# Do ALL conversion factors have an infinite number of sigfigs?

Conversion factors have an infinite number of sigfigs, right? They should be exact values; but not all of them are, are they?

I looked at http://www.webassign.net/question_assets/buelemphys1/chapter01/section01dash2.pdf and it agreed with the fact that all conversion factors have an infinite amount.

I know this is true with a conversion factor such as $\frac{1\ \text{cal}}{\mathrm{g\ ^\circ C}}$, because this is how a calorie is defined. Similarly, it’s the same for $\frac{1\ \mathrm{kg}}{1000\ \mathrm g}$, $\frac{4\ \text{quarts}}{\text{gal}}$, etc. We know these are exact, and in dimensional analysis, if you multiplied by the first conversion factor, you wouldn't end up with one sigfig because of the single sigfig in $1\ \text{cal}$.

However, what is the case with measured conversions, such as $\frac{4.184\ \mathrm J}{\text{cal}}$ or $\frac{1.60934\ \mathrm{km}}{\text{mi}}$? These were measured using sigfigs, so it would make sense to round with sigfigs when multiplying (dimensional analysis), but the above link says otherwise.

What should I do, especially with the below type of conversions?

On a side note, I'm not sure if Chemistry Stack Exchange is the best site to place this under. Please flag to a different site if there is a better one.

• I think this should be physics? – Jan Sep 28 '15 at 23:33
• Nah, I think Chem.SE's a good choice. – hBy2Py Sep 29 '15 at 0:04

Your intuition is correct. There are two types of conversion factor.

One kind of conversion factor is based on definitions. These conversions are exact (infinite significant figures).

In addition to your examples of the specific heat of water (in calories), the kilogram/gram conversion, and the quart/gallon conversion, other conversions are exact, including:

• The speed of light - $c=299,792,458\ \mathrm{m/s}$ the length of the meter is then derived from this constant.
• The price of gasoline/petrol.
• The inch is defined as exactly $25.4\ \mathrm{mm}$, and thus the conversions from all imperial to metric lengths (including your are exact as well.
• The specific heat of water in joules - the relationship between the calorie and the joule is defined exactly as $\mathrm{1\ cal = 4.184\ J}$

Note that most conversions of this type are unit conversions. Many units have been defined exactly in terms of SI-units.

Other conversion factors are based on measurements. One common example is atomic mass. Atomic masses are valuable conversion factors that are based on measurements. However, some unit conversions live in this category.

The electronvolt is based on the measurement of the elementary charge of the electron (which is not exact). Thus the electronvolt is $\mathrm{1\ eV=1.6021766208(98)\times 10^{−19}\ J}$

• To clarify, how should I round with measurement-based conversions? – Jonathan Lam Sep 29 '15 at 1:27
• @jlam55555 Most measurement-based conversion factors have a functionally-infinite number of significant figures for the purposes of routine chemistry work. That being said, to be rigorous, one would have to track down the original source of the measurements (or a reputable secondary source such as Perry's Chem Eng Handbook or the CRC Handbook), and see how many sigfigs were reported there. That would then become the number of sigfigs to use in figuring out the final rounding, same as if it were a measurement you made yourself. – hBy2Py Sep 29 '15 at 3:06