Lecture Notes on Number Bases

Introduction

• Presenter: Bella, YouTube channel host
• Topic: Understanding number bases, specifically converting between them
• Objective: Learn how numbers can be represented in different bases other than base 10

Number Base Concepts

• Base 10: The standard number system using digits 0-9
• Other Bases:
  • Base 2 (Binary): Uses digits 0 and 1
  • Base 3: Uses digits 0, 1, 2
  • Base 4: Uses digits 0, 1, 2, 3
  • Bases 5-9: Uses respective digits up to n-1

Understanding Place and Digit Values

• Example: Number 213 in base 10
  • Place Value:
    • 3: 1
    • 1: 10
    • 2: 100
  • Digit Values:
    • 3: 1x3 = 3
    • 1: 10x1 = 10
    • 2: 100x2 = 200
  • Number Value: Sum of digit values: 200 + 10 + 3 = 213

Conversion Between Bases

Conversion to Base 10

• General Method: Multiply each digit by its place value and sum them
• Example 1: Convert 1101 from base 2 to base 10
  • Place values: 2^3, 2^2, 2^1, 2^0
  • Digit values: 1x8 + 1x4 + 0x2 + 1x1
  • Result: 13 in base 10
• Example 2: Convert 432 from base 5 to base 10
  • Place values: 5^2, 5^1, 5^0
  • Digit values: 4x25 + 3x5 + 2x1
  • Result: 117 in base 10

Conversion from Base 10

• General Method: Divide the base 10 number repeatedly by the target base and track remainders
• Example 1: Convert 32 from base 10 to base 5
  • Divide by 5, track remainders
  • Final result: 113 in base 5
• Example 2: Convert 408 from base 10 to base 8
  • Divide by 8, track remainders
  • Final result: 635 in base 8

Conversion Between Non-decimal Bases

• Methodology: Convert to base 10 as an intermediate step, then to the target base
• Example: Convert 22012 from base 3 to base 7
  • Convert to base 10 first, then to base 7
  • Result: 263 in base 7

Using Tables and Calculators for Base Conversion

• Tables: Can be used to quickly match base 2 to base 8 conversions
• Calculators: Useful for verifying conversions by switching modes (BIN for binary, OCT for octal, DEC for decimal)

Addition and Subtraction in Different Bases

Addition

• Example 1: Add 2331 (base 4) + 230 (base 4)
  • Perform base-specific addition, handling carryovers
  • Result: 3221 in base 4

Subtraction

• Example 1: Subtract 2034 (base 5) - 142 (base 5)
  • Borrow as needed, using base-specific borrowing
  • Result: 1342 in base 5

Conclusion

• Key Takeaways: Understanding and converting between number bases is crucial
• Next Steps: Practice these conversions to gain familiarity
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Note: The lecture provides a comprehensive understanding of number bases, conversion techniques, and practical examples to illustrate each concept.