Converting Decimal Fraction 0.375 to Binary

To convert a decimal fraction, like 0.375, into its binary (base-2) equivalent, we use a method of repeated multiplication by 2.

Step-by-Step Conversion Process

We take the fractional part of the decimal number and multiply it by 2. The integer part of the result becomes the binary digit, and we continue the process with the new fractional part. We repeat this until the fractional part becomes 0.

1. Start with the decimal fraction: $0.375$

   Multiply by 2: $0.375 \times 2 = 0.750$

   The integer part is 0. This is the first binary digit after the binary point.

2. Take the new fractional part: $0.750$

   Multiply by 2: $0.750 \times 2 = 1.500$

   The integer part is 1. This is the next binary digit.

3. Take the new fractional part: $0.500$

   Multiply by 2: $0.500 \times 2 = 1.000$

   The integer part is 1. This is the next binary digit.

4. The fractional part is now 0.000, so we stop the process.

Determining the Binary Fraction

We read the integer parts obtained in each step from top to bottom to form the binary fraction.

The integer parts obtained were 0, 1, and 1.

Therefore, the binary representation of the decimal fraction 0.375 is $0.011$.

Verification

We can verify this by converting the binary fraction $0.011$ back to decimal:

$0.011_2 = (0 \times 2^{-1}) + (1 \times 2^{-2}) + (1 \times 2^{-3})$

$= (0 \times 0.5) + (1 \times 0.25) + (1 \times 0.125)$

$= 0 + 0.25 + 0.125$

$= 0.375_{10}$

The calculation confirms that the binary form of 0.375 is indeed 0.011.