How Many Decimals to Keep When Converting Units

Every converter here answers with more decimals than most readers need, and it is right to. It cannot see the scale, the tape or the thermometer that produced your number, so it shows a generous answer and leaves the trimming to you.

What it can promise is arithmetic that adds no error of its own. That promise is real, and it is narrower than it sounds: the factor being exact says nothing whatever about the answer being exact.

An exact factor does not make an exact answer

Type 70 into the kilograms to pounds converter and it returns 154.32 lb. Reverse it on pounds to kilograms, or run any pair on the weight converter, and the same question arrives with different numbers. The arithmetic is 70 ÷ 0.45359237, and that divisor is the definition of the pound rather than a measurement of one, fixed by the 1959 International Yard and Pound Agreement. The division contributes nothing at all to the uncertainty.

Now look at the 70. A bathroom scale reading to the nearest kilogram displays 70 for anything from 69.5 up to 70.5. That is not a fault in the scale. It is what "to the nearest kilogram" means.

Push both edges of that range through the same exact factor. 69.5 kg is 153.2213 lb and 70.5 kg is 155.4259 lb. So the honest reading of the display is 154.32 lb give or take 1.10 lb, and the full width of the band is 2.20 lb.

That band is drawn at the top of this page. The display steps in hundredths of a pound while the scale behind it steps in whole kilograms, and every digit inside the shaded region is arithmetic rather than measurement.

The display resolves 0.01 lb. The reading behind it resolves 2.20 lb, which is 220 times wider. Two decimal places are being printed underneath a figure whose units column is already in doubt.

One detail in that 2.20 is worth stopping on. It is not a property of 70 kg. It is the number of pounds in one kilogram, so the band stays exactly 2.20 lb wide at 5 kg and at 500 kg alike. What set it was the size of your input's last digit, expressed in the output unit.

The rule: one of your units, expressed in theirs

That is the whole rule, and it fits in a sentence. The last digit you can defend in the answer is the last digit of your input, converted. Take one unit of whatever you measured in, express it in the unit you converted to, and you have the real step size of your result.

Which turns "how many decimals should I keep" into arithmetic. If one input unit comes to less than one output unit, you need extra decimals or you throw away precision you genuinely had. If it comes to more, every decimal is decoration.

Formally the count is the ceiling of −log₁₀ of the factor, floored at zero. In practice nobody computes that. You glance at whether the factor is above or below 1, and the answer is nearly always none or one.

Read the table down the right-hand column and the symmetry is the useful part. Every pair answers 0 in one direction and 1 in the other, and which is which never depends on the units — only on whether you are heading toward a smaller unit or a larger one.

Toward the smaller unit, one of your steps covers several of theirs, so the answer is coarser than it looks and decimals add nothing. Toward the larger unit, one of your steps is a fraction of theirs, and dropping to a whole number would discard a measurement you actually made.

Why the converter shows more than the rule asks for

There are 104 directional converters on this site running a plain multiplication, and their display settings can be checked against the rule directly. On 69 of them the display carries more decimals than a whole-number input pays for. On 30 the two agree. On 5 the display is the coarser of the two.

That distribution is the right one, and it is worth saying why rather than treating it as a flaw. A converter is handed a number with no history attached. It does not know whether your 70 came off a bathroom scale or a laboratory balance, and those two deserve very different answers.

Faced with that, the only safe default is to show more than most people need. Rounding is something a reader can do afterwards, and precision the tool discarded is gone for good. Generosity in the display costs a little clutter; stinginess costs data.

The five pages where the display is the coarser one are the same argument from the other side. Seconds into years shows three decimals, and one second is 0.0000000317 of a year, so anybody typing an exact number of seconds is losing digits at the display. The rule assumed a whole-number input, and at extreme ratios that assumption is what breaks first.

Significant figures say the same thing, more loosely

The school version of this rule counts significant figures instead of decimal places, and it is worth seeing where the two agree and where they part company.

The sig-fig rule says a result keeps as many significant figures as the least precise value that went into it. A conversion factor that is a definition rather than a measurement has no uncertainty at all, so it never gets a vote, and the count falls entirely to your measurement.

Applied to a tape reading of 175 cm, that gives three significant figures and an answer of 68.9 inches. The decimal-place rule agrees: one centimetre is 0.39370079 inches, the factor is below 1, so one decimal. Both routes land on 68.9.

Applied to 70 kg they disagree, and the sig-fig rule is the one that loses. Two significant figures of 154.3235835 is 150 lb, but the band behind the reading only ran from 153.2213 to 155.4259, so 150 throws away a digit that was real.

The reason is that counting digits is a proxy for relative precision, and the proxy is coarse near a change of leading digit. Reading 70 to the nearest kilogram is precise to 0.71 per cent, while two significant figures at 150 admits 3.3 per cent. Converting one input step is the exact version of what sig figs approximate.

Two conversions in a row

Chaining conversions is where the rule earns its keep, because the intermediate unit is a distraction and it is easy to let it set the precision.

A ceiling measured to the nearest foot at 8 ft is 2.4384 m, and that is 2438.4 mm. The millimetre figure looks like a survey. It came from a reading good to a foot.

Apply the rule to the ends rather than the middle. One foot — the step your tape actually offered — is 304.8 mm, so the honest answer is 2438 mm give or take 152 mm, and even the hundreds digit is uncertain. Nothing about passing through metres changed that.

You have already met this three times

Once the rule is named, several familiar oddities turn out to be the same thing wearing different clothes, and this site documents three of them on separate pages without ever connecting them.

A height of 5 ft 11 in is 180.34 cm. But a height recorded to the nearest inch could be anything from 70.5 to 71.5 inches, which is 179.07 cm to 181.61 cm — a band 2.54 cm wide, exactly one inch converted. The centimetres to feet page puts it well: converting cannot add precision, and two decimals on a converted height imply a measurement nobody took.

Normal body temperature is quoted as 98.6 °F, which looks like a figure measured to a tenth of a degree. It is 37 °C converted, and 37 read to the nearest whole degree spans 36.5 to 37.5, or 97.7 °F to 99.5 °F. The band is 1.8 °F, one Celsius degree, and the decimal in 98.6 is an artefact of the unit change rather than a measurement. The Celsius to Fahrenheit page has the history behind it.

A marathon is 42.195 km, and the famous 26.2 miles is that rounded down from 26.218757. The gap between the two is about 30 metres of course, which is nothing to a runner and everything to whoever has to measure the route out. The kilometres to miles page carries the race distances in full.

Three different shelves, three different pages, one rule. In each case the suspicious digit came from the conversion rather than the measurement, and in each case the width of the doubt is one unit of the original scale, converted.

The exception: when the factor is a power of ten

Metric-to-metric conversions behave differently, and they are the reason the rule needs stating carefully. One metre is exactly 1000 millimetres, so the factor is a power of ten and the conversion is a decimal shift with no arithmetic in it.

Run 175 cm to metres and the answer is 1.75 m. Three significant figures went in and three came out. Nothing was gained and nothing was lost, because moving a decimal point cannot change how many digits were ever real.

The rule still applies, and it simply returns a tidy answer: one millimetre is 0.001 metres, so a millimetre reading converted to metres needs three decimals — not two, and not four. The centimetres to metres page works through the prefix ladder, and the same shift runs through millimetres to centimetres, centimetres to millimetres, metres to centimetres and millimetres to metres.

This is also why the metric side of an argument about precision is usually the quiet one. A conversion within a system is bookkeeping. A conversion between systems is where a number acquires digits it did not earn, and where the decision about which of them to write down actually falls to you.

Common mistakes when trimming a converted number

• Reading the display as the answer's precision. Two decimals is a formatting choice, not a claim about your measurement. The tool would show the same two decimals for a reading off a certified balance and a reading off a luggage scale.

• Rounding the factor instead of the answer. These are opposite moves. Rounding the answer discards digits you never had; rounding the factor to 2.2 or 1.6 introduces an error of its own, on top of whatever your instrument contributed.

• Rounding in the middle of a chain. Feet to metres to millimetres should be one calculation with one rounding at the end. Rounding at the metre step and then converting again writes the rounding error into every digit downstream.

• Assuming the round trip returns. Convert a number, round it, convert it back, and you do not get the original. Whichever unit the measurement was actually taken in is the one carrying no rounding, and that is the one to keep on the record.

• Adding decimals because the number looks too round. A converted 20 °C really is 68 °F, exactly. Writing 68.0 to make it look calculated claims a tenth of a degree that nothing in the chain ever measured.

• Trimming before a comparison. If two figures are going to be subtracted or compared, round them at the end rather than at the start. Two readings that genuinely differ by 0.8 of a unit can round to the same whole number and appear to differ by nothing at all.

The one habit worth keeping from all of this is small. Before you write a converted figure down, ask what one step of the instrument that produced it is worth in the unit you are writing in. That number is the width of your answer, and every digit narrower than it belongs to the arithmetic rather than to the measurement.