Solved Examples: Base Conversions

Example 1: Converting to Mayan

Convert 45 to Mayan Numerals.

Solution:

1. Determine the base: 20.
2. Divide 45 by 20: $45 \div 20 = 2 \text{ remainder } 5$
3. Top position (20s): 2 (Two dots $\bullet\bullet$)
4. Bottom position (1s): 5 (One bar —)

Example 2: Converting to Base-5

Convert 34 to Base-5.

Solution:

1. Powers of 5: 1, 5, 25.
2. Largest power $\le$ 34 is 25. $34 \div 25 = 1 \text{ remainder } 9$
3. Next power is 5. $9 \div 5 = 1 \text{ remainder } 4$
4. Last power is 1. $4 \div 1 = 4$
5. Result: $114_5$ ($1 \times 25 + 1 \times 5 + 4 \times 1$).

Example 3: Mesopotamian Addition

Add $12$ and $50$ in Base-60.

Solution:

1. Convert to Base-10 sum: $12 + 50 = 62$.
2. Convert 62 to Base-60. $62 = 1 \times 60 + 2$
3. Representation:
   • 60s place: 1 Wedge ($\triangledown$)
   • 1s place: 2 Wedges ($\triangledown\triangledown$)
   • Written: $\triangledown \quad \triangledown\triangledown$