How does this work exactly, since you can pick a real number to be as close to zero as you want?
In Non-Standard Calculus the notion of an 𝒊𝒏𝒇𝒊𝒏𝒊𝒕𝒆𝒔𝒊𝒎𝒂𝒍 is formalized
One such formalization is the extension of ℝ → *ℝ
*ℝ is called the hyperreals
In *ℝ an infinitesimal d𝑥 is a number that is greater than zero, but smaller than any other real numberpic.twitter.com/m6c1462ccf
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Informally, you can think of this as saying "no you actually get as close as you want. The closest you can get is infinitesimally close" That is there is a "smallest" number that is nonzero in the hyperreals
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I got questions! Are higher powers of infinitesimals defined in *R and also strictly smaller than lower powers? Are the Functions in your example defined viantaylor expansion?
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Right.. I believe in the hyperreals the differentials are not nilpotent.. so they always hang around. My understanding is that infinitesimal*infinitesimal = infinitesimal
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i actually learned calculus through hyperreal numbers. to this day i still can’t help but see 𝜀,𝛿 limits as anything but a special case
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That's awesome! I think it's way better pedagogically
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Worth mentioning that there is an algebraic approach to infinitesimals too, as nilpotent elements of certain fields. I could not find an open access version, but you get the gist of the idea:https://link.springer.com/chapter/10.1007/3-540-26474-4_7 …
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My understanding is that the approach of nilpotent differentials is not widely used because it takes some fancy logical footwork to avoid contradictions
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