What is the meaning of infinitesimal? - Mathematics Stack Exchange In the early days of calculus, a lot of the ideas were defined in terms of an intuitive idea of infinitesimals, but in the 19th century, as mathematics became more and more driven to make sure the foundations of mathematics made sense, they found problems with infinitesimals, and a way to do calculus without needing the infinitesimal numbers
calculus - Whats an example of an infinitesimal? - Mathematics Stack . . . One can also name concrete infinitesimals, in different manner, in typical pre-Robinson non-archimedean fields like Levi-Civita In the other direction, one can probably ask if there is a non-archimedean extension of $\mathbb R$ with no real-ordinal definable (ROD) infinitesimals, and I wonder if one can get such an extension in the Solovay model
How do you understand Infinitesimals? - Mathematics Stack Exchange On the other hand, the hyperreal number system ${}^{\ast}\mathbb R$ contains both infinite numbers and infinitesimals, and there you can rigorously speak about a number with an infinite tail of $3$'s falling infinitesimally short of $\frac13$, or of a number with an infinite tail of $9$'s falling infinitesimally short of $1$, precisely as you
Are infinitesimals equal to zero? - Mathematics Stack Exchange In can make sense to denote length using hyperreal infinitesimals, and you can have a line segment of infinitesimal length In fact, non-standard analysis, the principle application of hyperreal numbers, defines things like derivatives and limits using infinitesimals in a way that is equivalent to the standard definition**
Do limits leave residual infinitesimals, or do they resolve exactly? Thus, "residual infinitesimals" are indeed involved in evaluating a limit, but they disappear once we apply the standard part For an elementary axiomatic approach to infinitesimal analysis (that does not involve either the axiom of choice or ultrafilters), see this introduction
What problems arise when using infinitesimals in calculus? $\begingroup$ the reason mathematicians considered infinitesimals as absurd was for a very different reason than initial objections to negative or complex numbers For the latter there was no problem in presenting formal systems with such numbers, but these new numbers went against the intuition of mathematicians, born by prejudice perhaps
calculus - Justification of algebraic manipulation of infinitesimals . . . Leibniz wrote in 1695 that his infinitesimals violate Euclid V 4 In modern terminology, this means that Leibnizian infinitesimals are "non-Archimedean" (this certainly does not mean that Leibniz was either familiar with or "anticipated" modern non-archimedean theories)
What is the product of infinitely many infinitesimals? Then according to the rule of the limits product, $\lim\limits_{n \to \infty}[a_1(n)a_2(n)]=0$, which shows that the product of two infinitesimals is an infinitesimal Thus, by induction, we can generalize the conclusion to the case when a finite number of infinitesimals multiply But what about the product of infinitely many infinitesimals?
Are infinitesimals still being used in calculus? $\begingroup$ @ParamanandSingh Oh, I agree with that completely, perhaps the way I said that was unclear When I see arguments involving infinitesimals, I convert them into arguments involving extremely tiny but finite numbers, and I replace the equals signs with approximately equals signs, and once we derive a formula that holds approximately I hope that "in the limit" the approximate