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 ...
How do you understand Infinitesimals? - Mathematics Stack …
2017年8月3日 · 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 ...
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).
Differential and Infinitesimals - Mathematics Stack Exchange
Differential Equations and Infinitesimals. 5. Does 3Blue1Brown's series on Calculus : Essence of Calculus ...
Newest 'infinitesimals' Questions - Mathematics Stack Exchange
For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also ...
What is the answer to the paradox of the infinitesimal?
Take a line one unit long. You can divide it into n parts each of length 1/n. As n increases the parts get smaller, but at the same time to total length is always 1 unit.
number theory - Properties of infinities, infinitesimals
2019年7月14日 · $\begingroup$ @Eliot : There is the Nelson variant of NSA where infinitesimals and infinities are just "not observable" members of the real number line, numbers that will never occur in numerical computation, regardless of what multi-precision mechanism you use. Astonishingly, such a separation of numbers in standard and non-standard ones can ...
calculus - Book recommendation on infinitesimals - Mathematics …
2015年5月13日 · Infinitesimals tend to get used loosely by physicists and there isn't really a rigorous theory behind how they are, in practice, used. It's something you pick up "on the job" so to speak. Having said that, many lines of reasoning involving infinitesimals can be made rigorous in various ways.
calculus - Is $0$ an Infinitesimal? - Mathematics Stack Exchange
2014年12月31日 · Since these numbers (infinitesimals) are very small than most of the real numbers but not equal to zero, these numbers fall into a unique category of real numbers "Hyperreal Numbers" which tend to either zero or infinity. So, the answer is maybe Yes.
What is the product of infinitely many infinitesimals?
2018年12月19日 · 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?