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I was given an online assignment problem today. The answer was to be given to 2 decimal places. The answer I gave was $$P_2=0.06$$

This to me seems like 2 decimal places. However the answer was incorrect. I then wrote $$P_2=0.064$$

It was correct. I know I'm being knit-picky but I genuinely just want to know your thoughts on this.

Would you call this 3 decimal places and 2 significant figures, or 2 decimal places and 2 significant figures. In the context of the assignment it doesn't matter a whole lot but I still think he should have written 2 significant figures.

What's your 2 pence on the matter?

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    $\begingroup$ The confusion might arise because the answer might be expected in scientific notation. $\endgroup$
    – matt_black
    Commented Oct 21, 2017 at 15:23
  • $\begingroup$ It's important to have both concepts, decimal places and significant digits, because logarithms convert between them. For example, if you know hydrogen ion activity to 2 significant digits, then you know pH to 2 decimal places, or vice versa. $\endgroup$
    – Toby Bartels
    Commented Oct 22, 2017 at 5:57

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My verdict on $0.064$ is three decimal places, two significant figures. The underlying philosophy is: how many figures would you give if you converted this to "scientific" notation - which would be $6.4 \cdot 10^{-2}$. Since the results must be independent of representation and nobody would argue about the number of significant figures or decimal places here, this is the way to go for me.

I received some chemistry education both in the US and Germany. Unfortunately, my experience is that in Germany these two concepts are often confused, even on university level and by lecturers. All high school level books on Chemistry I looked at in the US contained a section on this and were clear and adamant.

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  • $\begingroup$ Except that we shouldn't be talking about decimal places in scientific notation. Or at the very least it's ambiguous unless specified that the answer should be in scientific notation with a single digits in front of the decimal (which is the default, but other representations aren't de facto incorrect). I think it's kind of silly to talk about decimal places in scientific notation though. $\endgroup$
    – spacetyper
    Commented Oct 21, 2017 at 19:25
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There's no room for my two-pence on the matter. "Significant figures" and "decimal places" mean exactly what you think they do, and the online assignment is wrong. Pick out a random sampling of textbooks on Google Books, or even the Wikipedia article, and you can find the distinction clear as day, e.g. this math book from Cambridge or the Olmsted and Williams chemistry book.

If we're being generous, maybe the problem was originally written to have the answer 0.64, and a factor of 10 was introduced later without the question being appropriately updated.

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I have since a long time a non-professional interest in numerics and the answer is unequivocal: You and TAR86 are right, the given answer is wrong.

Decimal places are defined by the numbers after the decimal point:

  • 0.1/3456.8/2e-1 = 1 decimal place
  • 0.01/234.90/3e-2 = 2 decimal places
  • 0.001/12.912/4e-3 = 3 decimal places

Significant figures/digits are the effective number of digits (without leading zero)

  • 0.1/3/1e3/2e-1 = 1 significant figure/digit
  • 0.23/3.2e6/12 = 2 significant figures/digits
  • 78.9/8.12e-3 = 3 significant figures/digits
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