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Q: Where do the rules for “significant figures” come from?

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Significant digits are why driving directions don’t say things like “drive 8.13395942652 miles, then make a slight right”. Not coincidentally, this has two significant digits, just like “0.32” which had the least number of significant digits at the start of the calculation. When adding numbers using significant digits, you’re declaring that a=0.5 x 10-D1 and b=0.5 x 10-D2, where D1 and D2 are the number of significant digits each number has. When using sig figs the larger error is by far the more important, and the product once again has the same number of sig figs. Personally, I like to prepare everything algebraically, keep track of sig figs and scientific notation from beginning to end, then drop the last 2 significant digits from the final result.