Binary Fraction Converter
Convert decimal fractions to binary representation and vice versa with step-by-step explanations
Binary Fraction Converter
Enter a decimal number between 0 and 1 (e.g., 0.125, 0.25, 0.75)
Example: 0.5 to Binary
Example: 0.1101 to Decimal
Binary Fraction Basics
Decimal to Binary
- 1. Multiply fraction by 2
- 2. Take integer part as binary digit
- 3. Use fractional part for next step
- 4. Repeat until fraction is 0 or desired precision
Binary to Decimal
- 1. Each position = power of 2
- 2. Position 1: 2⁻¹ = 0.5
- 3. Position 2: 2⁻² = 0.25
- 4. Sum all weighted digits
Common Binary Fractions
Binary Limitations
Not all decimal fractions have exact binary representations
Fractions like 0.1, 0.2, 0.3 create repeating binary patterns
Exact representation when denominator is a power of 2
Computer precision limits cause truncation errors
Understanding Binary Fractions
What are Binary Fractions?
Binary fractions represent decimal numbers less than 1 using only 0s and 1s. Just as decimal fractions use powers of 10 (0.1 = 1/10, 0.01 = 1/100), binary fractions use negative powers of 2 (0.1₂ = 1/2, 0.01₂ = 1/4).
Why Use Binary Fractions?
- •Essential for computer science and digital systems
- •Understanding floating-point representation
- •Digital signal processing applications
- •Binary arithmetic and computer mathematics
Conversion Methods
Decimal to Binary Algorithm
- 1. Start with decimal fraction (0 ≤ x < 1)
- 2. Multiply by 2
- 3. Extract integer part (0 or 1) → binary digit
- 4. Keep fractional part
- 5. Repeat until fraction = 0 or desired precision
Binary to Decimal Formula
For binary fraction 0.b₁b₂b₃...bₙ:
Decimal = b₁×2⁻¹ + b₂×2⁻² + b₃×2⁻³ + ... + bₙ×2⁻ⁿ
Precision and Limitations
Important: Not all decimal fractions can be exactly represented in binary. Fractions like 0.1, 0.2, and 0.3 become repeating binary decimals, leading to approximation errors in computer calculations. Only fractions with denominators that are powers of 2 (1/2, 1/4, 3/8, etc.) have exact binary representations.