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Recently, I was watching this video and I began thinking about how much my arithmetic skills have declined in recent years due to over reliance of calculators in upper year (high school) math courses. I've tried to keep these skills practised (e.g doing simple calculations in my head during commute, trying to do homework w/o a calculator, etc). I have also always had an interest in how past mathematicians, scientists and engineers performed calculations (Although this part may be a question for the Math and Science History SE) I am looking for resources pertaining to arithmetic, specifically algorithms/techniques/tricks to performing calculations on paper/mentally (potentially from a historical perspective?). Thanks in advance.

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    $\begingroup$ I think it would be useful to distinguish between (1) calculation skills a typical mathematics student, even a mathematics researcher, had before calculators, and (2) specialized techniques used by those whose work involved a lot of arithmetic calculations. For the former (which sounds like what you're interested in), simply pick any book on a topic you've been wanting to learn (calculus, ODEs, linear algebra, etc.) published before 1975 or so (before early 1980s is probably good enough, but before 1975 will definitely nail it) and work through the book pretending calculators don't exist. $\endgroup$ – Dave L Renfro Mar 15 '20 at 18:25
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    $\begingroup$ My point is the the vast majority of people didn't employ various rapid calculation methods of the type you're possibly thinking of. They used logarithm tables, square root tables, linear interpolation (to roughly get an extra significant digit from using table values), etc. and not rapid/trick mental arithmetic "parlor trick" methods. But people were a bit quicker with basic arithmetical calculations due to constant practice. $\endgroup$ – Dave L Renfro Mar 15 '20 at 18:30
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You might be interested in the Trachtenberg system of rapid mental calculation. The Wikipedia page lists a summary of its multiplication and addition methods. More information can be found at Trachtenberg Speed Math, where a book about it is for sale.

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