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I have a question I picked on the internet, but I am not sure about the term truth-worthiness part of the question.

Find the product of 346.1 and 865.2. State how many figures of the result are trustworthy, given that the numbers are correct to four significant figures.

The product is 299445.72. Which becomes 299400(4 sig. fig).

In terms of how trustworthy, what are we looking at here? Since we have been restricted to the rule of 4 sig. fig, why is the answer not 299400?

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    $\begingroup$ All we know is that the first number is between $346.05$ and $346.15$ and the second between $865.15$ and $865.25$, so the product is between $346.05\times865.15=299385$ and $346.15\times865.25=299506$. $\endgroup$ – almagest May 19 '16 at 4:56
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    $\begingroup$ BTW where on earth do you get $39885.72$ from! Is this some kind of joke? $\endgroup$ – almagest May 19 '16 at 4:58
  • $\begingroup$ @almagest, sorry. I don't even understand myself how I computed that product :). But see the amendments next to the old values. What is the "truthworthiness" from your computation? $\endgroup$ – Sylvester May 19 '16 at 9:13
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    $\begingroup$ Sylvester: I see goblin has written out my comment more elaborately as an answer. Ask him! $\endgroup$ – almagest May 19 '16 at 9:22
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How about using interval arithmetic?

Define $$x = [346.1-0.05,346.1+0.05], \qquad y = [865.2-0.05,865.2+0.05]$$

Then $$xy = [(346.1-0.05)(865.2-0.05), (346.1+0.05)(865.2+0.05)].$$

You should be able to puzzle out the accuracy of the result from the width of the interval $xy$, and then translate this back into sig-fig language at the end.

Let me know how this goes.

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