Free Binary Fraction Converter

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Just as a decimal fraction uses negative powers of 10 after the decimal point (0.625 means 6/10 + 2/100 + 5/1000), a binary fraction uses negative powers of 2 after a "binary point" — so a digit immediately after the point represents halves (2⁻¹), the next represents quarters (2⁻²), the next eighths (2⁻³), and so on. This is exactly the same positional-notation idea used for whole binary numbers, just extended below zero. Understanding binary fractions matters directly for computing and electronics: floating-point numbers in every modern computer (the IEEE 754 standard used by virtually all programming languages) are stored internally using exactly this binary fractional representation, which is also why certain simple decimals like 0.1 or 0.2 cannot be represented exactly in binary and produce the small floating-point rounding errors that programmers occasionally have to work around. Digital signal processing, fixed-point arithmetic in embedded systems, and low-level debugging of floating-point behavior all rely on understanding how binary fractions actually work under the hood.

How it works

Choose a conversion direction. Converting binary to decimal: the whole-number part before the binary point is read as an ordinary binary integer, and each digit after the point is multiplied by 2 raised to a successively more negative power (2⁻¹ = 0.5, 2⁻² = 0.25, 2⁻³ = 0.125, and so on) and all the results are summed together. Converting decimal to binary: the whole-number part is converted with ordinary repeated division by 2, while the fractional part is converted by repeatedly multiplying by 2 and recording whether the result crosses 1 or not (a 1 bit if it does, a 0 bit if it doesn't), then subtracting that whole part and continuing with what remains — a mirror-image process of the standard decimal-to-binary integer method.

Enter conversion direction.
Enter binary number (for binary → decimal).
Enter decimal number (for decimal → binary).
Click Calculate to see your results.

Examples

101.101 (binary) → decimal

The integer part 101₂ = 4+0+1 = 5. The fractional part .101₂ = 1×2⁻¹ + 0×2⁻² + 1×2⁻³ = 0.5 + 0 + 0.125 = 0.625. Total: 5.625.

11.01 (binary) → decimal

The integer part 11₂ = 2+1 = 3. The fractional part .01₂ = 0×2⁻¹ + 1×2⁻² = 0 + 0.25 = 0.25. Total: 3.25.

2.5 (decimal) → binary

The integer part 2 converts to 10₂. The fractional part 0.5, multiplied by 2, gives exactly 1.0 — a single bit, 1, with nothing left over. Combined: 10.1₂.

Common mistakes to avoid

Forgetting that the fractional part uses negative exponents (2⁻¹, 2⁻², 2⁻³...) rather than positive ones, which are used for the whole-number part.
Assuming every decimal fraction has an exact, terminating binary equivalent — many common decimals (like 0.1) actually repeat forever in binary.
Mixing up the order of bits after the binary point, reading the first digit as representing quarters instead of halves.
Confusing this positional binary-fraction notation with unrelated binary encodings like binary-coded decimal (BCD), which represents each decimal digit separately rather than using true place values.

Frequently asked questions

Just like 1/3 never terminates when written in decimal, some decimal fractions — including the very common 0.1 and 0.2 — never terminate when written in binary, since their denominators (as fractions) involve prime factors other than 2. The calculator caps the fractional expansion at a fixed number of bits rather than continuing forever.

Because 0.1 in binary is an infinitely repeating fraction (0.0001100110011...), just as 1/3 is an infinitely repeating decimal. Since computers store only a finite number of bits, 0.1 gets silently rounded to the nearest representable binary fraction, which is the root cause of many famous floating-point "0.1 + 0.2 ≠ 0.3 exactly" surprises in programming.

Only decimals whose fractional part can be written as a sum of powers of 1/2 — that is, fractions whose denominator (in lowest terms) is itself a power of 2, like 1/2, 1/4, 3/8, or 5/16. Any other denominator (like 1/10 or 1/3) produces a repeating binary fraction.

The first position after the point represents halves (2⁻¹ = 0.5), the second represents quarters (2⁻² = 0.25), the third represents eighths (2⁻³ = 0.125), and so on — each successive position is worth exactly half of the one before it.

It's the same underlying binary-fraction concept, though IEEE 754 (the standard used by virtually every modern programming language for float/double types) adds additional structure on top — a sign bit, an exponent field, and a normalized mantissa — rather than storing a simple fixed binary point directly.

Yes — the same fractional binary-point notation applies regardless of sign; the sign is typically handled separately (with a leading minus sign or, in computer hardware, a dedicated sign bit) rather than changing how the fractional digits themselves are interpreted.