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I have some idea of what the main concepts of calculus are about but I have never actually taken a calculus class or studied myself. My general understanding is that before the concept of limits were formally defined by later mathematicians, the first people that actually invented calculus explained the idea of derivatives and integrals using the hyperreal numbers, something for which they were actually criticised by some for the concept being more intuitive rather than mathematically provable/valid. I also believe that modern, standard calculus does not deal with hyperreals at all but rely solely on real numbers in explaining all of its concepts.

So I was thinking whether it would be a good idea to get the gist of what hyperreals are and how they relate to limits as a foundation before actually taking on calculus, or would it be not worth it. I'd appreciate any comment/suggestion you guys would make.

Mikhail Katz
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jacob78
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    I am not very knowledgeable about hyperreals. So, I want to say that I believe the following just from the priors together with my extremely fleeting knowledge of hyperreals: most probably, there exists no good introductory material on hyperreals that will work for you if you haven't studied calculus before. Therefore, even if hyperreals are a good foundation to study before actually studying calculus, there probably is no viable study route for you. – CrabMan Sep 12 '23 at 13:52
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    Also, I am not sure about hyperreals, but if you're thinking about infinitesmals, there is an introductory calculus textbook that uses them - Full Frontal Calculus by Seth Braver. Its preface says "Full Frontal Calculus covers the standard topics in single-variable calculus, but in a somewhat unusual way. Most notably, in developing calculus, I favor infinitesimals over limits. ... Why infinitesimals rather than limits? ... The subject’s proper historical name is the calculus of infinitesimals. Its basic notation refers directly to infinitesimals. ..." – CrabMan Sep 12 '23 at 13:54
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    The hyperreals are a branch of non-standard analysis, and are an extension of the standard real numbers. You get all the tools of standard analysis, plus some extra ones that can make tedious arguments easier to express. But you should still study and master standard analysis before looking into ways to change or improve it. It acts like a foundation. – user3257842 Sep 12 '23 at 13:57
  • I agree the comments : It is better to begin with standard analysis , and you can (although personally , I see no merit by doing it) additionally learn non-standard analysis. But beginning with non-standard analysis is very probably not a good idea. – Peter Sep 12 '23 at 14:05
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    Keisler's "Elementary Calculus: An Infinitesimal Approach" basically uses unproved facts about hyperreals instead of usually-unproved facts about Cauchy's definition of limit like other informal intro Calculus texts. If you're looking for an informal treatment, I don't think it matters much whether there are infinitesimals or not, and the pedagogy of not-infinitesimal texts may be better-developed. – Mark S. Sep 12 '23 at 14:22
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    Liebnitz and Newton did not use hyperreals, they used a vague notion of infinitesimal number. – Paul Sep 12 '23 at 14:49
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    If you're doing it for your own curiosity, then sure, go ahead. But it's very unlikely to help you when you do take a standard calculus class. – JonathanZ Sep 12 '23 at 14:53
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    @Paul "Leibniz" :) – Peter Sep 12 '23 at 16:07
  • Apologies, mortifying! – Paul Sep 12 '23 at 17:03
  • Mathematicians didn't start with hyperreals. They used infinitesimals but in an informal way. Approch with infinitesimals has been later pushed out by more formal approach that we know nowadays. Hyperreals (which are formal way to define infinitesimals) were made after that. But anyways, hyperreals aren't today used as commonly as standard analysis. You may start to learning about this but, it's a very cool thing to learn but 1) It would be better to start with standard analysis (most people don't know hyperreals that much) 2)the formal def. of hyperreals might be very hard for a beginner. – Antares Sep 13 '23 at 15:33
  • @Paul, it is true that Leibniz and Newton did not use the hyperreals, but this is not the point of the OP's question; see my answer. – Mikhail Katz Oct 04 '23 at 09:21
  • @Antares, you wrote that "Mathematicians didn't start with hyperreals". This is true, but arguably they did start with infinitesimals; see my anwer. – Mikhail Katz Oct 04 '23 at 10:56
  • @Mikhail the OP says "...the first people that actually invented calculus explained the idea of derivatives and integrals using the hyperreal numbers". I commented on this in this comment section. I'm sure your answer in the answer section is super. – Paul Oct 04 '23 at 11:10
  • @MikhailKatz I wrote that they used infiniesimals, but in informal way because they hadn't yet formalized notion of infinitesimals. – Antares Oct 05 '23 at 08:45
  • @Antares, anything done in mathematics before advent of formalisations such as ZFC would be "informal" by your definition. – Mikhail Katz Oct 05 '23 at 08:52
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    @MikhailKatz Before ZFC mathematics could cause some contradictions. But anyways, definition of limit occured a long before ZFC, because it was already used in xix. century. Proofs using infinitesimals before formalized motions of infinitesimals can't be treated as a rigorous proofs.Before formalized notions of infinitesimals there weren't any meaningful definition of what is infinitesimal that's why these can't be formal proofs. – Antares Oct 05 '23 at 21:46
  • @Antares, you are certainly entitled to your personal opinion, but what historical or mathematical scholarship are your claims based upon? Why would the use of infinitesimals before ZFC be any less meaningful than the use of limits before ZFC? Your views are likely consistent with triumvirate scholarship but this does not mean that they are historically sound. – Mikhail Katz Oct 08 '23 at 12:27

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"the first people that actually invented calculus explained the idea of derivatives and integrals using the hyperreal numbers": This is not strictly speaking true. Leibniz, for example, did not use "hyperreal numbers". He did use infinitesimal numbers, and made it clear in a 1695 publication (a response to Nieuwentijt) that such numbers were in violation of the Archimedean property.

Whether or not "it would be a good idea to get the gist of what hyperreals are and how they relate to limits as a foundation before actually taking on calculus" really depends on your goals. One advantage of learning the infinitesimal approach first (using for example Keisler's textbook) is that it will give you a direct intuitive access to the key notions of the calculus such as derivatives, continuity, and integral. Once you understand these key notions, it will be easier to understand the epsilon-delta definitions thereof that are usually presented in the (non-infinitesimal) calculus textbooks. However, if your goal is eventually to enroll in a university calculus course, you should be careful about actually using infinitesimals on a test (or even in the classroom): this may be misunderstood and will not necessarily facilitate your academic progress toward a degree.

Mikhail Katz
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  • I actually meant infinitesimal numbers when I said hyperreal numbers since it it my understanding that infinitesimals are a part of hyperreal numbers. However I did not realize it was wrong to use the two terms interchangeably. Now I know :) My "main" goal is to take a college enterance exam in which I am not required to provide proofs or any type of reasoning for that matter. However I also like to learn mathematical subjects in depth and actually understand the inner workings. – jacob78 Oct 04 '23 at 16:41
  • For college entrance exam, you probably need a good knowledge of pre-calculus (i.e., mosty basic algebra) rather than calculus, whether infinitesimal or non-infinitesimal! Note however that Keisler has a good introductory chapter on the pre-calculus material. @jacob78 – Mikhail Katz Oct 05 '23 at 08:25
  • Where I'm from (Turkey), the maths test of the college entrance exam consists of 40 questions, 10 of which are questions on limits, derivatives and integrals. I've found the book of Keisler you mentioned and it has amazing content. Can't wait to look into it once I'm done reviewing some pre-calculus material. One last thing I would ask is, do you think it is reasonable to learn standard calculus and infinitesimal calculus "hand in hand", meaning I will study Keisler's book and I will also learn standard definitions from another source as I go? – jacob78 Oct 05 '23 at 09:03
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    @jacob78, I think you would necessarily have to do that, because Keisler does not provide enough material for mastering the epsilon-delta techniques. – Mikhail Katz Oct 05 '23 at 09:06