Binary To Fraction Decimal Converter

Binary Fraction Formula:

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

Bit 1 (b₁):

(0 or 1)

Bit 2 (b₂):

(0 or 1)

Bit 3 (b₃):

(0 or 1)

Bit 4 (b₄):

(0 or 1)

Decimal Value:

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1. What is Binary Fraction to Decimal Conversion?

The binary fraction to decimal conversion transforms fractional binary numbers (numbers after the binary point) into their decimal equivalents. This is essential in computer science and digital systems where binary representations need to be interpreted as decimal 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:

\( b_1, b_2, b_3, \ldots \) — Binary digits (bits) after the binary point (each 0 or 1)
The denominators are powers of 2 (2¹, 2², 2³, ...)

Explanation: Each bit represents a fractional component where the first bit after the point is worth 1/2, the second 1/4, the third 1/8, and so on.

3. Importance of Binary Fraction Conversion

Details: Understanding binary fractions is crucial for working with floating-point numbers in computers, digital signal processing, and any application requiring precise numerical representation in binary systems.

4. Using the Calculator

Tips: Enter binary digits (0 or 1) for each bit position. At least two bits are required, but you can calculate up to four bits for more precision. The calculator will sum the weighted values of each bit.

5. Frequently Asked Questions (FAQ)

Q1: Why are only four bits shown in the calculator?
A: Four bits provide reasonable precision for demonstration purposes. In practice, computers use many more bits (typically 23 or 52 for floating-point fractions).

Q2: How do I represent numbers greater than 1?
A: This calculator handles only the fractional part. For whole numbers, use standard binary-to-decimal conversion for the integer part.

Q3: What's the maximum precision possible?
A: With n bits, you can represent fractions with precision up to 1/2ⁿ. More bits mean higher precision but require more storage.

Q4: Are there rounding errors in this conversion?
A: Some decimal fractions cannot be represented exactly in binary (just as 1/3 can't be represented exactly in decimal), leading to rounding errors.

Q5: How is this used in floating-point numbers?
A: Floating-point numbers combine a binary integer (significand) with a binary exponent, where the fractional part is typically normalized.