$d_k$ = The digit in position k (must be 0, 1, or 2)
$3^k$ = The place value of position k

How It Works

In base 10 (decimal), place values go 1, 10, 100, 1000, and so on — each a power of 10. In base 3, place values go 1, 3, 9, 27, 81, and so on — each a power of 3. To read a ternary number, multiply each digit by its place value and add the results. To convert a decimal number to ternary, repeatedly divide by 3 and record the remainders from last to first.

Worked Example

Problem: Convert the ternary number 1202₃ to decimal.

Step 1: Write out the place values from right to left: 3⁰ = 1, 3¹ = 3, 3² = 9, 3³ = 27.

Step 2: Multiply each digit by its place value.

1 \times 27 + 2 \times 9 + 0 \times 3 + 2 \times 1

Step 3: Add the results.

27 + 18 + 0 + 2 = 47

Answer: 1202₃ = 47 in decimal.

Why It Matters

Understanding base 3 builds fluency with place value and positional notation, which is essential when studying modular arithmetic and number theory. Ternary also appears in computer science (balanced ternary logic) and in classic math puzzles like the counterfeit coin problem, where base-3 reasoning gives the optimal solution.

Common Mistakes

Mistake: Using the digit 3 in a ternary number (e.g., writing 130₃).

Correction: Base 3 only allows the digits 0, 1, and 2. If a digit reaches 3, you carry 1 to the next place, just like carrying in base 10 when a digit reaches 10.