Binary Fraction Converter — Decimal Fractions ↔ Binary
Convert numbers with a decimal point between decimal and binary, both ways, live in your browser. See the repeated-division and repeated-multiplication working, handle non-terminating fractions like 0.1, and read the positional expansion after the binary point.
How to Use
- Pick the direction: are you typing a decimal number or a binary number?
- Type a value with a fractional part — e.g. <code>5.75</code> or <code>101.11</code>. The other side updates instantly.
- Set “Max fraction bits” to control how far a non-terminating fraction (like 0.1) is expanded before it is truncated.
- Open the worked steps to see the integer part by repeated division ÷2 and the fraction by repeated multiplication ×2.
- Click any result (or its Copy button) to copy it.
Fractions, not just whole numbers
Most base converters stop at the decimal point — they happily turn 42 into 101010 but have nothing to say about 5.75 or 0.1. This tool fills that gap. It converts numbers with a fractional part between decimal and binary in both directions, and — crucially — it is honest about the fact that many everyday decimal fractions do not have an exact binary form. Everything runs in your browser; nothing is uploaded.
A number like 5.75 splits into an integer part (5) and a fractional part (0.75). The two parts are converted by different methods and then joined around the binary point: 101.11₂.
The two methods, side by side
Decimal → binary. The integer part uses repeated division by 2: divide, keep the remainder, repeat, and read the remainders from the bottom up. The fractional part uses repeated multiplication by 2: multiply the fraction by 2, write the digit that lands in front of the point (0 or 1) as the next bit, drop that digit, and repeat with what is left. For 0.625 that is 0.625→1.25 (bit 1), 0.25→0.5 (bit 0), 0.5→1.0 (bit 1) = 0.101.
Binary → decimal. This is plain positional expansion, except the columns after the point carry negative powers of two: 2⁻¹ = 0.5, 2⁻² = 0.25, 2⁻³ = 0.125, and so on. So 1011.01₂ = 8 + 2 + 1 + 0.25 = 11.25.
When binary never terminates. The repeated-multiplication method only stops if the fraction eventually hits exactly 0. That happens only when the decimal fraction is a sum of halves, quarters, eighths… — i.e. its lowest-terms denominator is a power of two. 0.5, 0.25, 0.75 and 0.625 all terminate; 0.1, 0.2 and 0.3 do not. 0.1 becomes the endlessly repeating 0.0001100110011…₂. The tool truncates at your “max fraction bits” limit and flags the result as repeating, which is the same approximation your computer makes — and the reason 0.1 + 0.2 famously isn’t quite 0.3.
Quick reference
About the Binary Fraction Converter — Decimal Fractions ↔ Binary
The Binary Fraction Converter — Decimal Fractions ↔ Binary gives you a fast, free answer for everyday tasks without sending anything off your device. Convert numbers with a decimal point between decimal and binary, both ways, live in your browser. See the repeated-division and repeated-multiplication working, handle non-terminating fractions like 0.1, and read the positional expansion after the binary point.
How it works
Type a value, then pick what you want to change it into. The answer appears straight away. It all happens on your own device, so it is fast and nothing you type is sent away. Just check that you picked the right “from” and “to” so you get the answer you wanted.
Want the deeper story? The Knowledge Base explains the ideas behind the tools in more detail.
Frequently Asked Questions
How do I convert a decimal fraction to binary by hand?
Split the number at the point. Convert the integer part by repeated division by 2 (read the remainders bottom-to-top). Convert the fractional part by repeated multiplication by 2: each time, the digit before the point (0 or 1) is the next bit, and you carry the part after the point into the next multiplication. Stop when the fraction reaches 0 or you hit your bit limit. For 0.625: 0.625×2=1.25 → 1, 0.25×2=0.5 → 0, 0.5×2=1.0 → 1, giving 0.101.
Why does 0.1 not have an exact binary form?
A fraction terminates in binary only when its denominator (in lowest terms) is a power of two. 0.1 = 1/10, and 10 = 2×5 has a factor of 5, so it never terminates — it repeats as 0.0001100110011… forever. This is exactly why 0.1 + 0.2 ≠ 0.3 in most programming languages. The tool truncates such fractions at your chosen bit limit and labels them as repeating.
How do I read a binary fraction back to decimal?
Use positional values, but the columns after the point are negative powers of two: the first is 2⁻¹ = 0.5, then 2⁻² = 0.25, 2⁻³ = 0.125, and so on. So 0.101₂ = 0.5 + 0 + 0.125 = 0.625, and 1011.01₂ = 8 + 0 + 2 + 1 + 0 + 0.25 = 11.25.
What is the “max fraction bits” setting?
Because many decimal fractions repeat forever in binary, the conversion has to stop somewhere. This control caps how many bits are produced after the binary point (default 32). When the cap is reached before the fraction terminates, the result is marked as truncated so you know it is an approximation.
Is anything uploaded?
No. The whole conversion runs in your browser with JavaScript — nothing is sent to a server.
How do I use the Binary Fraction Converter — Decimal Fractions ↔ Binary?
Just type or paste your value. The answer shows up right away — there is no button to press. Change anything and it updates by itself.
Does it cost anything or need an account?
No. The tool is completely free, there is no account to create, and it keeps working offline after the page first loads.
Is anything I type uploaded?
No. The tool works entirely on your device, so the values you enter never leave your browser.
Common Use Cases
Understanding floating-point error
See for yourself why 0.1 and 0.2 cannot be stored exactly, and where the rounding creeps in.
Learning number systems
Watch the repeated-multiplication method build a binary fraction bit by bit, with each carry shown.
Fixed-point and DSP work
Convert real-valued constants to binary fractions of a known bit width for fixed-point arithmetic.
Exams and homework
Check radix-point conversions and reproduce the textbook ÷2 / ×2 working step by step.
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