Convert Binary Fraction To Decimal

Binary Fraction Conversion:

\[ 0.b_1b_2..._2 = \frac{b_1}{2} + \frac{b_2}{4} + \frac{b_3}{8} + \cdots \]

Decimal Equivalent:

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1. What is Binary Fraction Conversion?

Binary fraction conversion transforms fractional binary numbers (numbers less than 1) into their decimal equivalents. This is essential for understanding how computers represent fractional numbers.

2. How Does the Calculator Work?

The calculator uses the binary fraction formula:

\[ 0.b_1b_2..._2 = \frac{b_1}{2} + \frac{b_2}{4} + \frac{b_3}{8} + \cdots \]

Where:

\( b_1, b_2, b_3, \) etc. — Binary digits (0 or 1)
The first digit after the point represents halves (1/2)
The second digit represents quarters (1/4)
The third digit represents eighths (1/8), and so on

Explanation: Each binary digit represents a negative power of 2, with the first digit being 2^-1, the second 2^-2, etc.

3. Importance of Binary Fraction Conversion

Details: Understanding binary fractions is crucial for computer science, digital electronics, and floating-point number representation. It helps in understanding precision limitations in computer arithmetic.

4. Using the Calculator

Tips: Enter the binary fraction part (without the leading "0.") using only 0s and 1s. For example, to convert "0.101", just enter "101".

5. Frequently Asked Questions (FAQ)

Q1: What's the maximum precision this calculator can handle?
A: The calculator can handle up to about 20 digits, though precision may be limited by PHP's floating-point implementation.

Q2: How do I represent repeating binary fractions?
A: This calculator doesn't handle symbolic representation of repeating fractions. Enter as many digits as needed for your desired precision.

Q3: Can I convert decimal fractions back to binary?
A: This calculator only converts binary to decimal. A separate tool would be needed for decimal to binary conversion.

Q4: Why are some decimal results repeating?
A: Just like 1/3 = 0.333... in decimal, some fractions have repeating representations in binary (e.g., 0.1 in decimal is 0.0001100110011... in binary).

Q5: How is this related to floating-point representation?
A: Floating-point numbers use binary fractions in their mantissa/significand component, making this conversion fundamental to understanding floating-point precision.