The ancient Indians likely used base 10 for the Hindu number system because humans have 10 fingers, and so we can use our fingers to count. But what if we had only 8 fingers? How would we be writing numbers then? What would the Hindu numerals look like if we were using base 8 instead? Base 5? Try writing the base-10 Hindu numeral 25 as base-8 and base-5 Hindu numerals, respectively. Can you write it in base-2?
Context: If humans had 8 fingers, we would likely use base-8 (octal) with digits 0-7.
Given: Base-10 numeral 25
Base-8 (Octal) Conversion: 25 = + 1 = 31 (octal)
Base-5 Conversion: 25 = + + = 100 (base-5)
Base-2 (Binary) Conversion: 25 = + + + + = 11001 (binary)
Verification: 31 (base-8) = + 1 = 25 ✓ 100 (base-5) = + + = 25 ✓ 11001 (base-2) = 16 + 8 + 0 + 0 + 1 = 25 ✓
Answer:
- 25 in base-8 = 31
- 25 in base-5 = 100
- 25 in base-2 = 11001
Note: In base-8, we would need only 8 digits (0-7). In base-5, we would need only 5 digits (0-4). In base-2 (binary), we need only 2 digits (0 and 1), which is why computers use binary.
Explanation
The chapter explains that base-n systems use digits 0 to (n-1) and positions representing powers of n. The conversion process involves expressing a number as sums of powers of the base. For base-8: 25 = ¹ + ⁰ = 31. For base-5: 25 = ² + ¹ + ⁰ = 100. For base-2: 25 = + + + + = 11001. The chapter notes that ancient Indians likely chose base-10 because humans have 10 fingers for counting.
Solution Steps
Step 1: For base-8 conversion of 25: 25 ÷ 8 = 3 remainder 1, so 25 = 31 in base-8
Step 2: For base-5 conversion of 25: 25 ÷ 25 = 1 remainder 0, 0 ÷ 5 = 0 remainder 0, so 25 = 100 in base-5
Step 3: For base-2 conversion of 25: 25 = 16 + 8 + 1 = 2⁴ + 2³ + 2⁰, so 25 = 11001 in binary
Step 4: Verify: 31 (base-8) = + 1 = 25 ✓
Step 5: Verify: 100 (base-5) = + + = 25 ✓
Step 6: Verify: 11001 (base-2) = 16 + 8 + 0 + 0 + 1 = 25 ✓