(a) Convert 0.11 into base-2 and store the result in a 10-bit word of the follow

Question

(a) Convert 0.11 into base-2 and store the result in a 10-bit word of the following format: (b) Convert the stored number back into base-10, and calculate the true percent relative error caused by the storage process. What is the smallest positive number that can be accurately stored in a 10-bit word of the same format as the one in Problem 2? The quantity 0.375 can be stored exactly in the 10-bit word from Problem 2. What is the next larger quantity that can be stored exactly in that 10-bit word? In other words, what is the smallest number greater than 0.375 that can be accurately stored in the 10-bit word of Problem 2?

Explanation / Answer

Solution:

Problem 2:

a)

Let's convert 0.11 to binary representation first

which is 0.00011100001010001111

Now let's write it in normalized format 1.1100001010001111* 2^-4

Mantissa can be only five digits here because the magnitude of the mantissa is only 5.

1.11000* 2^-4

So our representation will be

b)

Now representation is

1.11000* 2^-4= 0.000111000

let's convert this to decimal to calculate error percentage

in decimal 0.000111000 is 0.109375

Error (%)= ((0.11-0.10935)/0.11)* 100= 0.5909090909 %

Problem 3:

For smallest positive number our table will look like this:

The representation of that number would be

1.00000* 2^-7= 0.000000100000

which is in decimal will be 0.0078125

Problem 4:

0.375 will be stored like this

0.011= 1.1* 2^-2

the number which is the smallest number greater than 0.375 will be

1.10001 * 2^-2 = 0.011001

in decimal, it is 0.40625

I hope this helps if you find any problem. Please comment below. Don't forget to give a thumbs up if you liked it. :)

Sign of number Sign of Exponent Exponent Mantissa 0 1 100 11000

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