Significant figures: Is the answer on this question correct?

Question

A google form question

So I got this question in a test and options 2, 3 and 4 were wrong, in my view. So I chose a random option and I got it wrong. Upon further enquiry, the teacher explained that the second and fourth option was actually correct because it could be ROUNDED to 4 significant figures and 3 significant figures respectively. However, the third option was impossible to be rounded to 2 significant figures. So my question is, was option three really the right answer, or options 2, 3 and 4 were all the right answer?

It may not really be math but its just the wording of the questions and answers are ambiguous.

Answer 1

All the proposed answers are wrong except the first. $97.040$ has five significant figures because the trailing zero is not needed to set the decimal point, so it is just there as a significant figure. $21000$ has only two significant figures. All the zeros are there to set the scale. $0.03$ has only one and $970$ has two. All the zeros in the answers except the first are there to set the decimal point. The question does not have a unique answer. I suspect the writer lost track of the not in the question when proposing the answers. I don't think the wording is ambiguous, the question and answers do not match.

Answer 2

The question is flawed in my opinion.

Let's think of the options as being "rounded off" figures and try to see if we can decide how many significant figures the original number (usually the result of a measurement or calculation incorporating measured quantities) had to begin with. But first let me express these using standard form (also known as scientific notation) as I often find it easier to decide on significant figures this way.

When you express these choices in standard form, you get:

1. $9.7040 \times 10^4$

2. $2.1 \times 10^4$ (but this is ambiguous, explained below)

3. $3 \times 10^{-2}$

4. $9.7 \times 10^2$ (but this is ambiguous, explained below)

Now, of those choices, only with the first and the third can you explicitly state the number of significant figures the number has. This is because with the first choice, there is a trailing zero - if this were not significant, it would have been clipped off (truncated). You must represent this trailing zero in the standard form as well. The fact that it's left intact means it is significant. So the first option is definitely true.

The third option definitely has just the one significant figure because you only have the single digit in the standard form, without any trailing zeroes. Remember, zeroes that "just" indicate the magnitude of a number do not count as significant digits. They disappear in standard form.

OK, now on to the second option. Now, what I wrote is one perfectly reasonable standard form for the result (with two significant figures), which will give $21000$ in "longhand" form. The problem is that I could also have written $2.10 \times 10^4$ or $2.100 \times 10^4$ or $2.1000 \times 10^4$. These are all numerically equivalent to $21000$ but all have different numbers of significant figures. Note that the representation $2.100 \times 10^4$ does have $4$ significant figures, so $21000$ could have $4$ significant figures. But there are other representations that have different numbers of significant figures. To me it's impossible to decide simply based on the representation $21000$, so I would leave it as ambiguous (but $4$ s.f. is possible).

Similarly, the last one has two significant figures in the representation I chose but it could also have been represented as $9.70 \times 10^2$, which would mean it had $3$ significant figures. Again, there is ambiguity here. But $3$ s.f. is a possible answer.

So, I would have gone with choice number three based on it being the only unequivocally incorrect choice. But it is a poor question as options two and four are ambiguous.

Some might think to take the "longhand" representations as exact numbers, and ask how many significant figures they possess based on the nonzero digits they had (excluding trailing zeroes). On this basis, they might state that the options had (respectively) $5, 2, 1, 2$. In my opinion, this is an incorrect way of thinking about it. Fundamentally, significant figures are based on measurement or derived from measured values in some way. They are much more intimately linked to the physical sciences than to pure mathematics. So if those values are taken as "exact", that implies that some measuring instrument returned a value that was exactly equal to that, in which case, you would actually have $5, 5, 1, 3$ s.f. as every "non-magnitude determining" digit in a value returned by a measurement method is to be considered significant. We are basically positing here that the putative measuring instrument in the second option could have returned a value like $21002$ but instead returned "exactly" $21000$ (in which case the value $21000$ has exactly five signficant figures) and that the putative measuring instrument in the last option could have returned $973$, but instead returned "exactly" $970$. It is not sensible in my view to talk about significant figures in a context outside measurement (or calculated values derived from measured values).