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Questions tagged [infinitesimals]

For questions about infinitesimals, both in an intuitive sense as well as more rigorous settings (see also [nonstandard-analysis]).

Intuitively, an infinitesimal is an infinitely small number. They played a significant role in both Leibniz's and Newton developments of calculus, as well as in the work of Archimedes.

For a more rigorous treatment of infinitesimals, also see .

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Do Cantor's infinities imply a multitude of infinitesimals?

Cantor showed that a multitude of infinities exist. $\aleph_0, \aleph_1, \aleph_2, \ldots$ and so on. Are there a multitude of infinitesimals as well? My notion would be that the "normal" infinitesimal, $\epsilon$, would really be $\epsilon_0$ in a…
MaxW
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Non-Standard analysis and infinitesimal

Can someone please explain how Non Standard Analysis is used to justify infinitesimals? I am not very clear about this but apparently it has something to do with hyperreals.
jjjesse
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How does exponentiation work with infinitesimal hyperreal numbers

Given a hyperreal infinitesimal number $\epsilon$ , is it meaningful to take its square root, $\sqrt{\epsilon}$ or any other root? What about using it as an exponent, as in $2^{\epsilon}$ ? And what about something like $\epsilon^{\epsilon}$ . I've…
bowsersenior
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Ceiling function of an infinitesimal

I was working with infinitesimals and I came across the problem: what is the ceiling function of an infinitesimal? Wolfram Alpha says an infinitesimal equals zero, so therefore the ceiling function of it should equal zero. But what I've seen is that…
Revilo
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Dealing with double infinitesimal quantities?

I have been working on some physics problems, and have realised that I sometimes write a single infinitesimal (or delta) quantity that actually is a product of two independent delta quantities, e.g. $dA=r dr d\phi$ for an area, but equally we have…
Meep
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How do you understand Infinitesimals?

I've often heard that $\dfrac{1}{3} = 0.\overline{3} $, and $\dfrac{1}{3} \cdot 3 = 1$, yet $0.\overline{3}\cdot 3 = 0.\overline{9} \neq 1$. There is an $\epsilon$ (infinitesimal) thrown in there as well. How do you understand these extremely small…
Shreyas Shridharan
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The area $A$ of metric elements $(dx)^2$

I define a metric: $$ (d s)^2=(d x)^2+(d y)^2 $$ As per the pythagorean theorem, all terms of the infinitesimal metric $(ds)^2$, $(dx)^2$ and $(dy)^2$ are (infinitesimal) areas. I am trying to connect the terms to their corresponding area, but I am…
Anon21
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Suggestions for defining the exponential function for infinitesimals

I am currently working on "my own" system of infinitesimals. They are pretty much analogous to dual numbers, except for the fact that mine are not nilpotent. My main question concerns possible extensions of the exponential function, but before that,…
naytte2
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Can infinitesimals be limits to sequences?

[EDIT] Is zero the limit to the following sequence? $1/2, 1/4, 1/8, 1/(2^4),.....$ Why it shouldn't be $1/\aleph_0$? Since $\aleph_0$ is the limit to sequence: $2,4,8,2^4,..,2^n,...$ $1/ \aleph_0$ can be understood as an infinitesimal, i.e. larger…
Zuhair
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Equivalent infinitesimal for $\log(\cos(x))$

I recently came across a limit problem which replaces $\log(\cos(x))$ with an equivalent infinitesimal $\cos(x) - 1$. How do we prove that $\cos(x)-1$ is the equivalent infinitesimal for $\log(\cos(x))$?
Sandeep Deb
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Attempting to express infinitesimals using Arabic numerals

I've been researching infinitesimals in my spare time, and have come to an hypothesis. Is it fair to say that $\varepsilon = 0.\bar01$? In English: the infinitesimal $\varepsilon$ is equal to an infinite amount of zeros after the decimal point,…
HarrisonO
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Problem with recurring decimals

Let's take $ \frac{1}{3} $. Writing this out as a decimal, we say that $ \frac{1}{3} = 0.333\ldots $, but this is where my problem lies. No matter where you cut off the continuous stream of $ 3 $s, it's still inaccurate. $ 0.3333 $ is inaccurate,…
VortixDev
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proof that $ \lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = \infty $

I started learning infinitesimally math and I have the following question: Is the following sentence true $ \lim \limits_{n \to \infty} \sqrt[n]{n^5 -2n + 7} = 1 $ I can see that it tends to $\infty$ but I can't prove it. How can I prove that this…
MyNick
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If a square has a concrete area of $2 m²$, how long is its side?

If we make a square with an area of $2 m²$, its side is square root of $2$ then. Wouldn't that mean that the square root of 2 has a concrete length (and therefore point in the real numbers axis). We could measure the side and draw the conclusion…
Carlos Perez
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Using any number system, Is there a more interesting answer for 0.999...^infinite than undefined?

I was reading up on 0.999... = 1 and since my background is not in math, my mind started coming up with in my mind, intuitive, but misguided wrong problems. After some wolfram alpha i came to the following results: 1^infinite is either undefined or…
Mth
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