Decimal to Fraction: worked example
| Decimal | 0.688 |
|---|---|
| Largest denominator allowed | 1,000 |
| Numerator | 11 |
| Denominator | 16 |
| Approximation error | 0 |
Decimal to Fraction: formula
Enter the decimal, and the largest denominator you allow. A larger limit gives a closer approximation, and a smaller one gives a simpler fraction such as one third instead of 0.3333 as 3333 over 10000. The method uses continued fractions to find the best rational approximation within that limit.
For terminating decimals such as 0.6875, the exact fraction is found quickly. Repeating decimals like 0.333 need a limit to become 1 over 3.
Decimal to Fraction: changing Decimal
| Decimal | Numerator | Denominator | Approximation error |
|---|---|---|---|
| 0.55 | 11 | 20 | 0 |
| 0.619 | 237 | 383 | 0 |
| 0.688 (example) | 11 | 16 | 0 |
| 0.756 | 90 | 119 | 0 |
| 0.825 | 33 | 40 | 0 |
Decimal to Fraction: compare input ranges
| Input | Tested from | To | Numerator: low input | Numerator: high input | Swing |
|---|---|---|---|---|---|
| Decimal | 0.619 | 0.756 | 237 | 90 | 147 (1336%) |
| Largest denominator allowed | 900 | 1,100 | 11 | 11 | 0 |
Decimal to Fraction: individual input comparisons
| Input | Value | Numerator | Denominator | Approximation error |
|---|---|---|---|---|
| Decimal | 0.55 | 11 | 20 | 0 |
| 0.619 | 237 | 383 | 0 | |
| 0.688 | 11 | 16 | 0 | |
| 0.756 | 90 | 119 | 0 | |
| 0.825 | 33 | 40 | 0 | |
| Largest denominator allowed | 800 | 11 | 16 | 0 |
| 900 | 11 | 16 | 0 | |
| 1,000 | 11 | 16 | 0 | |
| 1,100 | 11 | 16 | 0 | |
| 1,200 | 11 | 16 | 0 |
Decimal to Fraction: which assumptions move the result?
Decimal to Fraction: values and definitions
| Quantity and definition | Value used |
|---|---|
| Decimal — The decimal to turn into a fraction. Negative values are allowed. | 0.688 |
| Largest denominator allowed — Caps how large the bottom number may be. 16 gives sixteenths, 100 gives hundredths. | 1,000 |
| Numerator | 11 |
| Denominator | 16 |
| Approximation error | 0 |
Decimal to Fraction: half and double Decimal
| Decimal | Numerator |
|---|---|
| 0.344 | 11 |
| 0.688 | 11 |
| 1.4 | 11 |
Common error: Rounding 0.333 to 333/1000 because it looks precise. The repeating value behind it is 1/3, and a cap of 10 or 100 finds it while the long form hides it.
Irrational numbers such as pi have no exact fraction, so the result is an approximation that improves with the denominator limit. Floating-point inputs like 0.1 are stored inexactly, which can lead to an odd fraction unless the limit is small. Negative decimals keep the sign in the numerator.
Algebra & Arithmetic calculators: Arithmetic Sequence Calculator, Geometric Sequence Calculator, Percent Difference Calculator, Proportion Calculator, Roman Numeral Calculator, Completing the Square Calculator.
Every decimal has many fractions close to it, and the useful one depends on the tool. A tape measure reads sixteenths, a drill index reads sixty-fourths, a price reads hundredths. The denominator cap is the real input here.
Cap it at 16 and 0.6875 becomes 11/16 exactly. Cap it at 8 and the same number becomes 2/3, with an error of about 0.02 that the third output states in plain decimals.
Approximation error is the difference between the entered decimal and the returned fraction. Exact conversions show zero. A nonzero error means the fraction is the nearest one your cap allows, not the true value.
Simplify a fraction on the fraction simplifier calculator, and operate on fractions on the fraction calculator. The rounding calculator goes the other way, and the mixed number page handles improper fractions. Unit conversion pages give decimals that often need fractions for tools and drawings.
Does a limit of 16 mean sixteenths only?
No. It allows every denominator from 1 to 16. For example, it can return 1/3.
Is 0.333 exactly one third?
No. The exact input is 333/1000; 1/3 is a nearby approximation when the denominator limit excludes 1000.
How do I convert a decimal to a fraction by hand?
Write the decimal over a power of ten, such as 0.6875 as 6875 over 10000, then divide by the greatest common factor to reduce it.
Why is the fraction for pi only approximate?
Pi is irrational, so no fraction equals it exactly, but 22 over 7 and 355 over 113 are close approximations.
What denominator limit should I use for inches?
Use 16, 32 or 64, depending on the smallest fraction of an inch marked on your ruler.