Is a bar over a significant figure needed for rounding?

Question

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?

Answer 1

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 $10,000\text{ cm}^2 = 1\text{ m}^2$.