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Section 2.4 Exercises

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. \(\binary{1010} \; \binary{1011} \; \binary{1100} \; \binary{1101}\)

  2. \(\binary{0001} \; \binary{0010} \; \binary{0011} \; \binary{0100}\)

  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. \(31658\)

  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. \(21845\)