Question

: Do the following conversion problems:
(a) Convert decimal 27.315 to binary.
(b) Calculate the binary equivalent of 2/3 out to eight places. Then convert from binary to
decimal. How close is the result to 2/3?
(c) Convert the binary result in (b) into hexadecimal. Then convert the result to decimal. Is the
answer the same?​

Answer 1

Answer:

Which class???????????

Answer 2

Explanation:

Given:

(a) Convert decimal 27.315 to binary.

(b) Calculate the binary equivalent of 2/3 out to eight places. Then convert from binary to

decimal. How close is the result to 2/3?

(c) Convert the binary result in (b) into hexadecimal. Then convert the result to decimal. Is the answer the same?

Solution:

(a) Convert decimal 27.315 to binary:

Ans: We know that a decimal can be converted to its equivalent binary number by division of 2 before decimal point and multiplying by 2 after decimal point.

2) 27 (1

2)13(1

2)6(0

2)3(1

1

Binary of 27 is (11011)

Now convert 0.315 into binary.Multiply the no by 2,if result is greater than 1,keep it and multiply again with 2. The process is shown below

0.315×2=0.630; =>0

0.630×2=1.260;=> 1

0.260×2=0.520;=>0

0.520×2=1.040;=>1

0.040×2=0.080;=>0

Binary of 27.315 is 11011.01010...

(b) Calculate the binary equivalent of 2/3 out to eight places. Then convert from binary to

decimal. How close is the result to 2/3?

Ans:

2/3 is non terminating recurring decimal expansion

Take recurring digit upto 8 places

0.66666666

Now convert 0.66666666 into binary.Multiply the no by 2,if result is greater than 1,keep it and multiply again with 2. The process is shown below

0.66666666×2=1.33333333; =>1

0.33333333×2=0.66666666;=> 0

0.66666666×2=1.33333333; =>1

0.33333333×2=0.66666666;=> 0

0.66666666×2=1.33333333; =>1

0.33333333×2=0.66666666;=> 0

0.66666666×2=1.33333333; =>1

0.33333333×2=0.66666666;=> 0

Binary of 2/3 is (0.10101010...)

Convert 0.10101010... into decimal

$0.1 \times {2}^{ - 1} + 0 \times {2}^{ - 2} + 1 \times {2}^{ - 3} + \\0 \times {2}^{ - 4} + 1 \times {2}^{ - 5} + 0 \times {2}^{ - 6} +\\ 1 \times {2}^{ - 7} + 0 \times {2}^{ - 8} \\ \\ = 0. \frac{1}{2} + \frac{1}{8} + \frac{1}{32} + \frac{1}{128} \\ \\ = 0.(0.5 + 0.125 + 0.03125 + 0.0078125) \\ \\ = 0.6640625$

On converting it again into decimal one will get 0.6640625.

The difference of result starts from 3 digit after decimal place.

(c) Convert the binary result in (b) into hexadecimal. Then convert the result to decimal. Is the answer the same?

Ans: Hexadecimal conversation group 4-4 digits of binary numbers

(0.10101010...)in binary = (0.AA...) in hexadecimal

Convert 0.AA... into decimal

$0.(10) \times {16}^{ - 1} + (10) \times {16}^{ - 2} ... \\ \\ = 0.(0.625 + 0.0390625) \\ \\ = 0.6640625$

Yes, the result is same in both cases.

Hope it helps you.

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