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I have heard this stuff called non-standard analysis. It introduces hyper reals $-$ an extension to real numbers $-$ to deal with infinitesimals. Now if you extend the real number line, how do you extend it? Shall we create another coordinate on $y$ axis (if our real number line is $x$ axis) OR shall we create a whole new infinitesimal number line at the point zero on real number line like: enter image description here

The latter seems intuitive but is there any problem in imagining it that way?

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    $\begingroup$ @Wojowu ... except the zero, that is considered an infinitesimal. $\endgroup$ – Masacroso Mar 25 '17 at 13:20
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    $\begingroup$ @Wojowu If your goal is to redevelop simple real analysis imo the most motivated definition is a number whose standard part is 0 (thus including both 0 and negatives). This opinion coming mostly from having done so myself and trying out various definitions (although I have references that use the same convention). $\endgroup$ – GPhys Mar 25 '17 at 13:36
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    $\begingroup$ @Masacroso I am aware of all the subtelties regarding the definitions of nonstandard analysis, but in any fixed such system saying "bigger than zero and smaller than all positive reals" does make sense. On the second thought though, I do have to agree with GPhys that taking his definition of an infinitesimal does seem to be the most natural one. I can't remember where, but I swear I have seen a definition somewhere in which positivity was required. $\endgroup$ – Wojowu Mar 25 '17 at 13:42
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    $\begingroup$ In the hyperreal number line, every real number (not just zero) is surrounded by a “halo” of hyperreal numbers which lie infinitesimally close to it. But the hyperreal line is not just “denser”, it's also “longer”, since there are also numbers greater than any real number, such as $\omega=1/\epsilon$ if $\epsilon$ is a positive infinitesimal. $\endgroup$ – Hans Lundmark Mar 25 '17 at 15:29
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    $\begingroup$ @Joe Try first chapter of math.wisc.edu/~keisler/calc.html $\endgroup$ – A---B Mar 26 '17 at 15:16
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The correct/usual way to think about it is the following number line \begin{equation*} [\, \underbrace{\cdots\quad-\nu-1\quad\phantom{+}-\nu\phantom{+\,}\quad-\nu+1\quad\cdots}_{\text{negative "infinite" hyperreals}} \,]\,\cdots\,[\, \underbrace{\cdots\,\,\,-1\quad0\quad1\quad \cdots}_{\text{usual reals}} \,]\,\cdots\,[\, \underbrace{\cdots\quad\nu-1\quad\phantom{+}\nu\phantom{+\,}\quad\nu+1\quad\cdots}_{\text{positive "infinite" hyperreals}} \,] \end{equation*} with the additional note that every point $c$ on the above number line is surrounded by a family of points infinitesimally close to it: \begin{equation*} \cdots\quad c-2\varepsilon\quad c-\varepsilon\quad\phantom{+}c\phantom{+\,}\quad c+\varepsilon\quad c+2\varepsilon\quad\cdots \end{equation*}

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  • $\begingroup$ +1 This is what I was trying to imagine and put as an answer. $\endgroup$ – user335907 Mar 26 '17 at 15:48
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By the transfer principle, anything you do with the standard reals looks exactly the same when done internally to the hyperreals.

In particular, there is a hyperreal number line. And (internally) it looks exactly like the standard number line.

The value of nonstandard analysis comes in comparing the real and hyperreal number lines. When overlaid atop one another, you'll find:

  • Every standard real number is also a hyperreal number
  • Every standard real number has a halo of hyperreal numbers surrounding it, and this halo doesn't contain any standard reals.
  • There are no "gaps"; if $a<x<b$ where $a,b$ are standard real and $x$ is hyperreal, then $x$ is in the halo of some standard real number
  • There are more hyperreals off to the right and left, larger in magnitude than anything in the halo of a standard real.

If you take the picture of the extended reals (and extended hyperreals) instead — that is, add $\pm \infty$ as the endpoints of the number line, then the hyperreals of the last bullet can be gathered up into the halos of $+\infty$ and $-\infty$.

(note, still, that every hyperreal is still smaller than $+\infty$; it's just that they're 'closer' to $+\infty$ than any standard real, or even any hyperreal in the halo of a standard real)

So, the picture you are trying to imagine looks fairly reasonable. Keisler's book uses something like that a lot, where you look at the standard number line, and then when desired, you use a "telescope" to "zoom in" on some point to see an infinitesimal segment of the hyperreal line.

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