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

Question

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

UpStudy AI · Accepted Answer

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!
Here are the decimal equivalents of the given binary numbers: 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** If you need more detailed explanations for any specific conversion, feel free to ask!

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