Topic 1d
Floating-point Representation

Learning Outcomes

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

  • Explain how a computer stores decimal numbers in the floating-point format using [8 | 32] bits.

 

Learning Materials

  • Readings
    • Floating point representation - pp 63-66
    • NOTE: There is an error on page 65. The second line of text should read "the value represented would have been -2 3/4" That is, it should be a NEGATIVE number.
  • Videos
  • Extension
    • While we expect you to understand the structure of converting binary to floating point values and vice versa, we do not actually expect that you can complete the calculation itself.
    • However, we found that it was beneficial to our own understanding of the structure of the encoding to play with the following tools.
    • Your book describes a simplified 8-bit format.
      • This tool will model this format if you set the precision to 5 and the exponent to 3
      • On the website linked, you can click on the "bits" in the colored boxes to toggle them between 1 and 0 and see how the decimal value changes.
    • The 8-bit described above is a simplification to help us understand the process. Computers don't ever use this format. But they do use 16, 32, and 64 bit encodings.

    •  

Checking for Understanding

We are mostly concerned about the following:

  • Can you explain how floating point values are stored in the computer standard of 32 bits
    • For each of the following terms, what does it represent, and how much of the 32 bits is allocated to this portion
      • Mantissa (What does it represent? How much memory?)
      • Exponent Field (What does it represent? How much memory?)
      • Sign Bit
  • Can you explain how floating point values are stored in the textbook example of only 8 bits

 

While we would not ask you do to this on a competency demo, we think that it will help your overall understanding to be able to complete:

  • p68, 1-2
    • Again, neither 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 even after you try these and watch the videos below if you feel you still could not do this on your own - you won't be asked to do so.

 

Answers & Guidance

The answers to the CFU questions above are provided in the back of the book. The following videos explain how we would arrive at some of those answers.