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Section2.4Exercises

1

Referring to Equation (2.3.3), what are the values of \(r\text{,}\) \(n\) and each \(d_{i}\) for the decimal number \(29458254\text{?}\) The hexadecimal number \(\hex{29458254}\text{?}\)

Answer

Decimal number: \(r = 10, n = 8, d_{7} = 2, d_{6} = 9, d_{5} = 4, d_{4} = 5, d_{3} = 8, d_{2} = 2, d_{1} = 5, d_{0} = 4\)

Hexadecimal number: \(r = 16, n = 8, d_{7} = 2, d_{6} = 9, d_{5} = 4, d_{4} = 5, d_{3} = 8, d_{2} = 2, d_{1} = 5, d_{0} = 4\)

2

Convert the eight-digit binary number \(\binary{1010} \; \binary{0101}\) to decimal.

Answer

\begin{align*} \binary{1010} \; \binary{0101}_{2} &= 1 \times 2^{7} + 0 \times 2^{6} + 1 \times 2^{5} + 0 \times 2^{4} + 0 \times 2^{3} + 1 \times 2^{2} + 0 \times 2^{1} + 1 \times 2^{0}\\ &= 128 + 0 + 32 + 0 + 0 + 4 + 0 + 1 \\ &= 165_{10} \end{align*}
3

Convert the following 8-bit binary numbers to decimal by hand:

  1. \(\binary{1010} \; \binary{1010} \)

  2. \(\binary{0101} \; \binary{0101} \)

  3. \(\binary{1111} \; \binary{0000} \)

  4. \(\binary{0000} \; \binary{1111} \)

  5. \(\binary{1000} \; \binary{0000} \)

  6. \(\binary{0110} \; \binary{0011} \)

  7. \(\binary{0111} \; \binary{1011} \)

  8. \(\binary{1111} \; \binary{1111} \)

Answer

  1. \(170\)

  2. \(85\)

  3. \(240\)

  4. \(15\)

  5. \(128\)

  6. \(99\)

  7. \(123\)

  8. \(255\)

4

Convert the following 16-bit binary numbers to decimal by hand:

  1. \(\binary{1010} \; \binary{1011} \; \binary{1110} \; \binary{1101}\)

  2. \(\binary{0001} \; \binary{0011} \; \binary{0011} \; \binary{0100}\)

  3. \(\binary{1111} \; \binary{1110} \; \binary{1101} \; \binary{1100}\)

  4. \(\binary{0000} \; \binary{0111} \; \binary{1101} \; \binary{1111}\)

  5. \(\binary{1000} \; \binary{0000} \; \binary{0000} \; \binary{0000}\)

  6. \(\binary{0000} \; \binary{0100} \; \binary{0000} \; \binary{0000}\)

  7. \(\binary{0111} \; \binary{1011} \; \binary{1010} \; \binary{1010}\)

  8. \(\binary{0011} \; \binary{0000} \; \binary{0011} \; \binary{1001}\)

Answer

  1. \(43981\)

  2. \(4660\)

  3. \(65244\)

  4. \(2015\)

  5. \(32768\)

  6. \(1024\)

  7. \(32170\)

  8. \(12345\)

5

Develop an algorithm to convert hexadecimal to decimal, and then convert the following 16-bit numbers to decimal by hand:

  1. \(\hex{a000}\)

  2. \(\hex{ffff}\)

  3. \(\hex{0400}\)

  4. \(\hex{1111}\)

  5. \(\hex{8888}\)

  6. \(\hex{0190}\)

  7. \(\hex{abcd}\)

  8. \(\hex{5555}\)

Hint

Review the algorithm to convert binary to decimal above.

Answer

  1. Set \(Result = 0\)

  2. For \(i = 0, \cdots, (n-1)\)

    • \(Result = Result + d_{i} * 16^{i}\)

  1. \(40960\)

  2. \(65535\)

  3. \(1024\)

  4. \(4369\)

  5. \(34952\)

  6. \(400\)

  7. \(43981\)

  8. \(21845\)