Evaluate (212)$_3$ - (121)$_3$ + (222)$_3$

To solve this problem, we need to perform arithmetic operations in Base 3. Remember that in base 3, the digits allowed are only 0, 1, and 2. When you add or subtract, any value that reaches 3 is carried over or borrowed as "10" in base 3.

Step 1: Perform the subtraction (212)$_3$ - (121)$_3$

Align the numbers:

  2 1 2
- 1 2 1
-------

• Units column: 2 - 1 = 1
• Threes column: 1 - 2. We cannot do this, so we borrow 1 from the "nines" column. The 2 becomes 1, and the 1 becomes (1 + 3) = 4. Now, 4 - 2 = 2.
• Nines column: 1 - 1 = 0.

Result of subtraction: (021)$_3$

Step 2: Perform the addition (021)$_3$ + (222)$_3$

Align the numbers:

  0 2 1
+ 2 2 2
-------

• Units column: 1 + 2 = 3. In base 3, 3 is written as 10. Write down 0 and carry over 1.
• Threes column: 2 + 2 + 1 (carried) = 5. In base 3, 5 is (1 × 3) + 2, which is written as 12. Write down 2 and carry over 1.
• Nines column: 0 + 2 + 1 (carried) = 3. In base 3, 3 is written as 10. Write down 10.

Result of addition: (1020)$_3$

Summary of Calculation

1. (212)$_3$ - (121)$_3$ = (021)$_3$
2. (021)$_3$ + (222)$_3$ = (1020)$_3$

Exam Tip: When working in any base $n$, if your sum or difference involves a number equal to or greater than $n$, always divide by $n$ to find the remainder (the digit to write) and the quotient (the carry-over). For example, in base 3, if you get 5, divide 5 by 3: $5 \div 3 = 1$ remainder $2$. So, you write 2 and carry 1.