Exercise 1.4: A) Express the following binary numbers in decimal form: \( \begin{array}{lll}\text { 1) } 1 & \text { 7) } 1001 & \text { 13) } 11111 \\ \text { 2) } 11 & \text { 8) } 101 & \text { 14) } 100010001 \\ \text { 3) } 111 & \text { 9) } 10100.01 & \text { 15) } 111111 \\ \text { 4) } 1111 & \text { 10) } 10001 & \text { 16) } 11011011 \\ \text { 5) } 11.10 & \text { 11 } 11101 & \text { 17) } 111011.1 \\ \text { 6) } 110 & \text { 12) } 1000110 & \text { 18) } 10001\end{array} \)

Sure! Let's convert each of the given binary numbers to their decimal (base-10) equivalents. ### Conversion Rules: - **For integer parts:** Multiply each binary digit by \(2^{n}\), where \(n\) is the position index starting from 0 on the right. - **For fractional parts:** Multiply each binary digit after the decimal point by \(2^{-n}\), where \(n\) starts at 1. ### Detailed Conversions: 1. **\(1\) (Binary)** \[ 1 \times 2^0 = 1 \] **Decimal:** **1** 2. **\(11\) (Binary)** \[ 1 \times 2^1 + 1 \times 2^0 = 2 + 1 = 3 \] **Decimal:** **3** 3. **\(111\) (Binary)** \[ 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 4 + 2 + 1 = 7 \] **Decimal:** **7** 4. **\(1111\) (Binary)** \[ 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 8 + 4 + 2 + 1 = 15 \] **Decimal:** **15** 5. **\(11.10\) (Binary)** \[ \text{Integer Part: } 1 \times 2^1 + 1 \times 2^0 = 2 + 1 = 3 \] \[ \text{Fractional Part: } 1 \times 2^{-1} + 0 \times 2^{-2} = 0.5 + 0 = 0.5 \] **Decimal:** **3.5** 6. **\(110\) (Binary)** \[ 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 4 + 2 + 0 = 6 \] **Decimal:** **6** 7. **\(1001\) (Binary)** \[ 1 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 8 + 0 + 0 + 1 = 9 \] **Decimal:** **9** 8. **\(101\) (Binary)** \[ 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 4 + 0 + 1 = 5 \] **Decimal:** **5** 9. **\(10100.01\) (Binary)** \[ \text{Integer Part: } 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0 = 16 + 0 + 4 + 0 + 0 = 20 \] \[ \text{Fractional Part: } 0 \times 2^{-1} + 1 \times 2^{-2} = 0 + 0.25 = 0.25 \] **Decimal:** **20.25** 10. **\(10001\) (Binary)** \[ 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 16 + 0 + 0 + 0 + 1 = 17 \] **Decimal:** **17** 11. **\(11101\) (Binary)** \[ 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 16 + 8 + 4 + 0 + 1 = 29 \] **Decimal:** **29** 12. **\(1000110\) (Binary)** \[ 1 \times 2^6 + 0 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0 = 64 + 0 + 0 + 0 + 4 + 2 + 0 = 70 \] **Decimal:** **70** 13. **\(11111\) (Binary)** \[ 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 16 + 8 + 4 + 2 + 1 = 31 \] **Decimal:** **31** 14. **\(100010001\) (Binary)** \[ 1 \times 2^8 + 0 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 = 256 + 0 + 0 + 0 + 16 + 0 + 0 + 0 + 1 = 273 \] **Decimal:** **273** 15. **\(111111\) (Binary)** \[ 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 16 + 8 + 4 + 2 + 1 = 63 \] **Decimal:** **63** 16. **\(11011011\) (Binary)** \[ 1 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 128 + 64 + 0 + 16 + 8 + 0 + 2 + 1 = 219 \] **Decimal:** **219** 17. **\(111011.1\) (Binary)** \[ \text{Integer Part: } 1 \times 2^5 + 1 \times 2^4 + 1 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 1 \times 2^0 = 32 + 16 + 8 + 0 + 2 + 1 = 59 \] \[ \text{Fractional Part: } 1 \times 2^{-1} = 0.5 \] **Decimal:** **59.5** 18. **\(10001\) (Binary)** \[ \text{Same as item 10)} \] **Decimal:** **17** ### Summary of Decimal Equivalents: | # | Binary | Decimal | |----|--------------|----------| | 1 | \(1\) | 1 | | 2 | \(11\) | 3 | | 3 | \(111\) | 7 | | 4 | \(1111\) | 15 | | 5 | \(11.10\) | 3.5 | | 6 | \(110\) | 6 | | 7 | \(1001\) | 9 | | 8 | \(101\) | 5 | | 9 | \(10100.01\) | 20.25 | | 10 | \(10001\) | 17 | | 11 | \(11101\) | 29 | | 12 | \(1000110\) | 70 | | 13 | \(11111\) | 31 | | 14 | \(100010001\)| 273 | | 15 | \(111111\) | 63 | | 16 | \(11011011\) | 219 | | 17 | \(111011.1\) | 59.5 | | 18 | \(10001\) | 17 | Feel free to ask if you need further explanations on any specific conversion!