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11) Evaluate the following problem: 311five - 24five

11) Evaluate the following problem: 311five - 24five

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Problem 11:** Evaluate the following problem: \( 311_{five} - 24_{five} \). This question involves subtraction in base 5 (quinary numeral system). To solve this, convert both numbers to base 10, perform the subtraction, and convert back to base 5 if needed. 1. **Convert to Base 10:** - \( 311_{five} \): - \( 3 \times 5^2 + 1 \times 5^1 + 1 \times 5^0 \) - \( 3 \times 25 + 1 \times 5 + 1 \) - \( 75 + 5 + 1 = 81 \) - \( 24_{five} \): - \( 2 \times 5^1 + 4 \times 5^0 \) - \( 2 \times 5 + 4 \) - \( 10 + 4 = 14 \) 2. **Subtract in Base 10:** - \( 81 - 14 = 67 \) 3. **Convert Back to Base 5:** - Divide 67 by 5 to convert back to base 5. - \( 67 \div 5 = 13 \) remainder \( 2 \) (rightmost digit) - \( 13 \div 5 = 2 \) remainder \( 3 \) (next digit) - \( 2 \div 5 = 0 \) remainder \( 2 \) (leftmost digit) - Result: \( 232_{five} \) **Solution: \( 311_{five} - 24_{five} = 232_{five} \)**
Transcribed Image Text:**Problem 11:** Evaluate the following problem: \( 311_{five} - 24_{five} \). This question involves subtraction in base 5 (quinary numeral system). To solve this, convert both numbers to base 10, perform the subtraction, and convert back to base 5 if needed. 1. **Convert to Base 10:** - \( 311_{five} \): - \( 3 \times 5^2 + 1 \times 5^1 + 1 \times 5^0 \) - \( 3 \times 25 + 1 \times 5 + 1 \) - \( 75 + 5 + 1 = 81 \) - \( 24_{five} \): - \( 2 \times 5^1 + 4 \times 5^0 \) - \( 2 \times 5 + 4 \) - \( 10 + 4 = 14 \) 2. **Subtract in Base 10:** - \( 81 - 14 = 67 \) 3. **Convert Back to Base 5:** - Divide 67 by 5 to convert back to base 5. - \( 67 \div 5 = 13 \) remainder \( 2 \) (rightmost digit) - \( 13 \div 5 = 2 \) remainder \( 3 \) (next digit) - \( 2 \div 5 = 0 \) remainder \( 2 \) (leftmost digit) - Result: \( 232_{five} \) **Solution: \( 311_{five} - 24_{five} = 232_{five} \)**
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