Exercise 3.15 Binary numbers are used in the mantissa fi eld, but they do not have to be. IBM used base 16 numbers, for example, in some of their fl oating-point formats. There are other approaches that are possible as well, each with their own particular advantages and disadvantages. The following table shows fractions to be represented in various fl oating-point formats.

a. 1/2

b. 1/9 3.15.1 [10] <3.5, 3.6> Write down the bit pattern in the mantissa assuming a fl oating-point format that uses binary numbers in the mantissa (essentially what you have been doing in this chapter). Assume there are 24 bits, and you do not need to normalize. Is this representation exact?

3.15.2 [10] <3.5, 3.6> Write down the bit pattern in the mantissa assuming a fl oating-point format that uses Binary Coded Decimal (base 10) numbers in the mantissa instead of base 2. Assume there are 24 bits, and you do not need to normalize. Is this representation exact?

3.15.3 [10] <3.5, 3.6> Write down the bit pattern assuming that we are using base 15 numbers in the mantissa instead of base 2. (Base 16 numbers use the symbols 0–9 and A–F. Base 15 numbers would use 0–9 and A–E.) Assume there are 24 bits, and you do not need to normalize. Is this representation exact?

3.15.4 [20] <3.5, 3.6> Write down the bit pattern assuming that we are using base 30 numbers in the mantissa instead of base 2. (Base 16 numbers use the symbols 0–9 and A–F. Base 30 numbers would use 0–9 and A–T.) Assume there are 20 bits, and you do not need to normalize. Is this representation exact? Do you see any advantage to using this approach?

§3.2, page 229: 3.

§3.4, page 269: 3.