Binary Fraction Converter
Convert decimal fractions to binary and back. Repeating binary digits are found exactly and marked, with the reason a fraction ends or repeats, and the doubling method shown step by step.
How to Use
- Choose decimal to binary or binary to decimal.
- Type a decimal (0.625), a fraction (1/3), or a binary number with a point (0.1011). Put a repeating block in brackets: 0.0(0011).
- Set how many bits to show for long or repeating results.
- Read the binary, the decimal, whether it ends or repeats and how long the repeat is.
- Show Work doubles the fraction step by step, the way it is done by hand.
Worked Example
0.625 to binary. Double it and keep the whole part each time: 0.625 × 2 = 1.25 → 1; 0.25 × 2 = 0.5 → 0; 0.5 × 2 = 1.0 → 1, and nothing is left. So 0.625 = 0.101₂ = ½ + ⅛. It ends because 0.625 = 5/8, and 8 is a power of 2.
0.1 to binary. 0.1 → 0.2 (0), 0.4 (0), 0.8 (0), 1.6 (1), keep 0.6 → 1.2 (1), keep 0.2, and 0.2 has come up before, so from here the digits 0011 repeat for ever: 0.1 = 0.0(0011)₂ = 0.000110011001100…₂.
The common mistake: reading the digits in the wrong order. For whole numbers the remainders are read from the last to the first, but for fractions the whole parts are read from the first to the last. Reversing them turns 0.101 into the same digits by luck, but 0.0011 (0.1875) into 0.1100 (0.75).
Show Work
Formulas
Halving, from Egypt to Silicon
Binary fractions are older than binary numbers. Egyptian scribes measured grain with the “Eye of Horus” fractions ½, ¼, ⅛, 1/16, 1/32 and 1/64, and imperial units still split an inch into halves, quarters, eighths and sixteenths, all of which end in binary. Leibniz extended binary to fractions in 1703 and noticed that 1/3 becomes the repeating 0.010101….
The rule for when a fraction ends is the same in every base: it ends when the denominator’s prime factors all divide the base. Base 10 has the primes 2 and 5, so tenths and fifths end in decimal; base 2 has only 2, which is why so many everyday decimals repeat inside a computer.
About This Calculator
This converter turns decimals and fractions into binary, finding the exact repeating block and its length, and turns binary with a point, including bracketed repeating blocks, back into decimal and a fraction. It explains whether a value ends or repeats, gives the octal and hex forms, and shows the error if the binary is cut at the bit limit.
Everything runs in your browser; nothing is sent anywhere. The arithmetic is exact, so long repeating periods are found, not guessed.
Related calculators: Number Base Converter, Fixed-Point Converter, and IEEE 754 Float Converter.
Frequently Asked Questions
How do you convert a decimal fraction to binary?
Multiply the fraction by 2 and write down the whole-number part (0 or 1), then repeat with the part after the point. The digits, read from the first, are the binary fraction. 0.625 × 2 = 1.25 (1), 0.25 × 2 = 0.5 (0), 0.5 × 2 = 1.0 (1), so 0.625 = 0.101 in binary.
Why does 0.1 repeat in binary?
A fraction ends in binary only when its denominator, in lowest terms, is a power of 2. 0.1 = 1/10, and 10 = 2 × 5; the factor 5 can never be cleared, so the doubling never reaches zero. 0.1 = 0.0001100110011…, with the block 0011 repeating every 4 bits.
How do you convert a binary fraction to decimal?
Each place after the point is worth half the one before: ½, ¼, ⅛, 1/16 and so on. Add the places that have a 1. 0.1011 = ½ + ⅛ + 1/16 = 0.5 + 0.125 + 0.0625 = 0.6875.
How do I write a repeating binary fraction?
Put the repeating block in brackets: 0.(01) means 0.010101…, which is 1/3, and 0.0(0011) is 0.1. The value of a block of r bits repeating straight after the point is the block’s value divided by 2ʳ − 1, so 0.(01) = 1/3 and 0.(001) = 1/7.
What does this mean for computers?
Floating-point numbers store a fixed number of binary digits, so a repeating fraction such as 0.1 is cut off and rounded. That small error is why 0.1 + 0.2 is not exactly 0.3 in most programming languages. Fractions like 0.5, 0.25 and 0.375 end in binary and are stored exactly.
How do I use the Binary Fraction Converter?
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
Homework
0.625 = 0.101₂ and 0.1011₂ = 0.6875, with every doubling shown.
Floating point
0.1 = 0.0(0011)₂ repeats every 4 bits, so it can never be stored exactly.
Fractions
1/3 = 0.(01)₂ and 1/7 = 0.(001)₂: the period is set by the denominator.
Fixed-point design
See how many bits a value needs, and the error if it is cut at 8 or 16 bits.
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