# 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
b + (a + c) =
(b + a) + c =

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Solution
here (x) y = x subscript y
5)
a. Largest Positive: (011111) 2 (31) Smallest Negative:
(100000) 2 (-31)
b. Largest Positive: (011111) 2 (31) Smallest Negative:
(100000) 2 (-32)
6)
a. 111100
b. 10010011
c. 100111000
7)
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
8)
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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