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While you can build a calculus using infinitesimals, the usual construction does not use them.

https://en.wikipedia.org/wiki/Nonstandard_analysis




The original construction did though, if you look at the writings of Newton/Leibniz. But their work was totally hand-wavey. Cauchy discovered limits in the 1800s, and it wasn’t until the 20th century that the original idea of infinitesimals was formalized. In some sense we have come full circle.

I do think that infinitesimals are a much more intuitive way to understand calculus. It’s a shame it took so long to find positive numbers less than any real.




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