Solved Examples

Example 1: Convert 45 (base 10) to base 2 (binary). - 45 ÷ 2 = 22 remainder 1 - 22 ÷ 2 = 11 remainder 0 - 11 ÷ 2 = 5 remainder 1 - 5 ÷ 2 = 2 remainder 1 - 2 ÷ 2 = 1 remainder 0 - 1 ÷ 2 = 0 remainder 1 - Reading remainders bottom to top: 101101 - Check: 1×32+0×16+1×8+1×4+0×2+1×1 = 32+8+4+1 = 45 ✓

Example 2: Convert (1101)₂ to base 10. - 1×2³ + 1×2² + 0×2¹ + 1×2⁰ = 8+4+0+1 = 13

Example 3: Convert 100 (base 10) to base 5. - 100 ÷ 5 = 20 remainder 0 - 20 ÷ 5 = 4 remainder 0 - 4 ÷ 5 = 0 remainder 4 - Reading bottom to top: 400 (in base 5) - Check: 4×25 + 0×5 + 0×1 = 100 ✓

Example 4 (Base 16 / hexadecimal edge case): Convert 255 (base 10) to base 16. - 255 ÷ 16 = 15 remainder 15 (digit “F” in hex) - 15 ÷ 16 = 0 remainder 15 (digit “F” in hex) - Reading remainders bottom to top: FF (base 16) - Check: F×16 + F = 15×16 + 15 = 240 + 15 = 255 ✓

Example 5 (Chained conversion between two non-decimal bases): Convert (101)₂ to base 5. - First convert to base 10: 1×2² + 0×2¹ + 1×2⁰ = 4 + 0 + 1 = 5 - Now convert 5 (base 10) to base 5: 5 ÷ 5 = 1 remainder 0; 1 ÷ 5 = 0 remainder 1 - Reading bottom to top: 10 (base 5) Answer: (101)₂ = (10)₅ (Always route a base-to-base conversion through base 10 as an intermediate step unless a direct shortcut is obvious.)