Check: ( 2 \times 9 + 2 \times 3 + 1 = 18 + 6 + 1 = 25 ). Multiply each digit by its place value (power of 3) and sum.
| Decimal | Base 3 | Why? | |---------|--------|------| | 0 | 0 | | | 1 | 1 | | | 2 | 2 | | | 3 | 10 | (1 \times 3 + 0) | | 4 | 11 | (3 + 1) | | 5 | 12 | (3 + 2) | | 6 | 20 | (2 \times 3 + 0) | | 7 | 21 | (6 + 1) | | 8 | 22 | (6 + 2) | | 9 | 100 | (1 \times 9 + 0 + 0) | | 10 | 101 | (9 + 1) | | 11 | 102 | (9 + 2) | | 12 | 110 | (9 + 3) | | … | … | | Method: Repeated division by 3, reading remainders bottom to top . base 3
| Place (from right) | Power of 3 | Decimal value | |--------------------|------------|---------------| | 1st (units) | (3^0) | 1 | | 2nd | (3^1) | 3 | | 3rd | (3^2) | 9 | | 4th | (3^3) | 27 | | 5th | (3^4) | 81 | | … | … | … | Check: ( 2 \times 9 + 2 \times 3 + 1 = 18 + 6 + 1 = 25 )
( 1022_3 ) [ 1 \times 27 + 0 \times 9 + 2 \times 3 + 2 \times 1 = 27 + 0 + 6 + 2 = 35_10 ] 6. Arithmetic in Base 3 Addition Add digit by digit, carry when sum ≥ 3. | |---------|--------|------| | 0 | 0 | |
The base-3 number ( 210_3 ) means: [ 2 \times 3^2 + 1 \times 3^1 + 0 \times 3^0 = 2 \times 9 + 1 \times 3 + 0 = 18 + 3 = 21_10 ] 3. Counting in Base 3 Counting works like any base: increment the rightmost digit; when it exceeds 2, set it to 0 and carry 1 to the left.