[12 pts] Given a 10-bit floating-point representation in the IEEE format, with 3
ID: 3871062 • Letter: #
Question
[12 pts] Given a 10-bit floating-point representation in the IEEE format, with 3 bits for the exponent. Answer the following: 1. a. (4 points) What decimal value does 1011010010 represent? b. (4 points) What is the floating point representation for the real number 79/16 (seven nine-sixteenths)? (2 points) Using this representation, what is the smallest positive normalized number one can represent? c. 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 i..e 1001, The decimal Equivalanet for the given number (010010) is 8+1 = 9
b. 7 9/16(Seven 9 sixteenth) = 3.9316
Answer: 0 001 111110
Explanation:
First Convert the number into binary
.9316*2 = 1.8632 leaves 1
.8632 *2 = 1.7264 leaves 1
.7264 *2= 1.4528 leaves 1
.4528 *2 = 0.9057 leaves 0
.9057 *2 = 1.8114 leaves 1
.8114 *2 = 1.6228 leaves 1
0.6228 * 2= 1.2456 leaves 1
.2456 * 2 = 0.4912 leaves 0
.4912 * 2 = 0.9824 leaves 0
0.9824 * 2 = 1.9648 leaves 1
We can stop after these many iteration
Therefore 3.9316 = 011.1110111001
Now converting this to floating point
011.1110111001 * 20
01.11110111001 * 21
1st bit is sign bit which is 0 in case of positive
next 3 bits exponent ,here the exponent is 1 = 001
Remaining 6 bits is Mantessa which is 111110
c. smallest positve number with normalization, let the binary number be 0 000 000001
The number would be 0.000001 * 20 = 0.015625
d. Smallest positve number without normalization, let the binary be 0 0.00000001
The number is 0.00390625.
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