# Tagged Questions

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### infinitisimal part and the directional integral

In the paper A circle detection approach based on Radon Transform by Erman Okman and Gozde B. Akar. I have a few questions on some basics. first of all what does $$ds^2 = dx^2 + dy^2$$ ...
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### Newton's “Famous Blunder”?

On page $225$ of Isaac Newton on Mathematical Certainty and Method by Niccolo Guicciardini (see here for a link), I read In the following demonstration... Newton made a famous blunder... He wrote, ...
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### What problems arise when using infinitesimals in calculus?

In contemporary real analysis we use a limit definition in terms of deltas and epsilons. Before that, people used infinitesimals to calculate limits. Is there a specific non-philosophical reason why ...
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### Could we assign a numerical value to an infinitesimal?

The common definition of 'infinitesimal' is that they are quantities that are too small to be measured or perceived. If we base the mathematically definition entirely on the common definition it ...
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### Help with this simple limit question

We have this definition: $$f(h)=O(h^p) \quad\text{if} \quad \lim_{h\rightarrow0}\frac{f(h)}{h^p}=K\neq 0$$ show this: $$O(h^2)+O(h^3 )=O(h^2)$$ So what we need to do is follow the definition: ...
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I just read this article on npr, which mentioned the following question: You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're ...
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### Definition of tangent

What is the formal definition of a tangent to a curve? The only one I can find is that it is a straight line drawn between two infinitely close points on the curve.
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### In-depth explanation of infinitely small differences?

As in this question's title, could anyone give me an explanation of an infinitely small difference and how one would calculate it (if one could even do that)? I've been trying to learn calculus for a ...
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### Muddled Notions Regarding the Measurement of Quantities in Different Dimensions

In my calculus class, I've learned a lot about finding areas/volumes of various shapes by summing up infinitely small slices of 'something' and adding them all up. This is very interesting to me, but ...
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### Functions and Infinitesimals?

My textbook says the following: $ads = v dv$ where $a$, $s$, $v$ are functions. I was introduced to integral calculus a semester ago, and Riemann Sums (and integrals) are the only notion of ...
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I know differentials (in a way of standard analysis) are not very rigorous in mathematics, there are a lot of amazing answers here on the topic. But what about line element? $$ds^2 = dx^2 + dy^2 ... 2answers 803 views ### Are infinitesimals dangerous? Amir Alexander is a historian of mathematics. His new book is entitled "Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World". See here. Two questions: (1) In what sense are ... 1answer 236 views ### infinitesimal calculus I have been reading some non-standard analysis from Keisler's book and I think it is logically consistent till now but there are criticisms against it and why isn't non-standard analysis accepted more ... 1answer 112 views ### Non-standard analysis way of proving that derivative of e^x is e^x What is the non-standard (infinitesimal) analysis way of proving that the derivative of e^x is e^x? I tried to prove it myself, but I am having a hard time proving this without recourse to ... 7answers 250 views ### Intuition for a real line vs. a “hyperreal line” I am a student of pure mathematics but I have no formal background in nonstandard analysis. I came across the concept of a hyperreal field recently, read just a little about them, and followed the ... 3answers 465 views ### Who gave you the epsilon? Who gave you the epsilon? is the title of an article by J. Grabiner on Cauchy from the 1980s, and the implied answer is "Cauchy". On the other hand, historian I. Grattan-Guinness points out in his ... 3answers 146 views ### non-archimedean in lay terms I've dabbled with studying infinitesimals off and on for years ... Robinson, Keisler, Bell ("Smooth Worlds"), etc., even a bit of category theory. But I'm not a mathematician and tend to jump in way ... 1answer 98 views ### sum of legs of inscribed right triangle Consider a right and isoceles triangle ABC inscribed in a circle such as its hypotenuse forms the diameter AB of the circle (the right vertex is thus at the "apex" of the circle). If we procede to an ... 2answers 291 views ### Definition of the integral in non-standard calculus In the sources I've seen, the integral is defined in non-standard calculus as the hyperreal extension of a function related to Riemann sums. E.g., Let$$ S(\Delta x) = \sum_{a}^{b}f(x)\Delta x$$be a ... 6answers 994 views ### What is the meaning of infinitesimal? I have read that an infinitesimal is very small, it is unthinkably small but I am not quite comfortable with with its applications. My first question is that is an infinitesimal a stationary value? It ... 6answers 893 views ### What does limit actually mean? I have been in a deep confusion for about a month over the topic of limits! According to our book the limit at a is the value being approached by a function f(x) as x approaches a I have a ... 1answer 318 views ### Calculus over \mathbb{Q} The mismatch between the sensitivity of 'mathematical calculus' and the flexibility of 'real world calculus' has been bothering me a bit recently. What I mean is this: in the real world, I can trust ... 1answer 133 views ### Proof that 0 = 1 or that \frac1\infty = 0 [closed] Proof that 0.9\ldots = 1: Let x = 0.999\ldots$$10x = 9.999\ldots\\ 9x = 9.999\ldots - 0.999\\ x = 1$$Proof 0 = 1:$$1 - \frac{1}\infty = 0.999\ldots\\ 1 - \frac{1}\infty = 1\\ ...
Can such definition of infinitesimals hold? $$\mathrm{d} x :=a:(a>0 \;\And\; \forall b \in \mathbb{R}^+\backslash \{ a \}\;(a<b))$$ And, if the above definiton works, then obviously ...