Convert `(100101)_2` into decimal representation.

Options

A. 36
B. 35
C. 37
D. 38

Answer

To convert the binary number \( (100101)_2 \) into its decimal representation, we follow these steps:

1. **Identify the binary digits**: The binary number \( 100101 \) consists of the digits 1, 0, 0, 1, 0, and 1.

2. **Assign powers of 2**: Starting from the rightmost digit (which is the least significant bit), assign powers of 2 to each digit. The rightmost digit corresponds to \( 2^0 \), the next to \( 2^1 \), and so on, up to \( 2^5 \) for the leftmost digit.
- \( 1 \times 2^5 \) (for the first digit)
- \( 0 \times 2^4 \) (for the second digit)
- \( 0 \times 2^3 \) (for the third digit)
- \( 1 \times 2^2 \) (for the fourth digit)
- \( 0 \times 2^1 \) (for the fifth digit)
- \( 1 \times 2^0 \) (for the sixth digit)

3. **Calculate the contributions**:
- \( 1 \times 2^5 = 1 \times 32 = 32 \)
- \( 0 \times 2^4 = 0 \times 16 = 0 \)
- \( 0 \times 2^3 = 0 \times 8 = 0 \)
- \( 1 \times 2^2 = 1 \times 4 = 4 \)
- \( 0 \times 2^1 = 0 \times 2 = 0 \)
- \( 1 \times 2^0 = 1 \times 1 = 1 \)

4. **Sum the contributions**: Now, add all the contributions together:
\[ 32 + 0 + 0 + 4 + 0 + 1 = 37 \]

5. **Final result**: The decimal representation of \( (100101)_2 \) is \( 37 \).