To add the numbers \(1021_3\) and \(1102_3\) in base 3, we'll follow these steps:

### Step 1: Convert each base 3 number to base 10.

1. **Convert \(1021_3\) to base 10:**

   \[
   1021_3 = 1 \times 3^3 + 0 \times 3^2 + 2 \times 3^1 + 1 \times 3^0
   \]
   \[
   = 1 \times 27 + 0 \times 9 + 2 \times 3 + 1 \times 1
   \]
   \[
   = 27 + 0 + 6 + 1 = 34_{10}
   \]

2. **Convert \(1102_3\) to base 10:**

   \[
   1102_3 = 1 \times 3^3 + 1 \times 3^2 + 0 \times 3^1 + 2 \times 3^0
   \]
   \[
   = 1 \times 27 + 1 \times 9 + 0 \times 3 + 2 \times 1
   \]
   \[
   = 27 + 9 + 0 + 2 = 38_{10}
   \]

### Step 2: Add the two base 10 numbers.

\[
34_{10} + 38_{10} = 72_{10}
\]

### Step 3: Convert the result back to base 3.

To convert 72 from base 10 to base 3:

\[
72 \div 3 = 24 \quad \text{remainder} \, 0
\]
\[
24 \div 3 = 8 \quad \text{remainder} \, 0
\]
\[
8 \div 3 = 2 \quad \text{remainder} \, 2
\]
\[
2 \div 3 = 0 \quad \text{remainder} \, 2
\]

Reading the remainders from bottom to top, \(72_{10}\) converts to \(2200_3\).

### Final Answer:
\[
1021_3 + 1102_3 = 2200_3
\]

Would you like more details or have any questions?

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Here are 8 related questions you might find interesting:

1. How do you subtract two numbers in base 3?
2. How do you multiply two numbers in base 3?
3. How do you convert a base 10 number to any other base?
4. What is the process for dividing numbers in base 3?
5. Can you solve a similar problem for base 4 numbers?
6. What is the significance of using different bases in number systems?
7. How do you add numbers in hexadecimal (base 16)?
8. What is the binary (base 2) equivalent of the sum \(2200_3\)?

**Tip:** When adding or subtracting in different bases, converting to base 10 first can help prevent errors!

Learn how to add two numbers in base 3 (ternary system): 1021 and 1102. Follow a step-by-step guide with conversions between base 3 and base 10, including final result conversion. Suitable for students learning about number systems and base conversion in Grades 7-9.

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