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I'm revisiting Calculus and learning more about infinitesimals. As I concluded one of the reasons they introduced infinitesimals is because of "rounding real numbers problem" (read third paragraph). For example, $2+10^{-(10^{10^{10}})}$ is close to $2$, but still it is equal $2+10^{-(10^{10^{10}})}$ not $2$. In other words we can say $\displaystyle2+10^{-(10^{10^{10}})}{\simeq2}$ but not $\displaystyle2+10^{-(10^{10^{10}})}=2$.

In contrast, if $\mathrm{d}x$ is infinitesimal or, by definition, number which is larger than zero and less than any real number, then we can neglect that term and say $\displaystyle2+\mathrm{d}x=2$. Let's imagine for a moment that $\mathrm{d}x=10^{-(10^{10^{10}})}$. Isn't it sufficiently good to use such number to be neglected, so we can obtain "slope of a curve at a point"? In other words if $f(x)=x^2$ and $\mathrm{d}x=10^{-(10^{10^{10}})}$ then $\displaystyle\frac{f(x_0+\mathrm{d}x)-f(x_0)}{\mathrm{d}x}=2\cdot x_0+\mathrm{d}x\simeq2\cdot x_0$. We rounded number $2\cdot x_0+\mathrm{d}x$ to $2\cdot x_0$.

This might sound weird in context but what kind of effects does rounding real numbers (such as number $\pi$ to $3.14$) have on whole idea of real number system? If it doesn't have any, then why do we introduce infinitesimals at all? We could use any sufficiently good $\mathrm{d}x$ where $\mathrm{d}x$ is real number (such as $\mathrm{d}x=10^{-(10^{10^{10}})}$) and get slope of a curve at each point using only real numbers and not including hyperreals! My amateur opinion is: if you could round every number then there will be "gaps" on real number line (and I know in layman terms that real numbers represent continuous number line). Hence we introduce infinitesimals as extension to real numbers where infinitesimals are not in a domain of real numbers.

Krushe
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  • 10^10^10^10 is a very big number, not a very small one – spaceisdarkgreen Feb 28 '18 at 17:49
  • @spaceisdarkgreen edited, thanks! – Krushe Feb 28 '18 at 17:53
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    Also, standard rigorous treatments of calculus don't have infinitesimals at all. They just talk about sufficiently small and sufficiently large things. Usually with statements of the form "for all $\epsilon>0,$ there exists a (sufficiently small) $\delta>0$ such that..." This is usually called the epsilon-delta framework. (Rigorous development of calculus using infinitesimals has been done, but it's not usually the way it's taught.) – spaceisdarkgreen Feb 28 '18 at 17:56
  • @spaceisdarkgreen but how much is "sufficiently small" and "sufficiently large" things? in standard rigorous treatments are they real numbers or? – Krushe Feb 28 '18 at 17:57
  • yes, $\delta$ is a real number. However you can always find a rational number smaller than any real number so you can usually choose it to be a rational number if you want to for some reason. – spaceisdarkgreen Feb 28 '18 at 18:03
  • It was largely due to the work of Karl Weierstrass that infinitesimals were eliminated from analysis. The use of $\epsilon, \delta$ proofs eliminated nonsense such as "dx is the smallest positive number greater than $0$." – John Wayland Bales Feb 28 '18 at 18:09
  • Sounds like you might be interested in non-standard analysis: https://en.wikipedia.org/wiki/Non-standard_analysis. – Adrian Keister Feb 28 '18 at 18:14
  • @spaceisdarkgreen aren't rational numbers at the same time real numbers also? – Krushe Feb 28 '18 at 18:28
  • @AdrianKeister thanks! – Krushe Feb 28 '18 at 18:29
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    If $dx$ is an infinitesimal then you can't write $dx=10^{-10^{10^{10}}}$. Infinitesimals are not small real numbers but are roughly sequences of real numbers which tend to $0$. Most of the confusion here is primarily the result of treating infinitesimals on par with real numbers. – Paramanand Singh Feb 28 '18 at 19:02
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    The idea of rounding off is a valid concept in infinitesimal calculus and more properly goes by the name "standard part" function but it applies to hyperreal numbers and gives real numbers as output. The rounding of real numbers is something different and is more related to practical application of mathematics. – Paramanand Singh Feb 28 '18 at 19:06
  • @ParamanandSingh I understand you bro, that's why I wrote: "If we let (for a moment)..". I'll look that up! – Krushe Feb 28 '18 at 19:13
  • While I am a fan of nonstandard analysis myself, I'd actually strongly caution against going too far down that rabbit hole before learning standard real analysis. There are some major conceptual difficulties with making nonstandard analysis coherent and I'd say it's better to wrestle with $\epsilon - \delta$ proofs before tackling nonprincipal ultrafilters and the like needed to construct the hyperreals. – Christian Sykes Mar 01 '18 at 03:30
  • Also I think you'll find that any attempt to pin down what "sufficiently good" means in general will lead you right to limits, which is how we get around all the conceptual strangeness you're puzzling over. – Christian Sykes Mar 01 '18 at 03:36
  • @ChristianSykes Thank you! – Krushe Mar 01 '18 at 09:59

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This is at least in part a philosophical question as well as a mathematical one.

Mathematicians have worked for centuries to build a logical foundation for the mathematical object we refer to as "the real line". The traditional view today is that the way to deal with small numbers is to use "epsilon-delta" proofs for limits. These replace the idea of "infinitely small" by "arbitrarily small" - infinitely many inequalities instead of single equalities involving infinitesimal numbers. There's a way to do it with actual infinitesimals you hint at in your last sentence. It's called "nonstandard analysis".

In the meanwhile, mathematicians starting as early as Archimedes and continuing through Leibniz and Newton to the present day have developed calculus for both theoretical and practical purposes even while the foundations were under construction. Sometimes that's referred to as "sufficient unto the day is the rigor thereof."

When actual numerical calculations are required, whether by Gauss computing orbits or today in computers, mathematicians carefully analyze the rounding errors that can occur when finite decimals are used to approximate theoretical "real numbers". Sometimes that's a bit like choosing a very small value for $dx$.

Related: Why can't the second fundamental theorem of calculus be proved in just two lines?

Ethan Bolker
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