Topic 1c
Binary Fractions, Floating-point representation, and numerical "problems"

Learning Outcomes

By the end of this topic students should be able to:

• Convert a length 8 bit-string into its decimal/fractional equivalent assuming the first four bits store the integer, and the second four bits store the fraction.
• Explain how a computer stores decimal numbers in the floating-point format using [8 | 32] bits.
• Explain overflow errors.
• Explain truncation (round-off) errors.
• Discuss situations where inexactness of computer calculations matter and don't matter explaining the source of the inexactness.

Learning Materials

• Readings
  • Fractions in binary - pp 54-56
  • Floating point representation - pp 63-66
  • Truncation Errors - pp 66-68
  • Overflow Errors - pp 59-60
• Videos
  • Crash Course Computing #4
    • Note, we are only going to be dealing with the floating part at this time. That runs from roughly 5:50-6:45.
  • https://www.youtube.com/watch?v=PZRI1IfStY0 (Why rounding errors occur in binary.)

 

Checking for Understanding

Answer the following questions from your textbook. The answers to all Q&E questions are in the back of your book in Appendix F.

• p56, #3
• p56, #4
• p56, #5 b and d

To really test your understanding

• p68, 1-3
  • None of these are directly related to a learning outcome.
  • BUT, the process of attempting these problems will help you with elements of other learning outcomes.
  • Do NOT worry if you don't understand how to do these.

Additional Guidance

• If you are still struggling with the material in this topic, the following videos may help.
  • Fractions in Binary
    • Explanation - binary strings to fractions
    • CFU p56, #3
    • Explanation - fractions to binary strings
    • CFU p56, #4
  • Floating Point Representations
    • Explanation
    • CFU p68, #1
    • CFU p68, #2
  • Storage Errors
    • Explanation

Further Information

• NA