How to find a point that corresponds to hyperreal number?

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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.

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

Thanks.
 
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It seems trivial to me that if this is a hyperreal line then they are hyperreal numbers.
 
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  • #3
Hill said:
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?
 
  • #4
Mike_bb said:
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 ##.
 
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pbuk said:
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?
 
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  • #8
Mike_bb said:
It needs infinitesimal microscope to see this infinitesimal number.
Yes it does.

Mike_bb said:
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.
 
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pbuk said:
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##.
 
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  • #10
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)##.
al
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
 
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  • #11
Erland said:
Smallest infinitesimal?? There is no such thing.
Yes of course you are right, I am not sure why I posted that.
 

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