Binary Fraction Converter
Convert fractional binary numbers into decimal fractions and vice versa with negative powers of 2. Designed for software developers, electrical engineers, students, and computer architecture researchers requiring deterministic client-side accuracy.
Enter a binary number with an optional fractional point (e.g. 0.1011).
Calculated Output
Step-by-Step Mathematical Proof
Active DerivationWhat Is Binary Fraction Converter?
A binary fraction represents real numbers with fractional components in base 2. Digits to the right of the radix point represent successive negative powers of two: 2^-1 = 0.5, 2^-2 = 0.25, 2^-3 = 0.125, 2^-4 = 0.0625, and so on.
How Does It Work?
Multiply each bit to the right of the point by 2^-i and sum them: Value = b_1(0.5) + b_2(0.25) + b_3(0.125) + ...
Mathematical Algorithm
Step-by-Step Example
Important Rules & Edge Cases
- Each position to the right of the binary point is half the value of the previous position.
- Some decimal fractions (like 0.1) cannot be represented finitely in binary and repeat infinitely.
Practical Applications in Engineering
- Fixed-point arithmetic in digital audio signal processors (DSPs).
- Understanding floating-point roundoff errors in numerical software.
- Embedded microcontroller sensor calibration.
Common Mistakes to Avoid
- Caution: Treating fractional bits as integers (e.g. thinking 0.1 is 0.1 in decimal instead of 0.5).
- Caution: Using powers of 10 instead of powers of 2.
Frequently Asked Questions
What is 0.1 in binary equal to in decimal?
0.1 in binary equals 2^-1 = 0.5 in decimal.
Why does 0.1 decimal produce an infinite repeating binary fraction?
Because 10 contains a prime factor of 5, which is not a factor of the binary base 2. In base 2, 0.1 is 0.0001100110011... repeating forever.
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