5 Given a (very) tiny computer that has a word size of 6 bits, what

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5. Given a (very) tiny computer that has a word size of 6

bits, what are the smallest negative numbers and the

largest positive numbers that this computer can represent

in each of the following representations?

a. One\'s complement

b. Two\'s complement

6. Perform the following binary multiplications, assuming

unsigned integers:

a. 1100 × 101 b. 10101 × 111 c. 11010 × 1100

Ans. a. 111100 b. 10010011 c. 100111000

7. If the floating-point number representation on a certain

system has a sign bit, a 3-bit exponent and a 4-bit

significand:

a. What is the largest positive and the smallest positive

number that can be stored on this system if the storage is

normalized? (Assume no bits are implied, there is no

biasing, exponents use two\'s complement notation, and

exponents of all zeros and all ones are allowed.)

b. What bias should be used in the exponent if we prefer

all exponents to be non-negative? Why would you choose

this bias?

8. Let a = 1.0 × 29, b = 1.0 × 29 and c = 1.0 × 21. Using

the floating-point model described in the text (the

representation uses a 14-bit format, 5 bits for the exponent

with a bias of 16, a normalized mantissa of 8 bits, and a

single sign bit for the number), perform the following

calculations, paying close attention to the order of

operations. What can you say about the algebraic

properties of floating-point arithmetic in our finite model?

Do you think this algebraic anomaly holds under

multiplication as well as addition?

b + (a + c) =

(b + a) + c =

Solution

here (x) y = x subscript y

a. Largest Positive: (011111) 2 (31) Smallest Negative:

(100000) 2 (-31)

b. Largest Positive: (011111) 2 (31) Smallest Negative:

(100000) 2 (-32)

a. 111100

b. 10010011

c. 100111000

a. Largest Positive: 0.1111 2 x 2^3 = (111.1) 2 = 7.5

Smallest Positive: (0.1) 2 x 2^-4 = (.00001) 2 = 1/32 =

0.03125

b + (a + c) = b + [1.0 x 2^9 + 1.0 x 2^1 ]

= b + [1.0 x 2^9 + .000000001 x 2^9 )

=b + 1.000000001 x 2^9

=b+ 0.1000000001 x 2^10 (but we can only have 8 bits

in the mantissa)

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5. Given a (very) tiny computer that has a word size of 6 bits, what are the smallest negative numbers and the largest positive numbers that this computer can represent in each of the following representations? a. One\'s complement b. Two\'s complement 6. Perform the following binary multiplications, assuming unsigned integers: a. 1100 × 101 b. 10101 × 111 c. 11010 × 1100 Ans. a. 111100 b. 10010011 c. 100111000 7. If the floating-point number representation on a certain system has a sign bit, a 3-bit exponent and a 4-bit significand: a. What is the largest positive and the smallest positive number that can be stored on this system if the storage is normalized? (Assume no bits are implied, there is no biasing, exponents use two\'s complement notation, and exponents of all zeros and all ones are allowed.) b. What bias should be used in the exponent if we prefer all exponents to be non-n ...
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