Binary Fraction to Decimal Converter
Fractional Binary Support
Convert binary fractions (e.g., 1011.0101) to their decimal equivalents instantly. See how each bit after the radix point maps to negative powers of 2 — with full step-by-step breakdowns.
5Number Systems
∞Precision
0msLatency
Interactive Tool Module
Understanding Fractional Binary Positions
Toggle the bits past the radix point below to dynamically witness how negative powers of 2 scale exponentially to form the decimal fraction.
20
.
2-1 (0.5)
2-2 (0.25)
2-3 (0.125)
2-4 (0.0625)
Current Value
0.0000
More
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FAQ
Frequently Asked Questions
What is a binary fraction?
A binary fraction is a binary number that includes a radix point (the binary equivalent of a decimal point) to represent values between whole numbers. Digits to the right of the radix point represent negative powers of 2: the first is 2⁻¹ (0.5), the second is 2⁻² (0.25), and so on. For example, 101.11₂ = 5.75₁₀.
How do you convert a binary fraction to decimal?
Split the binary number at the radix point. For the integer part, multiply each bit by its positive power of 2 (2⁰, 2¹, 2²…). For the fractional part, multiply each bit by its negative power of 2 (2⁻¹, 2⁻², 2⁻³…). Then sum all the values. Example: 11.101₂ = (1×2¹ + 1×2⁰) + (1×2⁻¹ + 0×2⁻² + 1×2⁻³) = 2 + 1 + 0.5 + 0 + 0.125 = 3.625₁₀.
Can all decimal fractions be represented exactly in binary?
No. Many common decimal fractions like 0.1 and 0.2 cannot be represented exactly in binary because they produce infinitely repeating binary fractions. For instance, 0.1₁₀ = 0.0001100110011…₂ (repeating). This is why floating-point arithmetic in computers can introduce tiny rounding errors.
What is a radix point in binary?
A radix point is the dot that separates the integer part from the fractional part in any number system. In decimal, it's called a decimal point; in binary, it's called a binary point or radix point. Everything to the left represents positive powers of the base, and everything to the right represents negative powers.
How many decimal places does a binary fraction produce?
Each binary fractional digit (bit after the radix point) doubles the precision. With n fractional bits, you can represent values in increments of 2⁻ⁿ. For example, 4 fractional bits give precision to 1/16 = 0.0625. 8 fractional bits give precision to 1/256 ≈ 0.00390625.
What is 0.1 in binary?
The decimal value 0.1 in binary is 0.000110011001100110011… (repeating). It cannot be represented exactly in binary with a finite number of bits, which is why computers use IEEE 754 floating-point approximations for this value.
How do binary fractions relate to IEEE 754?
IEEE 754 floating-point format stores numbers as a sign bit, an exponent, and a mantissa (significand). The mantissa is essentially a binary fraction — a 1 followed by fractional bits. Single precision uses 23 mantissa bits (about 7 decimal digits of precision), while double precision uses 52 bits (about 15-16 decimal digits).