I How to find a point that corresponds to hyperreal number?

[Mike_bb]

TL;DR Summary

How to take random point that corresponds to hyperreal number on the hyperreal line?

Hello!

I considered number line and I couldn't take random points that correspond to hyperreal numbers on the hyperreal line. Instead it was points A, B, C that correspond to real numbers.

How to take random point that corresponds to hyperreal number on the hyperreal line?

Thanks.

 

[Hill]

It seems trivial to me that if this is a hyperreal line then they are hyperreal numbers.

 

[Mike_bb]

It seems trivial to me that if this is a hyperreal line then they are hyperreal numbers.

Ok. How to take infinitesimal numbers on hyperreal line?

 

[pbuk]

How to take random point that corresponds to hyperreal number on the hyperreal line?

Using the definition of a (finite) hyperreal as $r + k \epsilon$ where $\epsilon$ is the smallest infinitessimal, take a random $r \in \mathbb R$ and $k \ne 0 \in \mathbb Z$.

Edit: I realise that I have used a non-standard definition of the hyperreals above: for the standard definition omit $r$: there are no non-real hyperreals $h: \epsilon < h < \frac 1 \epsilon$.

 

[Mike_bb]

Using the definition of a (finite) hyperreal as $r + k \epsilon$ where $\epsilon$ is the smallest infinitessimal, take a random $r \in \mathbb R$ and $k \ne 0 \in \mathbb Z$.

It's algebraic definition of hyperreal numbers. But how will it look on the hyperreal line?

 

[pbuk]

It's algebraic definition of hyperreal numbers. But how will it look on the hyperreal line?

Like any other number on the hyperreal line. You do realise that the hyperreal "line" cannot be represented by a single line: it looks more like this:
https://en.wikipedia.org/wiki/Hyperreal_number#/media/File:Números_hiperreales.png

 

[Mike_bb]

Like any other number on the hyperreal line. You do realise that the hyperreal line is not a single line and looks more like this:
https://en.wikipedia.org/wiki/Hyperreal_number#/media/File:Números_hiperreales.png

It needs infinitesimal microscope to see this infinitesimal number.

 

[pbuk]

It needs infinitesimal microscope to see this infinitesimal number.

Yes it does.

View attachment 343874

I don't think that is the standard representation of the hyperreal line: where would $\frac \omega 2$ be found?

In the standard representation the reals are all clustered around 0.

 

[Erland]

Using the definition of a (finite) hyperreal as $r + k \epsilon$ where $\epsilon$ is the smallest infinitessimal, take a random $r \in \mathbb R$ and $k \ne 0 \in \mathbb Z$.

Edit: I realise that I have used a non-standard definition of the hyperreals above: for the standard definition omit $r$: there are no non-real hyperreals $h: \epsilon < h < \frac 1 \epsilon$.

Smallest infinitesimal?? There is no such thing.

More precisely, there is no smallest positive infinitesimal. If there was one, say $\varepsilon$, then $\varepsilon/2$ would be a smaller positive infinitesimal.

And of course there are non-real hyperreals between the positive infinitesimal $\varepsilon$ and the infinite $1/\varepsilon$. For example: $2\varepsilon$.

 

[Erland]

It is not clear to me what the OP means by "taking random points that corresponds to hyperreal numbers on the hyperreal line". In the literal sense, we need a probability distribution to do this, and no one is given.

I think that what the OP wants is an explicit example of a (positive) infinitesimal. To give such an example, we need a model of the hyperreal line. We can talk about the hyprerreals figuratively by the microscope and telescope in a previous post, but although this is a useful intuitive way of thinking of hyperreals, it is not rigorous.

The most common way of constructing a model of the hyperreal line is the ultrapower construction.
In this construction, we must use the Axiom of Choice (or at least some weaker version of it, I don't know exactly which one is the weakeast possible). This means that we cannot specify the hyperreals completely, but there is a non-constructive element in the model.

We use the Axiom of Choice to establish the existence of a nonprincipal ultrafilter on the set $\Bbb N$ (natural numbers). Given such an ultrafilter $\cal U$, we define an equivalence relation on the set of infinite sequences of real numbers, such that two sequences $(r_1, r_2, r_3,\dots)$ and $(s_1, s_2, s_3,\dots)$ are considered as equivalent if the set of numbers $n$ such that $r_n=s_n$ belongs to the ultrafilter $\cal U$. A hyperreal is then defined as an equivalence class given by this equivalence relation.
We identify each real number $r$ by the equivalence class containing the constant sequence $(r,r,r,\dots)$.
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An example of an infinitesimal is then (the equivalence class containing) $(1, 1/2, 1/3, 1/4, \dots)$. For any positive real number $r$, $1/n < r$ for all but finitely many $n$, and the set of all $n$ for which this holds belongs to $\cal U$, which means that $(1, 1/2, 1/3, \dots) < (r,r,r,\dots)$ as hyperreals (more precisely, this holds for their equivalence classes). Since this holds for all positive reals $r$, (the equivalence class of) $(1, 1/2, 1/3, \dots)$ is infinitesimal.

For the details, see https://en.wikipedia.org/wiki/Hyperreal_number

 

[pbuk]

Smallest infinitesimal?? There is no such thing.

Yes of course you are right, I am not sure why I posted that.