Are inches and centimetres an exact conversion?

The imperial units are defined in metric

People generally assume that converting inches to centimetres involves an approximation, in the way that converting a currency does. It does not, and the reason is a small piece of history worth knowing.

Before 1959 the imperial units were defined independently in each country that used them, against physical reference objects. The result was that a British inch and an American inch were very slightly different lengths, and a British pound and an American pound very slightly different masses. The differences were tiny — parts per million — and entirely irrelevant to daily life, while being a genuine nuisance in engineering and in international trade.

The International Yard and Pound Agreement of 1959, signed by the United States, the United Kingdom, Canada, Australia, New Zealand and South Africa, resolved this by redefining the imperial units *in terms of* metric ones. The yard became exactly 0.9144 metres. The pound became exactly 0.45359237 kilograms. Everything else follows arithmetically: a foot is a third of a yard, so exactly 0.3048 metres; an inch is a twelfth of a foot, so exactly 25.4 millimetres or 2.54 centimetres.

The consequence is that these are not measured relationships that might be refined by better instruments. They are definitions, and the numbers are final. An inch is exactly 2.54 centimetres in the same way that a dozen is exactly twelve. Any imprecision in a conversion comes from rounding the result, never from the factor.

The same applies to mass. A pound is exactly 0.45359237 kilograms, an ounce exactly a sixteenth of that, and a stone exactly fourteen pounds, so a stone is exactly 6.35029318 kilograms. The long chain of decimals is not an approximation creeping in; it is what the exact value happens to look like in decimal.

The foot that was two parts per million too long

There is one genuine exception to all of this, and it survived until very recently.

When the United States adopted the 1959 definitions, land surveying presented a problem. Decades of survey records, property boundaries and state plane coordinate systems had been computed using the older definition, in which a foot was 1200/3937 of a metre. That is 0.30480061 metres against the new 0.3048 — a difference of about two parts per million.

Two parts per million sounds negligible and is, over the length of a room. Over the length of a state it is not. Across a hundred kilometres the two definitions disagree by about twenty centimetres, which is more than enough to move a property line or misplace a control point. Rather than recompute a century of survey data, the United States retained the old definition under the name the US survey foot, for surveying only, while everything else used the international foot.

This left two feet of slightly different length in simultaneous legal use in the same country, distinguished only by context, which is exactly the sort of arrangement that eventually causes an expensive mistake. The National Institute of Standards and Technology and the National Geodetic Survey announced its retirement, and the US survey foot was deprecated at the end of 2022. From 2023 the international foot is the only one, and historical survey data carries its own notation.

The practical relevance for anyone not surveying land is nil, but it illustrates the general point well: where a conversion factor is genuinely contested, it is contested at the sixth decimal place and for a specific technical reason, not in the range where anybody notices.

Temperature, volume and the ones with offsets

Temperature is exact too, and it is the only common conversion that is not a simple multiplication, which is why it is the one people get wrong.

The relationship is exact: Fahrenheit equals Celsius times nine fifths plus thirty-two. The nine fifths is the ratio of degree sizes — a Celsius degree is exactly 1.8 Fahrenheit degrees — and the thirty-two is the offset between the zero points. Both are definitional.

The offset is the source of the classic error, which is converting a temperature *difference* as though it were a temperature. A rise of ten degrees Celsius is a rise of eighteen degrees Fahrenheit, not fifty. Differences use only the ratio; absolute temperatures use the ratio and the offset. The same distinction applies to any scale with a non-zero origin.

One curiosity worth remembering because it is a useful sanity check: the two scales cross at minus forty. Minus forty Celsius is minus forty Fahrenheit exactly. If a conversion is behaving strangely, testing it at minus forty catches an offset applied in the wrong direction immediately.

Volume is exact as well, and is the place where the word gallon does the most damage. The United States gallon is defined as exactly 231 cubic inches, which via the exact inch works out to exactly 3.785411784 litres. The imperial gallon is defined directly as exactly 4.54609 litres. These are two different units sharing a name, differing by about twenty percent, and neither is a rounding of the other. Every unit derived from them inherits the split — a US fluid ounce and an imperial fluid ounce differ by about four percent, in the opposite direction to the gallon, because the two systems divide their gallon into different numbers of ounces.

Where the error actually comes from

If the factors are exact, every real conversion error comes from one of three other places.

The first is rounding at the wrong moment. Converting through several steps and rounding at each one accumulates error that a single calculation would not produce. The rule is to carry full precision through the whole chain and round once, at the end, to the precision the answer actually needs. This matters most when a converted value is then multiplied by something large.

The second is false precision, which is the opposite failure and more common in written work. A measurement of six feet, taken with a tape measure, converts to 182.88 centimetres — and reporting it that way claims a precision of a hundredth of a centimetre that the original measurement never had. Two significant figures in, two significant figures out. A converted number that is more precise than its source is misleading regardless of the arithmetic being correct.

The third is round-tripping. Converting a value and converting it back should return the original, and often does not because of the rounding applied in between. If a stored value needs to survive conversion, store it in its original unit and convert for display only. A system that converts on save and again on load will drift, slowly and invisibly, and drifting stored data is far harder to notice than a wrong display.

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