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Question: Report the area of a rectangle that has a reported height of 7.5 cm and length of 10.5 cm.

According to an HMH Algebra 1 textbook, the product of two [reported] measurements should have no more significant digits than the least precise measurement." This goes against the usual convention of reporting the product using no more significant digits than the factor with the least number of significant digits.

Using the textbook's criteria for comparing precision, the length and width of the rectangle have the same precision, as they are both measured to the nearest tenth of a centimeter.

So, using the textbook's rule, my question is what happens in this case? Since both measurements have the same level of precision, how many significant digits should the reported area have?

(Or maybe yet, and dare I even ask, do you agree with the textbook's wording of this rule?)

Anthony N.
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    The height has two significant digits; the length has three significant digits. Now, use the quoted statement. – N. F. Taussig Jun 28 '22 at 13:56
  • Thank you N.F. Taussig. I know what you mean. Most sources would say to report the area with two significant digits. However, the quoted statement doesn't follow this convention -- to examine the precision of the given measurements (instead of examining the measurement with the least number of significant digits). According to the quoted statement then, how should one report the area? – Anthony N. Jun 28 '22 at 14:02
  • There’s an urban legend that the first measurement of the height of Mt. Everest gave exactly 28,000 feet. Not 28,000 +/- 500 but exactly 28,000. So somebody decided to change the number to 27,997 feet so people would believe the number. – gnasher729 Jun 28 '22 at 14:04
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    For many people, in many contexts, "precision" is measured by the number of significant digits. Therefore the height is less precise than the length. It is possible that your book has defined "precision" a different way elsewhere for some reason and the author forgot it when writing this passage. – David K Jun 28 '22 at 16:20
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    General rule for multiplication is using significant digits count, not position of last known digit. Answer of GEdgar shows mathematics behind this rule. $7.5\times 10.5=78.75$, according to N. F. Taussig comment, result should be rounded to two significant digits, that is $79$. More precise uncertainty calculation (I suppose $a$ and $b$ measurements are independent) gives: $a=7.5\pm 0.05$, $b=10.5 \pm 0.05$, $S=ab=78.8\pm 0.7$ – Ivan Kaznacheyeu Jun 29 '22 at 07:02

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Ignore, for a while, the "general rule" or "rule of thumb" quoted.
Let's say that the two dimensions are $H$ and $L$, and the reported measurements mean $$ 7.45 \le H \le 7.55,\qquad 10.45 \le L \le 10.55 . $$ We may then conclude that $$ 77.8525 \le HL \le 79.6525 . \tag1$$ Conclusion: Reporting the answer as "$78.7$" would mean between $78.65$ and $78.75$, which is certainly not justified by the information in $(1)$.

Reporting the answer as "$79$" would mean between $78.5$ and $79.5$, which is better, but still slightly more than what is known in $(1)$.

Reporting the answer "$80$" would mean between $75$ and $85$. This does follow from $(1)$, but has lost a lot of information.

The "Rule of thumb" about significant digits says we should report "$79$". It is a compromise, but I agree it is the best of our choices.

GEdgar
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7,5 * 10,5 = 78,75

But since the least precise digit is one digit after the comma, the answer must be modified as 78,7cm^2.

My math is not very good, so if there's anything wrong, please don't hesitate to correct me!

Kuro
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