Convert a binary number with a fractional part (such as 101.101) to decimal exactly – digits after the point use negative powers of two: 1/2, 1/4, 1/8 and so on.
Show calculation steps
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Bits below the point
The binary point works exactly like the decimal point: places continue rightward with NEGATIVE powers of the base. The first place after the point is 2⁻¹ = 0.5, then 0.25, 0.125, 0.0625 — halving forever. A binary fraction is the sum of the places holding a 1.
This page evaluates that sum exactly. Because 2 divides 10, every finite binary fraction has a finite decimal form — the friendly direction of a famously unfriendly pair.
Worked example
Input: 101.101 Integer: 1×4 + 0×2 + 1×1 = 5 Fraction: 1×0.5 + 0×0.25 + 1×0.125 = 0.625 Result: 5.625 (exact)
The fraction table
| Place | Weight |
|---|---|
| .1 | 0.5 |
| .01 | 0.25 |
| .001 | 0.125 |
| .0001 | 0.0625 |
| .00001 | 0.03125 |
Why engineers meet binary fractions
IEEE-754 mantissas ARE binary fractions — decoding one by hand is exactly this operation. Fixed-point registers in DSPs and FPGAs store values as binary fractions with an implied point. Audio sample scaling, PWM duty registers and sensor calibration constants all read this way once you know the method.
Guarantees and privacy
Results are exact, always: n fractional bits produce at most n decimal fractional digits, computed with rational arithmetic rather than floats. Invalid digits are reported with their position; the only impossible input is a second point. Everything computes in your browser — nothing is uploaded.
How to use the Binary Fraction to Decimal Converter
- Enter a binary number with a point, for example 101.101.
- Click "Convert to decimal".
- The steps show positive powers of two for the integer part and NEGATIVE powers – 1/2, 1/4, 1/8 – for the digits after the point.
- The result is exact: binary fractions always terminate in decimal.
Frequently asked questions
How do digits after the binary point work?
They continue the powers of two downward: the first place after the point is 2⁻¹ = 0.5, then 0.25, then 0.125. So .101 contributes 0.5 + 0.125 = 0.625, and 101.101 is 5 + 0.625 = 5.625.
Is the decimal result always exact?
Yes – every finite binary fraction has a finite decimal form, because 2 divides 10. The reverse is NOT true, which is exactly why 0.1 causes trouble in floating point; the companion converter demonstrates it.
How many fractional digits can the input have?
As many as you like – the arithmetic is exact rational (BigInt numerator and denominator), so twenty places after the point convert perfectly. Note that n binary places produce at most n decimal places.
Can the integer part be zero, like .011?
Yes – a missing integer part reads as zero, so .011 is 0.375. Writing the leading 0 explicitly is clearer but optional.
What errors can my input produce?
Only two kinds: a digit other than 0/1 (reported with its position), or more than one point. Everything valid converts – there is no overflow, since the arithmetic is unbounded.
What is this conversion used for?
Reading fixed-point register values, understanding IEEE-754 mantissas (which are binary fractions), audio/DSP scaling, and coursework – it is the foundation for understanding how machines store non-integers.
Does the calculation happen locally?
Yes – exact fraction arithmetic in your browser, nothing transmitted. The steps panel is generated from your digits on your device.
