Example: I have $\pu{64 cm^{3}}$ of milk. How much is that in gallons?

When I use the conversion factors $\frac{\pu{1 mL}}{\pu{1 cm^3}}$ $\frac{\pu{1L}}{\pu{1000 mL}}$ $\frac{\pu{1.0567 qt}}{\pu{1 L}}$ $\frac{\pu{1 gal}}{\pu{4 qt}}$.

I know the final answer should only have two significant figures, but how many significant figures should the intermediate conversion factors have?

I have heard that:

  • conversion factors should have one more sig fig than the least precise measurement. (So three in this case)

  • But I also heard that we shouldn't round anything until the end. (So use all 5, and round after conversions are complete)

The reason why I got confused is because of my textbook:

enter image description here

  • 1
    $\begingroup$ check your conversion factor. $1 \text{cm}^3$ is a milliliter which is much much less than a quart. $\endgroup$ – MaxW Sep 14 '16 at 20:14
  • 1
    $\begingroup$ With a modern calculator, I wouldn't round anything until the end. When doing calculations with a slide rule or log tables, then things were a bit different. $\endgroup$ – MaxW Sep 14 '16 at 20:15
  • $\begingroup$ This is case dependent. You need to realize what kind of mathematical object are you deal with. $\endgroup$ – user1420303 Sep 14 '16 at 20:52
  • $\begingroup$ @MaxW, so is my textbook wrong in rounding in this example? $\endgroup$ – eromod Sep 15 '16 at 23:28
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    $\begingroup$ As I said, if I had a calculator, I wouldn't round anything until the end. If I had to do all the calculations by hand, then I'd be inclined to round intermediate values. I'm not going to multiple two 8-digit numbers by hand and then round to 2 digits. $\endgroup$ – MaxW Sep 16 '16 at 3:36

Use significant figures as much as you can in intermediate conversion factors,and then round off the final answer to two significant figures,using more significant figures in intermediate conversion factors will lead to a accurate answer. I let you conclude.

  • $\begingroup$ that was my initial impression too, but have a look at my textbook example I attached. $\endgroup$ – eromod Sep 15 '16 at 23:26
  • $\begingroup$ you should use as much as sig.fig.,i have seen many examples and solved many exersises regarding this topic,and have never seen rounding off before the final answer unless significant figures are too much and cannot be taken under arithmatic operations. $\endgroup$ – Vidyanshu Mishra Sep 16 '16 at 5:12

I'm not sure where your conversion factor came from as it is incorrect but regarding the correct factor: Because the inch is defined using the metre, and the gallon defined using the inch, there is an exact conversion between them—1 US gallon is exactly 3.785411784 L. Thus, like any exact quantity or conversion factor, we treat the gallon to litre conversion factor as having infinite significant figures. Because we have defined it exactly, there is no implicit uncertainty in the number. There are other conversion factors that are not exact and do actually have uncertainties, but for most practical purposes, these have far more significant figures than a normal measured quantity. See this question for more details.

  • $\begingroup$ conversion factors were wrong, I updated them. But what is the conclusion though? How many sig figs should I use? That link says I should look it up in the CRC Handbook. That what you recommend too? $\endgroup$ – eromod Sep 14 '16 at 21:50
  • $\begingroup$ As my answer says, the conversion factor is exact. $\endgroup$ – Michael DM Dryden Sep 15 '16 at 3:16
  • $\begingroup$ So if my measured quantity is $\ 64 cm^{3}$ , how many significant figures should I use in the qt/L conversion? All of them because its an exact conversion? Or just three sig figs (one more that the two I have to work with) $\endgroup$ – eromod Sep 15 '16 at 3:27
  • $\begingroup$ @eromod The conversion factor is only exact if you use that exact conversion factor. For example, an inch is defined as 2.54 cm. If you use that number, it has infinite precision. However, if you say "ahh, 2.5 is close enough" then the conversion factor only has 2 significant digits. Likewise here. If you use the exact conversion factor, there's infinite precision. If you round it, it has the rounded amount of precision. $\endgroup$ – R.M. Sep 15 '16 at 22:19
  • $\begingroup$ @ R.M. But in my textbook, they rounded the conversion factor...I added a screenshot $\endgroup$ – eromod Sep 15 '16 at 23:24

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