Hyperreal numbers make Leibniz notation literal and algebraically consistent, rather than a convenient shorthand for limits.
replyAuthor suggests using them.
This is effectively how dual numbers work! https://en.wikipedia.org/wiki/Dual_number
replyDual numbers are basically big/little O notation. They combine beautifully with Robinson's NSA to give the most 18th century-like approach to deriving integral/derivative formulas that I know. And it's fully rigorous!
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