Convert `(0.3125)_(10)` to binary equivalent.

Answer

To convert the decimal fraction \( (0.3125)_{10} \) to its binary equivalent, we will follow a systematic approach of multiplying by 2 and noting the integer parts. Here’s a step-by-step solution:

### Step 1: Multiply by 2
Start with the decimal fraction \( 0.3125 \) and multiply it by 2.

\[ 0.3125 \times 2 = 0.625 \]

**Hint:** When you multiply by 2, the integer part (before the decimal) will be the next binary digit.

### Step 2: Record the Integer Part
The integer part of \( 0.625 \) is \( 0 \). This is the first binary digit after the point.

**Binary so far:** \( 0. \)

### Step 3: Repeat the Process
Now take the fractional part \( 0.625 \) and multiply it by 2 again.

\[ 0.625 \times 2 = 1.25 \]

**Hint:** Always focus on the fractional part for the next multiplication.

### Step 4: Record the Integer Part
The integer part of \( 1.25 \) is \( 1 \). This is the second binary digit.

**Binary so far:** \( 0.1 \)

### Step 5: Continue the Process
Now take the fractional part \( 0.25 \) and multiply it by 2.

\[ 0.25 \times 2 = 0.5 \]

**Hint:** Keep repeating the multiplication with the new fractional part.

### Step 6: Record the Integer Part
The integer part of \( 0.5 \) is \( 0 \). This is the third binary digit.

**Binary so far:** \( 0.10 \)

### Step 7: Multiply Again
Now take the fractional part \( 0.5 \) and multiply it by 2.

\[ 0.5 \times 2 = 1.0 \]

### Step 8: Record the Integer Part
The integer part of \( 1.0 \) is \( 1 \). This is the fourth binary digit.

**Binary so far:** \( 0.101 \)

### Step 9: Final Step
Now we have \( 0.0 \) left, which means we can stop here as we have reached a terminating point.

### Final Binary Representation
Combining all the binary digits we recorded, we get:

\[ (0.3125)_{10} = (0.1010)_{2} \]

### Summary
The binary equivalent of \( (0.3125)_{10} \) is \( (0.1010)_{2} \).

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