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If I multiplied $651\ \mathrm{cm} \times 75\ \mathrm{cm}$, it equals $48,825\ \mathrm{cm}^2$. But I need to round it to 2 significant figures. So I would write $49,000\ \mathrm{cm}^2$ as my answer. However, do I put a bar over the $9$ to show that it's rounded to two sig figs?

gt6989b
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Dylan
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    Alternatively, perhaps you could write the answer as $4.9 \times 10^4$ cm$^2$ instead. – John Omielan Oct 28 '19 at 22:43
  • Yes, but my teacher doesn't want us to use scientific notation for this worksheet, with this problem on it. So, do I put the bar over the 9? – Dylan Oct 28 '19 at 22:47
  • Thanks for explaining this. I suggest you use whatever your teacher prefers since, as my answer states, although an overline (i.e., bar) is the most common means, sometimes an underline is used instead. – John Omielan Oct 28 '19 at 23:04
  • I know how to use the overline (bar) but I don't know WHEN to use it. Do I put it over the 9 to show that it has two significant figures, or do I just leave it as the answer with no bars? – Dylan Oct 28 '19 at 23:07
  • Thanks for clarifying the issue. I've updated my answer to mention this specifically. – John Omielan Oct 28 '19 at 23:10
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    Is writing “4.9 m²” not an option? – Dan Oct 28 '19 at 23:15
  • I don't understand it because it seems contradictory. My sig fig rules say, "Any trailing zero to the LEFT of the decimal are NOT significant." Which means 1,500 only has two sig figs (1 and 5). But my other rules say exact measurements, like cm or kg, have infinite sig figs. So now I'm confused. – Dylan Oct 28 '19 at 23:15

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As I suggested in a comment, you could use scientific notation. Alternatively, since you state your teacher doesn't want you to use this, as Wikipedia's Significant rules explained section of its "Significant figures" article states:

An overline, sometimes also called an overbar, or less accurately, a vinculum, may be placed over the last significant figure; any trailing zeros following this are insignificant. For example, $13\bar{0}0$ has three significant figures (and hence indicates that the number is precise to the nearest ten).

Less often, using a closely related convention, the last significant figure of a number may be underlined; for example, "$2\underline{0}00$" has two significant figures.

In the combination of a number and a unit of measurement, the ambiguity can be avoided by choosing a suitable unit prefix. For example, the number of significant figures in a mass specified as $1300$ g is ambiguous, while if stated as $1.3$ kg it is not.

I don't know about the history & reasons for using one option compared to the other, but one small issue I can see with using an overbar is that it may be somewhat confusing with situations where this is also to indicate a repeating decimal, e.g., $2.3\bar{4} = 2.34444\ldots\;$ .

Note the first two options are also used in other Web sites, e.g., Significant figures, in its practice problems, an over line in the third & and an under line in its fifth.

As for whether or not something like this is required at all, the Wikipedia article says:

Zeros to the right of the significant figures are significant if and only if they are justified by the precision of their derivation.

Nonetheless, to be unambiguous & to clearly differentiate your answer from the case of there possibly being $5$ significant digits instead in your particular case of $49,000$, I suggest you should explicitly indicate $9$ is the last significant digit, with the most commonly used options (without scientific notation) being an overbar (i.e., so it's $4\bar{9},000$) or an underbar (i.e., so it's $4\underline{9},000$).

Alternatively, as suggested by the third option & which Dan stated in a comment, you can also use a different unit of measurement, in particular, you could say it's $4.9\text{ m}^2$ instead since $10,000\text{ cm}^2 = 1\text{ m}^2$.

John Omielan
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  • I know how to use the overline (bar) but I don't know WHEN to use it. Do I put it over the 9 to show that it has two significant figures, or do I just leave it as the answer with no bars? – Dylan just now Edit Delete – Dylan Oct 28 '19 at 23:07
  • @Dylan I've updated my answer to address this issue. – John Omielan Oct 28 '19 at 23:15
  • My sig fig rule says, "Any trailing zeroes to the LEFT of the decimal is NOT significant." 1,500 only has 2 sig figs (1 and 5). So why would 49, 000 have 5 sig figs then? And exact measurements, in cm and kg, for example, have infinite sig figs. I'm so confused. – Dylan Oct 28 '19 at 23:19
  • @Dylan I don't know what sig fig rule you're referring to. Note there's some ambiguity about trailing zeros. However, as the Wikipedia article says "In most contexts it is understood that trailing zeros are only shown if they are significant". Nonetheless, it also says earlier "Zeros to the right of the significant figures are significant if and only if they are justified by the precision of their derivation". If it's understood only a certain # of digits are significant, you may be able to avoid using an overbar, underbar or something else, but it's generally better to be unambiguous. – John Omielan Oct 28 '19 at 23:27
  • @Dylan As for your statement of "exact measurements, in cm and kg, for example, have infinite sig figures", if I understand you correctly, keep in mind it's not the units which are being checked to see how many significant digits, but rather the quantity of those units. Thus, for example, $4.9\text{ m}^2$, to me, means it's accurate to $2$ significant digits, i.e., the actual amount is at least most likely (if not guaranteed) to be between $4.85\text{ m}^2$ and $4.95\text{ m}^2$. Does this make sense & answer your concern? – John Omielan Oct 29 '19 at 00:03
  • What if it asks me to round 12,422 cm to TWO significant digits? Yes, I know it will be 12,000 cm. But would I put the bar over the 2 or not? – Dylan Oct 29 '19 at 02:24
  • @Dylan I would recommend putting a bar over the $2$. Even if the context indicates this is strictly not needed in a particular case, as far as I know it's never wrong to use the bar, so it's safest & best to use this. – John Omielan Oct 29 '19 at 02:26
  • Thank you, I have a test on this sig figs tomorrow and hopefully, my teacher won't take off points if I use the bar :) – Dylan Oct 29 '19 at 02:32
  • @Dylan I don't know your teacher & how this person will act, but I would be surprised if you lost any points for doing something which is generally understood in the math community. However, if that does happen for any reason, I suggest you at least ask your teacher for the reason why, if a reason is not provided already. Also, good luck on your test tomorrow. One final thing, in case it's not obvious, only use the bar if there are digits to the left of the decimal point to not use &, in particular, don't use it to the right, as it usually means repeating the digit (as I explain in my answer). – John Omielan Oct 29 '19 at 02:36