Convert Fractional Number To Binary

Fractional to Binary Conversion:

\[ \text{Binary} = \text{IntegerPartToBinary}(d) + \text{FractionalPartToBinary}(d - \lfloor d \rfloor) \]

Binary Result:

Unit Converter ▲

Unit Converter ▼

From:	To:

1. What is Fractional to Binary Conversion?

Fractional to binary conversion is the process of converting a decimal number with a fractional part into its binary (base-2) equivalent. This includes both the integer and fractional components of the number.

2. How Does the Conversion Work?

The conversion process involves two main steps:

\[ \text{Binary} = \text{IntegerPartToBinary}(d) + \text{FractionalPartToBinary}(d - \lfloor d \rfloor) \]

Where:

\( d \) — The decimal number to convert
\( \lfloor d \rfloor \) — Floor function (integer part of d)
\( \text{IntegerPartToBinary} \) — Standard decimal to binary conversion
\( \text{FractionalPartToBinary} \) — Repeated multiplication by 2 method

Explanation: The integer part is converted using division by 2, while the fractional part is converted by repeated multiplication by 2 and taking the integer parts of the results.

3. Importance of Binary Conversion

Details: Binary representation is fundamental in computer science and digital electronics. Understanding fractional binary conversion is essential for floating-point arithmetic, digital signal processing, and computer number representations.

4. Using the Calculator

Tips: Enter any positive decimal number (with or without fractional part) and specify the desired precision (number of binary digits after the point). The calculator will show both the integer and fractional parts in binary.

5. Frequently Asked Questions (FAQ)

Q1: Why do some fractional numbers have infinite binary representations?
A: Similar to how some fractions have infinite decimal expansions (like 1/3), some numbers have infinite binary expansions when the denominator isn't a power of 2.

Q2: How precise should I set the fractional part?
A: For most applications, 10-15 bits of precision is sufficient. More precision may be needed for scientific computing or high-accuracy requirements.

Q3: Can negative numbers be converted?
A: This calculator handles positive numbers only. Negative numbers would require additional sign bit representation.

Q4: What's the difference between this and IEEE 754 floating point?
A: IEEE 754 includes additional components like exponent and normalization, while this shows the direct binary representation.

Q5: Why does my fractional part seem to repeat?
A: Some fractions produce repeating binary patterns, just like some produce repeating decimals in base 10.