What this converter does and why the denominator is a choice
You are snapping a continuous number onto a discrete set of marks. A decimal inch such as 0.4062 can take any value; a steel rule can only show halves, quarters, eighths, sixteenths, and if it is a machinist's rule, thirty-seconds and sixty-fourths. The conversion multiplies your decimal by the denominator, rounds to a whole count of those marks, and reduces the resulting fraction. Everything interesting happens in the rounding, because that is where a real dimension turns into a mark you can actually cut to.
The denominator is not fixed by the math — it is fixed by your tools. A framing tape reads sixteenths, so a stud layout should be rounded to sixteenths and nothing finer; a number like 3 21/64 on a cut list is a false precision that will be read wrongly on site. A machinist's scale reads sixty-fourths, so a fitter can work to 0.0156 in. Below that you stop using fractions altogether and read a decimal micrometer or a dial indicator, because the whole point of the fraction system is that it maps onto graduations you can see.
This is why the converter reports the error alongside the fraction, and why the comparison table shows every denominator at once. The fraction on its own hides the question that matters: how much did you give away by rounding, and can the job tolerate it?
The formula explained, including the reduction step
Start with the decimal value x and the denominator d you have chosen. Multiply them. The product counts how many dths of an inch your measurement contains, and it will almost never be a whole number: 0.4062 × 16 = 6.4992. Round it to the nearest whole number, 6, and you have the count of sixteenths. Divide back by 16 and the fraction is 6/16 of an inch, or 0.375 in decimal.
Reduce next. Six and sixteen share a common factor of two, so 6/16 becomes 3/8. Reduction changes nothing about the value; it changes how the number is said and how it is found on a rule, since 3/8 is a labelled mark on almost every tape and 6/16 is not. The reduction is done with the greatest common divisor of the numerator and the denominator, which is why the answer can come back with a denominator smaller than the one you asked for. Asking for the nearest 1/64 of 0.5 in returns 1/2, not 32/64.
The error is what is left: your original value minus the fraction. Positive means the fraction falls short of your measurement, negative means it overshoots. Quoting it in thousandths of an inch — thou, or mils in older shop usage — is the convention because thousandths is the unit a micrometer reads and the unit machining tolerances are written in. Rounding to the nearest 1/16 can never be wrong by more than half a sixteenth, which is 1/32 in, or 31.25 thou. To the nearest 1/64 the worst case is 1/128 in, or 7.8125 thou.
Worked example: a 9.5 mm hole on an imperial drill index
A metric drawing calls for a 9.5 mm hole and your drill index is fractional. Find the closest drill.
- Convert to inches. 9.5 ÷ 25.4 = 0.374015748 in. Check the magnitude: 25.4 mm is one inch, so 9.5 mm must be a little over three eighths.
- Scale by the denominator. Fractional drill sets are marked in sixty-fourths, but a tape and most drill indexes label to sixteenths, so try 16 first: 0.374015748 × 16 = 5.98425.
- Round. 5.98425 rounds to 6, so the count is six sixteenths.
- Reduce. gcd(6, 16) = 2, so 6/16 = 3/8 in.
- Find the error. 3/8 = 0.375 exactly, so the error is 0.374015748 − 0.375 = −0.000984252 in, which is −0.98 thou. The drill is about one thousandth of an inch larger than the drawing asks.
Now test whether a finer denominator helps. At sixty-fourths: 0.374015748 × 64 = 23.937, which rounds to 24, and 24/64 reduces to 3/8 again. The same answer. That is the useful lesson from the comparison table — going finer only helps when the extra marks fall closer to your value, and here they do not. A 3/8 in drill is the right choice for a 9.5 mm hole, and you are one thousandth oversize, which for a clearance hole is nothing and for a press fit is a problem.
Reverse the example to check the arithmetic. 3/8 in × 25.4 = 9.525 mm, which is 0.025 mm above 9.5 mm. Multiply the inch error by 25.4: 0.000984252 × 25.4 = 0.025 mm. The two agree.
How to read the error and decide whether it matters
Compare the error against the tolerance of what you are building, not against zero. Rough carpentry works to about 1/16 in, or 62 thou, so any fraction the converter returns at sixteenths is far inside the tolerance. Cabinet and trim work runs to 1/32 in, 31 thou. Metal fabrication with a mill and a dial caliper works to 5 thou or better, and at that point the fraction system runs out of resolution — a 1/64 rounding can be off by nearly 8 thou, which is larger than your tolerance.
That gives a clean decision rule. If the reported error is smaller than a quarter of your working tolerance, use the fraction and forget the decimal. If it is comparable to your tolerance, either pick a finer denominator or abandon fractions and work in decimals. If a finer denominator does not reduce the error much, as in the drilling example above, the value simply does not want to be a fraction and you should carry the decimal through to the machine.
Watch the sign when the fit is one-sided. A shaft that must slide through a hole tolerates a hole rounded up but not one rounded down. The converter reports the error as your value minus the fraction, so a negative error means the fraction is the larger number and a hole cut to it will be oversize. Read the sign before you cut, because rounding to the nearest mark is symmetric and a clearance requirement is not.
Inch fractions, decimals and millimeters in sixteenths
| Fraction | Decimal (in) | Millimeters |
|---|---|---|
| 1/16 | 0.0625 | 1.5875 |
| 1/8 | 0.1250 | 3.1750 |
| 3/16 | 0.1875 | 4.7625 |
| 1/4 | 0.2500 | 6.3500 |
| 5/16 | 0.3125 | 7.9375 |
| 3/8 | 0.3750 | 9.5250 |
| 7/16 | 0.4375 | 11.1125 |
| 1/2 | 0.5000 | 12.7000 |
| 9/16 | 0.5625 | 14.2875 |
| 5/8 | 0.6250 | 15.8750 |
| 11/16 | 0.6875 | 17.4625 |
| 3/4 | 0.7500 | 19.0500 |
| 13/16 | 0.8125 | 20.6375 |
| 7/8 | 0.8750 | 22.2250 |
| 15/16 | 0.9375 | 23.8125 |
| 1 | 1.0000 | 25.4000 |
For thirty-seconds and sixty-fourths, halve and quarter the 1/16 row: 1/32 is 0.03125 in and 0.79375 mm, 1/64 is 0.015625 in and 0.396875 mm.
Mistakes that put the wrong number on the cut list
- Rounding to a denominator your tape does not have. Writing 2 13/32 on a list read with a sixteenths tape guarantees the cut lands on 2 13/32 by guesswork. Round to the graduation the person cutting can actually see.
- Forgetting to reduce. 10/16 is correct and unreadable. Reduce it to 5/8 and it becomes a labelled mark.
- Rounding twice. Converting millimeters to a rounded decimal and then that decimal to a fraction compounds two errors. Carry the full decimal into the fraction step, as this converter does.
- Ignoring the sign of the error on a fit. A hole rounded down and a shaft rounded up produce an interference you did not design.
- Assuming a finer denominator is always closer. It is never worse, but it is often no better: 0.374 in gives 3/8 at sixteenths, thirty-seconds and sixty-fourths alike.
- Confusing thou with millimeters. One thou is 0.0254 mm, so a 5 thou error is 0.127 mm. A metric drawing quoting ±0.1 mm is tighter than a 1/64 rounding can hold.
Key terms
- Thou (mil)
- One thousandth of an inch, 0.001 in, equal to 0.0254 mm. The working unit of machining tolerances in imperial shops. British usage says thou; older American usage says mil.
- Graduation
- The finest mark on a rule or tape. It sets the smallest fraction you can honestly specify, because a dimension you cannot see on the tool is a dimension the cutter will estimate.
- Reduced fraction
- A fraction whose numerator and denominator share no common factor, such as 3/8 rather than 6/16. The value is identical; only the labelling changes.
Where fractions fit among the other length conversions
The fractional inch survives because it maps onto a physical rule, not because it is convenient arithmetic. Halving is easy on a tape and on a saw fence, so the system doubles: 1/2, 1/4, 1/8, 1/16, 1/32, 1/64. Decimal inches took over wherever an instrument reads directly in thousandths, which is why machinists quote 0.4375 and carpenters quote 7/16 for the same dimension.
If your input is metric and you mainly want the decimal inch answer with the fraction as a secondary reading, the millimeters to inches calculator is arranged that way round and handles centimeters and meters as well. For a height rather than a shop dimension, the feet and inches to centimeters calculator splits the answer into feet and inches instead of a fraction. For room and site dimensions where feet are the natural unit, use the meters to feet calculator, which also reports the feet-and-inches form.
Fractions leave the picture entirely once you move to area and volume. Material takeoffs are quoted in whole units — see the cubic yards to cubic feet calculator for bulk ordering — because a supplier delivers in truckloads, not sixty-fourths.
One convention worth knowing: drill and wire sizes in imperial shops are not fractions at all but number and letter gauges, running from #80 up to #1 and then A through Z. A #7 drill is 0.201 in, which is between 13/64 and 7/32. When a fractional drill and a numbered drill both sit near your target, the numbered one is often the closer match, so check an index rather than defaulting to the nearest fraction.