Worksheet for converting binary numbers to decimal.

To solve the problem of converting binary numbers to decimal (denary), we need to understand how binary numbers work. Binary is a base-2 number system, where each digit represents a power of 2, starting from the rightmost digit (which represents \(2^0\)).

Conversion Process:

1. Write down the binary number.
2. Assign powers of 2 to each digit, starting from the rightmost digit as \(2^0\).
3. Multiply each binary digit by its corresponding power of 2.
4. Sum all the results to get the decimal equivalent.

Let's convert each binary number step by step:

---

Worksheet 1: Convert Binary to Decimal

#### 1) 11010101
- Binary: \(11010101\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 0 \cdot 32 + 1 \cdot 16 + 0 \cdot 8 + 1 \cdot 4 + 0 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 64 + 0 + 16 + 0 + 4 + 0 + 1 = 213
\]
- Decimal: 213

#### 2) 10100000
- Binary: \(10100000\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 0 + 32 + 0 + 0 + 0 + 0 + 0 = 160
\]
- Decimal: 160

#### 3) 11000111
- Binary: \(11000111\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 0 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 64 + 0 + 0 + 0 + 4 + 2 + 1 = 199
\]
- Decimal: 199

#### 4) 10100111
- Binary: \(10100111\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 0 + 32 + 0 + 0 + 4 + 2 + 1 = 167
\]
- Decimal: 167

#### 5) 11011111
- Binary: \(11011111\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 0 \cdot 32 + 1 \cdot 16 + 1 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 64 + 0 + 16 + 8 + 4 + 2 + 1 = 223
\]
- Decimal: 223

#### 6) 10111101
- Binary: \(10111101\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 1 \cdot 32 + 1 \cdot 16 + 1 \cdot 8 + 1 \cdot 4 + 0 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 0 + 32 + 16 + 8 + 4 + 0 + 1 = 189
\]
- Decimal: 189

#### 7) 01110000
- Binary: \(01110000\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
0 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 0 \cdot 128 + 1 \cdot 64 + 1 \cdot 32 + 1 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 0 + 64 + 32 + 16 + 0 + 0 + 0 + 0 = 112
\]
- Decimal: 112

#### 8) 10001110
- Binary: \(10001110\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 0 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 0 + 0 + 0 + 8 + 4 + 2 + 0 = 142
\]
- Decimal: 142

#### 9) 01101110
- Binary: \(01101110\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
0 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 0 \cdot 128 + 1 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 0 \cdot 1
\]
\[
= 0 + 64 + 32 + 0 + 8 + 4 + 2 + 0 = 110
\]
- Decimal: 110

#### 10) 11110000
- Binary: \(11110000\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 1 \cdot 32 + 1 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 64 + 32 + 16 + 0 + 0 + 0 + 0 = 240
\]
- Decimal: 240

#### 11) 11001000
- Binary: \(11001000\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 0 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 64 + 0 + 0 + 8 + 0 + 0 + 0 = 200
\]
- Decimal: 200

#### 12) 10000011
- Binary: \(10000011\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 0 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 0 + 0 + 0 + 0 + 0 + 2 + 1 = 131
\]
- Decimal: 131

#### 13) 00000001
- Binary: \(00000001\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
0 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 0 \cdot 128 + 0 \cdot 64 + 0 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 1 \cdot 1
\]
\[
= 0 + 0 + 0 + 0 + 0 + 0 + 0 + 1 = 1
\]
- Decimal: 1

#### 14) 10000010
- Binary: \(10000010\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 0 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 0 + 0 + 0 + 0 + 0 + 2 + 0 = 130
\]
- Decimal: 130

#### 15) 11101110
- Binary: \(11101110\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 1 \cdot 4 + 1 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 64 + 32 + 0 + 8 + 4 + 2 + 0 = 238
\]
- Decimal: 238

#### 16) 10101010
- Binary: \(10101010\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 0 + 32 + 0 + 8 + 0 + 2 + 0 = 170
\]
- Decimal: 170

#### 17) 01100011
- Binary: \(01100011\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
0 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 0 \cdot 128 + 1 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 1 \cdot 1
\]
\[
= 0 + 64 + 32 + 0 + 0 + 0 + 2 + 1 = 99
\]
- Decimal: 99

#### 18) 00011100
- Binary: \(00011100\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
0 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 0 \cdot 128 + 0 \cdot 64 + 0 \cdot 32 + 1 \cdot 16 + 1 \cdot 8 + 1 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 0 + 0 + 0 + 16 + 8 + 4 + 0 + 0 = 28
\]
- Decimal: 28

#### 19) 00110001
- Binary: \(00110001\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
0 \cdot 2^7 + 0 \cdot 2^6 + 1 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 0 \cdot 128 + 0 \cdot 64 + 1 \cdot 32 + 1 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 1 \cdot 1
\]
\[
= 0 + 0 + 32 + 16 + 0 + 0 + 0 + 1 = 49
\]
- Decimal: 49

#### 20) 11100011
- Binary: \(11100011\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 64 + 32 + 0 + 0 + 0 + 2 + 1 = 227
\]
- Decimal: 227

#### 21) 11011000
- Binary: \(11011000\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 0 \cdot 32 + 1 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 64 + 0 + 16 + 8 + 0 + 0 + 0 = 216
\]
- Decimal: 216

#### 22) 10010000
- Binary: \(10010000\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 0 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 0 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 1 \cdot 128 + 0 \cdot 64 + 0 \cdot 32 + 1 \cdot 16 + 0 \cdot 8 + 0 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 128 + 0 + 0 + 16 + 0 + 0 + 0 + 0 = 144
\]
- Decimal: 144

#### 23) 01010100
- Binary: \(01010100\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
0 \cdot 2^7 + 1 \cdot 2^6 + 0 \cdot 2^5 + 1 \cdot 2^4 + 0 \cdot 2^3 + 1 \cdot 2^2 + 0 \cdot 2^1 + 0 \cdot 2^0
\]
\[
= 0 \cdot 128 + 1 \cdot 64 + 0 \cdot 32 + 1 \cdot 16 + 0 \cdot 8 + 1 \cdot 4 + 0 \cdot 2 + 0 \cdot 1
\]
\[
= 0 + 64 + 0 + 16 + 0 + 4 + 0 + 0 = 84
\]
- Decimal: 84

#### 24) 11101011
- Binary: \(11101011\)
- Powers of 2: \(2^7, 2^6, 2^5, 2^4, 2^3, 2^2, 2^1, 2^0\)
- Calculation:
\[
1 \cdot 2^7 + 1 \cdot 2^6 + 1 \cdot 2^5 + 0 \cdot 2^4 + 1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0
\]
\[
= 1 \cdot 128 + 1 \cdot 64 + 1 \cdot 32 + 0 \cdot 16 + 1 \cdot 8 + 0 \cdot 4 + 1 \cdot 2 + 1 \cdot 1
\]
\[
= 128 + 64 + 32 + 0 + 8 + 0 + 2 + 1 = 235
\]
- Decimal: 235

---

Final Answer:

\[
\boxed{
\begin{array}{ll}
1) & 213 \\
2) & 160 \\
3) & 199 \\
4) & 167 \\
5) & 223 \\
6) & 189 \\
7) & 112 \\
8) & 142 \\
9) & 110 \\
10) & 240 \\
11) & 200 \\
12) & 131 \\
13) & 1 \\
14) & 130 \\
15) & 238 \\
16) & 170 \\
17) & 99 \\
18) & 28 \\
19) & 49 \\
20) & 227 \\
21) & 216 \\
22) & 144 \\
23) & 84 \\
24) & 235 \\
\end{array}
}
\]