Math
Binary Fraction Converter
Fractions behave differently in binary: 0.625 is a tidy 0.101, but 0.1 becomes 0.000110011… forever.
Enter a decimal or a ratio to see its binary expansion with the repeating block marked, or enter a binary fraction to get its exact decimal and fraction form. Every step of the multiply-by-2 method is listed.
Type a decimal (0.625), a ratio (5/8), or a mixed number (2 3/8). Exact arithmetic is used, so 1/3 is treated as exactly one third.
How far to expand before giving up on finding a repeat. 24 matches a single-precision float's mantissa; 53 matches double precision.
Try an example
Result
Result
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- Exact fraction
- —
- Decimal
- —
- Expansion
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Student quick launch
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Study path
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Follow these when you want the formula behind the answer, a short lesson, or nearby tools in the same topic.
What the binary fraction converter solves
A fraction terminates in binary only when its reduced The bottom number of a fraction; it says how many equal parts make one whole. is a power of 2. Any other denominator (3, 5, 10, 7 …) gives a repeating binary expansion, which is why 0.1 + 0.2 is not exactly 0.3 in floating point. This converter uses exact rational arithmetic to find the expansion and the repeating block, and converts binary fractions back to exact decimals.
Common fractions in binary
| Decimal | Fraction | Binary |
|---|---|---|
| 0.5 | 1/2 | 0.1 |
| 0.25 | 1/4 | 0.01 |
| 0.625 | 5/8 | 0.101 |
| 0.1 | 1/10 | 0.0(0011) |
| 0.2 | 1/5 | 0.(0011) |
| 0.3333… | 1/3 | 0.(01) |
| 0.142857… | 1/7 | 0.(001) |
How to use it
- Choose the direction.
- For decimal to binary, type a decimal or a ratio such as 1/3 (ratios are exact; 0.3333 is not one third).
- Set the maximum fraction bits if you want a longer search for a repeating block.
- Read the table: each row multiplies the leftover fraction by 2 and peels off one bit.
How to read the answer
Parentheses mark the repeating block: 0.0(0011) means 0.0 followed by 0011 repeated forever. The expansion status says whether the fraction terminated, repeated, or was cut off at the bit limit. Binary to decimal is always exact because 2 divides 10.
Common mistakes and edge cases
- Typing 0.333 for one third. That is 333/1000, which terminates in a different way; type 1/3 for the exact value.
- Stopping the multiplication early and calling the result exact. Keep going until the fraction hits 0 or a value you have seen before.
- Assuming a short decimal is a short binary: 0.1 needs infinitely many bits.
- A denominator of 0 is undefined and is rejected.
- Whole numbers work too; the fraction part is simply empty.
Worked examples
Terminating fraction
0.625 → binary
Result
0.101
Repeating fraction
0.1 → binary
Result
0.0(0011)
Exact ratio
1/3 → binary
Result
0.(01)
Mixed number
2 3/8 → binary
Result
10.011
Negative decimal
−0.75 → binary
Result
-0.11
Zero
0 → binary
Result
0
Binary to decimal
1101.101 → decimal
Result
13.625
Long binary fraction
0.0001100110011 → decimal
Result
0.0999755859375
Zero denominator
1/0
Result
—
Frequently asked questions
Why is 0.1 not exact in binary?+
0.1 = 1/10 and 10 = 2 × 5. The factor 5 is not a power of 2, so the binary expansion never ends: 0.0001100110011… Computers store a rounded version, which causes tiny errors like 0.1 + 0.2 ≠ 0.3.
Which decimals terminate in binary?+
Only those whose reduced fraction has a power-of-2 denominator: 0.5, 0.25, 0.75, 0.125, 0.375, 0.625, and so on.
How do I convert a decimal fraction to binary by hand?+
Multiply by 2. Write the whole part (0 or 1) as the next bit and keep the fractional part. Repeat. 0.625 × 2 = 1.25 → 1; 0.25 × 2 = 0.5 → 0; 0.5 × 2 = 1.0 → 1; so 0.625 = 0.101.
How do I convert a binary fraction to decimal?+
Add the place values of the 1 bits: the first place after the point is 1/2, then 1/4, 1/8, and so on. 0.101 = 1/2 + 1/8 = 0.625.
What does the block in parentheses mean?+
It repeats forever. 0.(01) is 0.010101…, which is exactly 1/3.
How many bits does a float keep?+
Single precision keeps 24 significant bits and double precision keeps 53. Set the bit limit to those values to see what gets cut off.
About this calculator
- Written by
- mathcheck editorial team
- Last reviewed
- September 4, 2026
Method
- Uses the values entered by the user and stable formulas documented on the page.
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Last updated: September 4, 2026