# Is the amount of cubic inches in a cubic foot exact up to infinite significant figures?

I know we are supposed to use SI units and I know they are defined very well.

But does that mean that Imperial units of length/volume aren't exact? Don't they have infinite significant figures in the equations? I couldn't find it in the SI brochure.

I'm asking because I need to convert from cubic feet to liters and trying to decide how many significant figures to use. My first thought is that the conversion factors are exact and I should use all the numbers my calculator comes up with. But the conversion factor wasn't in the back of my book so maybe cubic inches in a cubic foot aren't exact?

• Cubic inches in a cubic foot is an exact conversion. There may be some uncertainty in the conversion between SI and imperial -- I'm not 100% sure -- but if there is it's probably in the 8th or 9th decimal place. – Curt F. Jan 16 '17 at 4:43
• @CurtF. I guess the question boils down to: Is the amount of Liters in a cubic foot exact? Or if not, how many significant figures should we use? – eromod Jan 16 '17 at 4:56
• A short time before the French revolution, the royal rent collectors showed up at farmsteads with the largest 'bushel' measures ever seen in that country, to measure out the royal share of the crops. The problem with non-SI measurements is the multiple disagreeing sources of authority. – Whit3rd Jan 16 '17 at 7:01

Dealing with imperial measures in science is problematic. The main reason for that is that you have (at least) two different definitions (fixed values) for a lot of them, one based on the laws of the United States and one based on the laws of the United Kingdom. While the differences are quite insignificant for everyday use (as in baking a cake), in the science world these small deviations can cause further inaccuracies. On Wikipedia you can find more, but for the argument, let's look at the values for the yard:

United States    0.914401829 m
United Kingdom   0.9143993   m
International    0.9144      m


In principle the definition of cubic inch to cubic foot is exact, while the conversion to the SI system may carry an error.

Unit        Abbreviation    Definition  SI equivalent (rounded 4sf)
Cubic inch  cu in or in³                16.38 cm³
Cubic foot  cu ft or ft³    1728 cu in  0.02831 m³
Cubic yard  cu yd or yd³    27 cu ft    0.7646 m³


Most of the time this shouldn't affect your measurements more than the use of significant figures, which is used as stating an error implicitly. That means it is in itself inaccurate. In any scientific capacity you would state the error explicitly.
When a value is stated as $$1.00$$ it has three significant figures it actually means that the measurement falls within the interval $$0.99$$ to $$1.01$$, or $$1.00\pm0.01$$.

Let's have a quick example. You measure the volume of water in a beaker to be $$0.35~\pu{in^3}$$. The beaker doesn't specify how accurate the measurement is, so we simply assume $$\pm0.01~\pu{in^3}$$. You would state that value as $$0.35\pm0.01~\pu{in^3}$$, or $$0.34-0.36~\pu{in^3}$$, or simply $$0.35~\pu{in^3}$$ with implied error. On the other hand, when we know significant figures are used, we would interpret a value of $$0.35~\pu{in^3}$$ as $$0.35\pm0.01~\pu{in^3}$$.
When you convert this to the SI system, your value doesn't become more or less accurate. You cannot use significant figures to express the error in one system in the same way you would do it in the other, because the implicit error is different.
The exact conversion (of the international yard) is $$\frac{0.9144~\pu{m}}{1~\pu{yd}}=\frac{0.9144~\pu{m}}{3~\pu{ft}}=\frac{0.9144~\pu{m}}{36~\pu{in}}\implies1~\pu{in^3}=\left(\frac{0.9144}{36}\right)^3~\pu{m^3}.$$

The actual trust of the value is $$\pm0.01~\pu{in^3}=\pm0.16387064~\pu{cm^3}$$. When you convert your value $$0.35~\pu{in^3}=5.74~\pu{cm^3}$$ (keeping the same number of significant figures since the conversion is exact), you would imply an error of only $$0.01~\pu{cm^3}$$, which is far less than the actual trust. Instead your measured value falls somewhere between $$5.58-5.90~\pu{cm^3}$$. So the correct way of implicitly stating the error would be $$6~\pu{cm^3}$$.

### TL;DR

The conversions are exact, the system of significant figures is not. In doubt, state error margins explicitly.

• @Nicolau That is pretty much what I said in the last sentence. I also use the exact (international) definition (of the yard) in my answer. I only try to make the point that the implied error is completely different when using imperial units compared to SI units. – Martin - マーチン Jan 16 '17 at 9:06
• Note that the value you list for the US yard-to-meter conversion is archaic. To make conversions precise, (and as noted in the Wiki article), the inch was redefined in 1960 to be 2.54 centimeters, exactly. – Jon Custer Jan 16 '17 at 14:23
• @JonCuster on Wikipedia it says this value is exact. However, this exactness does not extrapolate to the implied error. – Martin - マーチン Jan 16 '17 at 14:35
• Yes and no. Using "significant figures" is a sloppy method of doing error propagation. Typically two significant figures in centimeters gives two significant figures in inches. // We could debate if 0.35 cm is +/- 0.01 cm or +/- 0.005 cm too. – MaxW Jan 16 '17 at 23:17
• @Max I agree and I don't like the significant figure method either. I think implying an error is ignoring the error. However, it is used extensively and one should be aware of its inaccuracies. – Martin - マーチン Jan 17 '17 at 6:22

Yes, there are exactly $12 \times 12 \times 12$ = 1728 cubic inches in a cubic foot.

Also, an inch is exactly 2.54 centimeters.

A foot is exactly 0.3048 meters.

A yard is exactly 0.9144 meters.

Within a certain system of measures, conversion factors are typically exact.

In imperial units, this means that a foot is always twelve inches, a yard is always three feet and a mile is always 1760 yards. With the exact conversion, we can use multiplication to see that:

$$1~\mathrm{yd} = 36'' \pm 0''\\ 1~\mathrm{m} = 5280' \pm 0'\\ 1~\mathrm{m} = 63360'' \pm 0''$$

Likewise, the conversion from cubic imperial units — which are defined as $\text{imperial unit}\times\text{imperial unit}\times\text{imperial unit}$ — to other cubic imperial units is equally exact.

$$1~\mathrm{yd^3} = 1~\mathrm{yd}\times 1~\mathrm{yd}\times 1~\mathrm{yd} = 3' \times 3' \times 3' = 27~\mathrm{ft^3}$$

This is akin to the exactness of the conversion of metric units, except that the metric factors are always multiples of 10. But there are also semi-metric units, such as the German ‘metric pound’, by definition $500~\mathrm{g}$. Conversion from grams to German metric pounds is as exact as conversion from grams to kilograms.

This becomes different if you leave your system of measures and compare different systems. Except in a few very special cases, where two systems chose physically identical starting and/or ending points but with different step sizes,[1] the conversion factors will be nonexact.

Any nonexact conversion factor will introduce an error of its own into the equation and how to deal with these is best shown in Martin’s answer.

Note that even if the transformation is exact at a starting and ending point, measured values will still have an inherent uncertainty in them, as Martin pointed out. The uncertainty is transformed across the conversion and does not change magnitude.

[1]: See for example temperature. A measured temperature of $15~\mathrm{^\circ C}$ can be transformed into an exact value in kelvin because the kelvin and celsius scales are identical except for an addition factor. Thankfully, also the error is identical.

$$(15.00 \pm 0.005)~\mathrm{^\circ C} = (288.16 \pm 0.005)~\mathrm{K}$$

Similarly, the Réamur temperature scale is defined such that the boiling point of water is $80~\mathrm{^\circ r}$ while the zero-point is identical to the celsius scale. Therefore, the step is different and $1~\mathrm{^\circ C} = 0.8~\mathrm{^\circ r}$. Upon converting, we get a similarly exact value but a different error.

$$(15.00 \pm 0.005)~\mathrm{^\circ C} = (12.00 \pm 0.004)~\mathrm{^\circ r}$$