🔄 Binary Fraction Converter
Convert between decimal and binary fractions
How to Use This Calculator
Set Precision
Choose the number of binary digits you want in the result. Higher precision gives more accurate results but may show non-terminating fractions. Default is 10 bits.
Enter Decimal Fraction
Input a decimal fraction between 0 and 1 (e.g., 0.625, .5, 0.75). You can enter it as "0.625" or just ".625". The calculator will convert it to binary fraction format.
Click Convert or Review Results
Press "Convert to Binary" to see the binary fraction result. For binary to decimal conversion, enter a binary fraction (e.g., 0.101) and it will automatically convert. Review the step-by-step breakdown to understand the conversion process.
Formula
Decimal to Binary: Multiply by 2, take integer part, repeat
Binary to Decimal: Σ (bit × 2^(-position))
How it works:
Converting decimal fractions to binary uses the "multiply by 2" method. You repeatedly multiply the fractional part by 2, take the integer part (0 or 1) as the binary digit, and continue with the remaining fractional part.
Conversion Algorithm:
- Start with decimal fraction (0 < x < 1)
- Multiply by 2
- If result ≥ 1, binary digit is 1; else binary digit is 0
- Take fractional part and repeat
- Continue until fractional part is 0 or desired precision reached
Example 1: Convert 0.625 to Binary
Step 1: 0.625 × 2 = 1.25 → 1 (result += '1'), remainder = 0.25
Step 2: 0.25 × 2 = 0.5 → 0 (result += '0'), remainder = 0.5
Step 3: 0.5 × 2 = 1.0 → 1 (result += '1'), remainder = 0.0
Result: 0.101 (binary)
Verification: 1×2^(-1) + 0×2^(-2) + 1×2^(-3) = 0.5 + 0 + 0.125 = 0.625 ✓
Example 2: Convert 0.1 to Binary
This is a non-terminating binary fraction:
0.1 (decimal) = 0.0001100110011... (binary repeating)
The pattern "0011" repeats indefinitely
Result: 0.0001100110011... (infinite)
Frequently Asked Questions
Why do some decimal fractions have infinite binary representations?
Just like some fractions in decimal are non-terminating (like 1/3 = 0.333...), some decimal fractions have non-terminating binary representations. For example, 0.1 in decimal equals 0.0001100110011... in binary, where the pattern "0011" repeats infinitely.
How do I know if a binary fraction is terminating or non-terminating?
A decimal fraction terminates in binary if and only if its denominator (when written as a fraction in lowest terms) is a power of 2. For example, 0.625 = 5/8, and 8 is 2³, so it terminates. But 0.1 = 1/10, and 10 is not a power of 2, so it doesn't terminate.
What is the precision setting for?
The precision setting determines how many binary digits to calculate. For terminating fractions, you'll get the exact result. For non-terminating fractions, higher precision gives more accuracy but the result will be truncated after the specified number of bits.
Can I convert binary fractions back to decimal?
Yes! Enter a binary fraction (like 0.101) in the binary fraction field, and the calculator will automatically convert it to decimal. The conversion uses the formula: decimal = Σ (bit × 2^(-position)).
Why is binary fraction conversion important?
Binary fractions are crucial in computer science for representing floating-point numbers. Understanding how decimal fractions convert to binary helps in understanding floating-point precision, rounding errors, and why some calculations in programming may have slight inaccuracies.
What's the relationship between binary fractions and floating-point numbers?
Floating-point numbers in computers are stored using binary fractions. A number like 12.625 is stored as a sign bit, exponent, and mantissa (the fractional part). The mantissa is essentially a binary fraction, which is why understanding binary fraction conversion is important for understanding floating-point arithmetic.