What is the proof of the rules of significant figures?

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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• If you are trying to add $x$ and $y$ but are in fact adding $x+\delta$ and $y+\epsilon$ then the absolute error will be $\delta + \epsilon$, which will be dominated by the larger of the absolute errors, most likely by the absolute error in the number with the fewer decimal places. Subtraction is similar.
• If you are trying to multiply $x$ and $y$ but are in fact multiplying $x\times(1+\delta)$ and $y\times(1+\epsilon)$ then the relative error will be $(1+\delta)(1 + \epsilon)-1 = \delta + \epsilon + \delta \epsilon$, which will be dominated by the larger of the relative errors, most likely by the by the relative error in the number with the fewer significant figures. Division is similar.