What is the decimal number representation of the binary number `(11101.001)_(2)`?

Options

A. `30.125`
B. `29.025`
C. `29.125`
D. `28.025`

Explanation

To convert the binary number \( (11101.001)_2 \) to its decimal representation, we will break it down into two parts: the integer part \( 11101_2 \) and the fractional part \( 0.001_2 \).

### Step-by-Step Solution:

1. **Convert the Integer Part:**
The integer part is \( 11101_2 \). We will convert it to decimal by using the formula:
\[ \text{Decimal} = b_n \times 2^n + b_{n-1} \times 2^{n-1} + \ldots + b_1 \times 2^1 + b_0 \times 2^0 \]
where \( b_i \) are the binary digits.

For \( 11101_2 \):
- \( 1 \times 2^4 = 1 \times 16 = 16 \)
- \( 1 \times 2^3 = 1 \times 8 = 8 \)
- \( 1 \times 2^2 = 1 \times 4 = 4 \)
- \( 0 \times 2^1 = 0 \times 2 = 0 \)
- \( 1 \times 2^0 = 1 \times 1 = 1 \)

Now, add these values together:
\[ 16 + 8 + 4 + 0 + 1 = 29 \]

2. **Convert the Fractional Part:**
The fractional part is \( 0.001_2 \). We will convert it to decimal using the formula:
\[ \text{Decimal} = b_{-1} \times 2^{-1} + b_{-2} \times 2^{-2} + b_{-3} \times 2^{-3} \]
where \( b_{-i} \) are the binary digits after the decimal point.

For \( 0.001_2 \):
- \( 0 \times 2^{-1} = 0 \times \frac{1}{2} = 0 \)
- \( 0 \times 2^{-2} = 0 \times \frac{1}{4} = 0 \)
- \( 1 \times 2^{-3} = 1 \times \frac{1}{8} = 0.125 \)

Now, add these values together:
\[ 0 + 0 + 0.125 = 0.125 \]

3. **Combine Both Parts:**
Now, we combine the integer and fractional parts to get the final decimal representation:
\[ 29 + 0.125 = 29.125 \]

### Final Answer:
The decimal number representation of the binary number \( (11101.001)_2 \) is \( 29.125 \).

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