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Let's define infinity ($\infty$) as the number larger than any finite number. Also, let's define the infinitesimal constant ($\epsilon$) as the smallest number greater than zero.

  1. What is $\frac{1}{\infty}$? Is it zero or $\epsilon$? And why?

  2. What is $\frac{1}{\epsilon}$? Isn't it infinity?

  3. Is $\epsilon$ considered finite?

EDIT: this isn't a duplicate of $\frac{1}{\infty}$ - is this equal $0$? because i'm trying to compare infinity with the infinitesimal. As far as I can see, the inverse of one yields the other and zero doesn't come into it. At least, that's what I'm asking about.

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DrZ214
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  • possible duplicate of $\frac{1}{\infty}$ - is this equal $0$? – Zev Chonoles Jun 12 '15 at 23:20
  • Neither even exist, so none of this can be answered? – theage Jun 12 '15 at 23:22
  • You might be interested in the Levi-Civita field. – Théophile Jun 12 '15 at 23:22
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    @ZevChonoles I'm not sure that this is actually a duplicate. Sure, the two questions are similar, but here an infinitesimal is considered, which makes this a somewhat different question. – A.P. Jun 12 '15 at 23:25
  • @theage abstract numbers "don't exist" either. The number 1, as an abstract value, doesn't exist unless you're talking about 1 apple, or some such thing. A similar thing can be said of the number zero. – DrZ214 Jun 12 '15 at 23:30
  • What do you mean by "finite"? Since you defined $\epsilon$ as an infintesimal it shouldn't be finite, should it? – A.P. Jun 12 '15 at 23:42
  • @A.P. In some sense no, but it also seems as if e should exist in R (set of real numbers), whereas infinity would not. The reason I'm asking these questions is because I don't have the answers yet and depending on which way I define them, I get contradictory results. – DrZ214 Jun 12 '15 at 23:47
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    @DrZ214 The difference is that the abstract quantities $1,0$ have formal definitions as real (and rational, and integral) numbers. In a field such as $\mathbb R$, it is explicitly provable that neither your $\epsilon$ nor your $\infty$ exist (eg. if $\epsilon$ existed then $\epsilon/2$ would be smaller, contradiction). However, amusingly in $\mathbb Z$ you have $\epsilon=1$, but presumably you're asking about fields. – theage Jun 12 '15 at 23:48

1 Answers1

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There is a problem with your definition of $\infty$.

You can't say that $\infty$ is the number larger than any finite number, because $\infty$ is not a number in the first place. You can't say that some set of numbers ($\mathbb N$ or $\mathbb R$, for example) contains $\infty$ if you want usual rules to hold; for example from $$\infty + a = \infty$$ Subtract $\infty$ to get

$$a = 0$$ for every $a$. This is clearly absurd.

So $\infty$ is not a number, and $\frac 1\infty$ does not mean anything. You can say that

$$\lim_{x\to \infty} \frac 1x = 0$$ but this has a very precise meaning and you should work with that.

This is defined as

$$\forall \epsilon > 0 \exists N: x > N \implies \left|\frac 1x\right| < \epsilon$$

It basically means: for very number greater than zero, if you take numbers that are big enough, eventually $\frac 1x$ will become smaller than the number you've chosen at the beginning.

You can see that this responds very well to what we think of infinity (numbers as big as you want!) but the actual object of $\infty$ is nowhere to be seen.

Note

I have talked about standard analysis. You can define this concepts in a meaningful way (see hyper reals) but again, you have to ask yourself: What am I trying to model? Will this be useful, other than consistent? To be honest, I will advise not to bother with such formulations at the beginning ;-)

Ant
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  • Then let me ask, can some set of numbers (N or R) contain $\epsilon$? In other words, is $\epsilon$ finite? – DrZ214 Jun 12 '15 at 23:28
  • @DrZ214 This is more interesting. In the usual analysis, there is no such thing as $\epsilon$ ("infinitesimal", one may say). But non-standard analysis (https://en.wikipedia.org/wiki/Non-standard_analysis) make this concept works, and it's fruitful. – Ant Jun 12 '15 at 23:30
  • I can't see why. "The smallest number greater than zero" seems pretty well-defined to me. Let me also ask what is the proper definition of infinity? It can't be the limit of 1/x as x goes to inifinity can it? That would be like a dictionary entry for the word "the" where its definition also contains the word "the". – DrZ214 Jun 12 '15 at 23:37
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    "The smallest number greater than zero" is not at all well-defined when speaking about real numbers. Suppose $x$ were such a number. Then $x/2$ is also a real number and $0 < x/2 < x$. Contradiction. – Simon S Jun 12 '15 at 23:38
  • @SimonS Same exact thing can be said of infinity. If x were that number, then x + 2 would be bigger. Yet infinity is considered a valid concept, but not the infinitesimal? – DrZ214 Jun 12 '15 at 23:40
  • Isn't the reciprocal of an infinity an infinitesimal in the hyper-real numbers? – A.P. Jun 12 '15 at 23:40
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    @DrZ214 As Simon S shows, it is not well defined. I also edited the answer and specified what $\lim_{x\to \infty}$ means; as you can see there is no circluar reasoning. I will suggest to stop trying to think of $\infty$ as a well-defined object, and accept that we define $\infty$ differently in different cases to suit our needs. $\infty$ is not a number. Is a valid concept that guides or intuition but we modify it from case to case to best suit our needs. – Ant Jun 12 '15 at 23:42
  • @A.P. I think that the OP is confused enough about this :-) I believe it's better to "forget" these different formulations at the beginning and to learn when one has a better grasp of math :) – Ant Jun 12 '15 at 23:44
  • @Ant can you also put an explanatory paragraph for that final definition? As far as I can interpret: For every number e greater than zero something something... – DrZ214 Jun 12 '15 at 23:44
  • @DrZ214 I did! Tell me if it's clear ;-) – Ant Jun 12 '15 at 23:48
  • I understand your point, but I do think that this answer doesn't address the question (although it addresses its spirit), which was about adjoining an infinity and an infinitesimal to an unspecified set of numbers. – A.P. Jun 12 '15 at 23:50
  • @Ant Okay it's clear. But then, can we take the mirror image of those symbols (or something similar like that) and arrive with a definition for the infinitesimal? I'm working on one as we speak. – DrZ214 Jun 12 '15 at 23:50
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    @DrZ214 Mirror images of what? Symbols don't mean anything. I can define $\infty$ to be equal to $4$ and all is good. Similar for infinitesimal. So the key point is: what do you want to model? Which are the properties that you think these new concepts should have? For the infinitesimal, I've already linked you the non-standard analysis page on wikipedia. For $\infty$, there are other constructions that more or less work (see A.P. 's comment) but I think that at this level you'll just confuse yourself. – Ant Jun 12 '15 at 23:56
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    The main point I want to make: There is no $\infty$ as a general entity. Just a multitude of definitions which try to grasp our intuition about what $\infty$ should mean in different contexts, an each comes with it's different definition – Ant Jun 12 '15 at 23:56
  • @A.P. Okay, I've added a link about hyperreals in the answer ;-) – Ant Jun 12 '15 at 23:58
  • @Ant Could we say, for every number greater than zero, if you take numbers that are small enough, eventually 1/x will become bigger than the number you've chosen at the beginning. Can this be used for the definition of an infinitesimal? Sorry I can't put it into math symbols since I don't know all of them yet. Also, can someone explain why all the downvotes? Was it because I asked 3 questions instead of 1? Sometimes 3 clear questions are better than 1 giant question, in order to separate concerns. – DrZ214 Jun 13 '15 at 00:06
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    @DrZ214 Of course we can say that. Formally, we write it $$\lim_{x \to 0} \frac 1x = \infty$$. But this is just a symbol; we haven't defined a "infinity" number only because it appears next to an equal symbol. It just means what you said; eventually, $\frac 1x$ will become as big as you want. That's it. The downvotes come because this has probably been asked many times on this site and in general reflect a poor understanding of what you're talking about (note that I didn't downvote, just supposing :) ) – Ant Jun 13 '15 at 00:09
  • @Ant That's too bad because I've learned a few things here. Of course I have a poor understanding of what I'm asking about. That's why I ask, because I don't know. It's funny cuz now I'm getting upvotes, and if every downvote is -2 but everyupvote is +5, ill actually have more rep than when I started lol. Also, thanks for that new limit. I can see now that reversing your original limit doesn't get me the infinitesimal, but rather something else involving infinity. – DrZ214 Jun 13 '15 at 00:16
  • @DrZ214 Glad to help :) Don't let downvotes discourage you ;-) math is awesome and this site is, too! :D – Ant Jun 13 '15 at 00:18
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    @DrZ214 Infinity, as "that which is bigger than any number" doesn't work well, algebraically, as has been shown here. Therefore it seems to only be an abbreviation for a certain behavior of limits. However, we can abandon the endeavor of making infinity be a number, and settle for it being an abstract "point" that ties up the reals in some sense. (...) – GPerez Jun 13 '15 at 00:28
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    @DrZ214 (...) It will not be a real number, rather a geometric (topological, to be precise) abstraction that makes the up-until-now infinite limits be definable without considering them as a special case. Also, these limits are all we needed infinity for to begin with, so there's no real let-down in not having infinity as an actual number. – GPerez Jun 13 '15 at 00:29