1. a) Convert decimal rational number 1/6 to the binary form, keep 8-bits in the
ID: 3879254 • Letter: 1
Question
1. a) Convert decimal rational number 1/6 to the binary form, keep 8-bits in the binary fraction.
b) Convert the binary form of 1/6 to decimal fraction by addition of bit weights, keep 5 decimal digits. Compare the result to 1/6 in the fraction form with 5 decimal digits, find the difference (conversion error).
c) What is the maximum conversion error for a fraction presented with 8-bits? Compare it to the conversion error in part b.
d) Identify the repeated pattern in part a, extend the presentation of 1/6 to 16 bits.
e) Obtain the binary presentations for decimal numbers 1/3 and 2/3 from your binary result for 1/6 by shifting the binary point, keep 16 bits.
f) Present the numbers in part e in hexadecimal form.
What would be today’s age of a) Claude Shannon, b) George Stibitz, c) George Boole? Present the answers in hexadecimal form.
Explanation / Answer
a) 1/6 = 0.16666666666...
Step 1) Multipy it with 2 and note the left part of the decimal point.
0.16666666666 *2 = 0.3333333332
Step 2) Now multiply the right part of the decimal point by 2
0.333333333 * 2 = 0.6666666
Step 3) keep doing step 1 and 2
0.6666666*2 = 1.333333332
now the right part is 0.333333332
0.333333333 * 2 = 0.6666666666
0.666666666*2 = 1.33333333332
0.3333333333 * 2 = 0.666666666
0.6666666666 * 2 = 1.3333333332
0.333333333332 * 2 = 0.66666666
The 8 bit representation of 1/6 is .00101010
b) Convert the binary form into decimal form(till 5 places)
0.00101
(0*2-1) + (0*2-2) + (1*2-3) + (0*2-4) + (1*2-5) = 1/8 + 1/32 = 5/32 = 0.15625
Conversion error in part b = 0.01041
c) (0*2-1) + (0*2-2) + (1*2-3) + (0*2-4) + (1*2-5) + (0*2-6) + (1*2-7) + (0*2-8) = 1/8 + 1/32 + 1/128 = 21/128 = 0.1640625
The maximum conversion error for a fraction 1/6 = 0.1666666 = 0.0026041
Error in part c = 0.0026041 Error in part b = 0.01041
Error in part c is less than in part b
d) The repeated pattern in part a is "10" extend it to 16 bits
.0010101010101010
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