Binary Fraction to Decimal Converter

2

Conversion Result

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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

What is Binary Fraction to Decimal Converter?

The Binary Fraction to Decimal Converter is a specialized tool designed to convert binary numbers that contain a radix point (binary point) into their decimal equivalents. Unlike whole-number binary converters, this tool handles fractional binary values like 101.1101 — numbers that represent values between integers.

This tool exists because fractional binary conversion involves negative powers of 2, which can be tricky to calculate manually. Each digit after the binary point represents 2⁻¹ (0.5), 2⁻² (0.25), 2⁻³ (0.125), and so on. Our converter instantly breaks down these calculations step by step, making it invaluable for students studying computer science, engineers working with fixed-point arithmetic, and anyone dealing with binary precision in programming.

Understanding binary fractions is essential for grasping concepts like floating-point representation (IEEE 754), precision loss in programming languages, and how computers store decimal values internally.

Interactive Demo

Binary Fraction to Decimal Converter Formula

Toggle the bits past the radix point to see how negative powers of 2 combine to form a decimal fraction.

Fractional Bit Explorer
20
.
2-1 (0.5)
2-2 (0.25)
2-3 (0.125)
2-4 (0.0625)
Value: 0.0000 ₁₀
0
Precision Explorer

Why 0.1 Can't Be Exact in Binary

Many common decimal fractions like 0.1, 0.2, and 0.3 produce infinitely repeating patterns in binary — just like 1/3 = 0.333… in decimal.

This is why 0.1 + 0.2 ≠ 0.3 in most programming languages. The tiny rounding error from truncating the repeating binary fraction accumulates.

Try It Enter a decimal fraction below to see its binary representation and whether it terminates or repeats.
Precision Tester
Binary Representation (20 bits)
0.00011001100110011001
Status
⚠ Repeating — cannot be represented exactly
Actual Stored Value
0.09999847412109375
Quick Reference

Common Binary Fractions Table

The fractions that can be represented exactly in binary are sums of negative powers of 2.

BinaryDecimalFractionPower of 2
0.10.51/22⁻¹
0.010.251/42⁻²
0.0010.1251/82⁻³
0.00010.06251/162⁻⁴
0.000010.031251/322⁻⁵
0.110.753/42⁻¹ + 2⁻²
0.1010.6255/82⁻¹ + 2⁻³
0.1110.8757/82⁻¹ + 2⁻² + 2⁻³
0.00110.18753/162⁻³ + 2⁻⁴
0.10010.56259/162⁻¹ + 2⁻⁴
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).
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