In binary system decimal number 0.3 can be expressed as

Options

A. `(0.01001...)_(2)`
B. `(0.10110....)_(2)`
C. `(0.11001...)_(2)`
D. `(0.10101...)_(2)`

Answer

To convert the decimal number 0.3 into its binary equivalent, we will follow a systematic approach of multiplying by 2 and keeping track of the integer parts. Here’s the step-by-step solution:

### Step 1: Start with the decimal number
We begin with the decimal number:
\[ 0.3 \]

### Step 2: Multiply by 2
Multiply 0.3 by 2:
\[ 0.3 \times 2 = 0.6 \]
The integer part is 0.

### Step 3: Record the integer part
We write down the integer part:
\[ 0 \]

### Step 4: Take the fractional part and multiply by 2 again
Now, take the fractional part (0.6) and multiply it by 2:
\[ 0.6 \times 2 = 1.2 \]
The integer part is 1.

### Step 5: Record the integer part
We write down the integer part:
\[ 0, 1 \]

### Step 6: Take the fractional part and multiply by 2 again
Next, take the fractional part (0.2) from the previous result and multiply it by 2:
\[ 0.2 \times 2 = 0.4 \]
The integer part is 0.

### Step 7: Record the integer part
We write down the integer part:
\[ 0, 1, 0 \]

### Step 8: Take the fractional part and multiply by 2 again
Now, take the fractional part (0.4) and multiply it by 2:
\[ 0.4 \times 2 = 0.8 \]
The integer part is 0.

### Step 9: Record the integer part
We write down the integer part:
\[ 0, 1, 0, 0 \]

### Step 10: Take the fractional part and multiply by 2 again
Next, take the fractional part (0.8) and multiply it by 2:
\[ 0.8 \times 2 = 1.6 \]
The integer part is 1.

### Step 11: Record the integer part
We write down the integer part:
\[ 0, 1, 0, 0, 1 \]

### Step 12: Continue the process
Now take the fractional part (0.6) and multiply it by 2 again:
\[ 0.6 \times 2 = 1.2 \]
The integer part is 1.

### Step 13: Record the integer part
We write down the integer part:
\[ 0, 1, 0, 0, 1, 1 \]

### Step 14: Repeat until a pattern emerges
Continuing this process will show that we start to repeat the values. The binary representation will be:
\[ 0.0100110011... \]

### Final Result
Thus, the binary representation of the decimal number 0.3 is:
\[ 0.0100110011... \] (which is a repeating binary fraction)