Elementary calculus

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1. augustusseizureAug 2026
   Depends what you're looking for. Full Frontal Calculus[0], Intuitive Infinitesimal Calculus[1], and Elementary Calculus[2] are all textbooks on the calculus sequence using an infinitesimal pov. The basic approach is to extend the Real numbers to include infinitesimals (greater than zero but smaller than every positive real number) and transfinites (greater than every positive real number), collectively called the Hyperreals.

   If you're looking for a more formal approach, ie the infinitesimal analogue to the usual real analysis, it's called nonstandard analysis and you could probably start with the original, eponymous book written by the creator, Abraham Robinson, for which I unfortunately don't have a link.

   If this stuff interests you btw I would also check out Knuth's book on surreal numbers[3], which I believe, in some sense, are the fullest possible extension of what we think of as numbers? But it's been a while since I read into those.

   [0] https://www.bravernewmath.com/ [1] https://intellectualmathematics.com/calculus/ [2] https://people.math.wisc.edu/~hkeisler/keislercalc-06-03-26.... [3] https://people.math.harvard.edu/~knill/teaching/mathe320_201...

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2. QemJan 2025
   Nice, I didn't know about this book. Did you try the Keisler book too, "Elementary Calculus: An Infinitesimal Approach"? See https://people.math.wisc.edu/~hkeisler/calc.html
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3. QemJan 2024
   > i've stumbled a few years ago on a math book that described the original concept of "infinitesimals" and how a whole different way of doing calculus exists.

   Was this book "Elementary Calculus: An Infinitesimal Approach", by Keisler? It's an awesome book. It's free to download at https://people.math.wisc.edu/~hkeisler/calc.html

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4. QemOct 2023
   Silvanus' book makes calculus simple by adopting an infinitesimal approach, like Newton and Leibniz did when they invented calculus. But that approach was shunned by mathematicians for a long time, because it was only made rigorous in the 60s. After Silvanus' book, I also recommend Elementary Calculus: An Infinitesimal Approach, by professor Jerome Keisler, for those interested in this alternative pathway to calculus. It can be freely downloaded at https://people.math.wisc.edu/~hkeisler/calc.html

   Relevant wikipedia entry: https://en.m.wikipedia.org/wiki/Nonstandard_analysis

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5. QemSep 2022
   For the sciences that depend on higher maths, a Calculus book worth mention is "Elementary Calculus: An Infinitesimal Approach", by Jerome Keisler. One thing that makes learning Calculus harder than necessary is the clumsy epsilon and delta formalism, that became sort of the COBOL of Calculus teaching. This formalism is not the intuitive approach Newton and Leibniz used to develop Calculus, based on infinitesimals, that was shunned later because it took time until Abraham Robinson made it rigorous in the 1960s. The book by Keisler showcases nicely a way to improve higher math education, if academia had less inertia. The author made the entire book available for free online: https://people.math.wisc.edu/~keisler/calc.html See also: https://en.wikipedia.org/wiki/Nonstandard_analysis
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6. QemSep 2022
   "Elementary Calculus: An Infinitesimal Approach", by Jerome Keisler. Learning calculus is made harder than necessary by the legacy of clumsy epsilon and delta formalism. This formalism is not the intuitive approach Newton and Leibniz used to develop Calculus, based on infinitesimals, that was shunned later because it took time until Abraham Robinson made it rigorous in the 1960s. The author made the entire book available for free online: https://people.math.wisc.edu/~keisler/calc.html See also: https://en.wikipedia.org/wiki/Nonstandard_analysis
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