1. [12 pts] Given a 10-bit floating-point representation in the IEEE format, wit

Question

1. [12 pts] Given a 10-bit floating-point representation in the IEEE format, with 3 bits for the exponent. Answer the following: a. (4 points) What decimal value does 1011010010 represent? b. (4 points) What is the floating point representation for the real number 7 9/16 (seven nine- sixteenths)? c. (2 points) Using this representation, what is the smallest positive normalized number one can represent? d. (2 points) Using this representation, what is the smallest positive denormalized number one can represent?

Explanation / Answer

a.

1011010010

Answer: -9

Explanation:

1st bit 1 denotes sign which is negative

Next 3 bits are exponent 011 which is 3, Therefore exponent is 23

Remaining bits are Mantessa 010010 ,

Therefore the binary number is

1.0010 * 23

10.010 * 22

100.10 * 21

1001.0 * 20

i..e 1001, The decimal Equivalanet for the given number (010010) is 8+1 = 9

b. 7 9/16(Seven 9 sixteenth) = 7.5625

Answer: 0 010 111001

Explanation:

First Convert the number into binary

.5625 * 2 = 1.125 leaves 1

.125 * 2 = .25 leaves 0

.25 * 2 = .5 leaves 0

.5 * 2 = 1.0 leaves 1

0111.10012

Now converting this to floating point

0111.1001 * 20

011.11001 * 21

01.111001 * 22

1st bit is sign bit which is 0 in case of positive, next 3 bits exponent ,here the exponent is 1 = 010

Remaining 6 bits is Mantessa which is 111001

The enitre number is 0 010 111001

c. smallest positve number with normalization, let the binary number be 0 000 000001

2-6 = 0.015625

The number would be 0.000001 * 20   = 0.015625

d. Smallest positve number without normalization, let the binary be 00.00000001

2-8 = 0.00390625

The number is 0.00390625

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