I wanted to know how do we know that the rules that we follow when doing arithmetic with significant figures are correct? Like why when adding or subtracting we keep the same number of decimal places as the original number with the least decimal places and when multiplying and dividing we have to retain the same number of significant figures as in the number with the least of them. What is the proof of that? I searched on the web about it but couldn't find anything except the rules themselves. Thanks.
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Intuitively it is not too difficult:
If you want to do this properly, use interval arithmetic.