Thread: ? about .999~=1
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Hurkyl
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Jan8-07, 04:02 PM
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Quote Quote by mr_homm View Post
BTW, could you give me a reference on the hypernaturals indexing the hyperreals?
It's a direct application of the transfer principle.

In the standard model, the decimal expansion of a real number s is nothing more than a function f:Z->{0, ..., 9} satisfying

[tex]s = \sum_{n \in \mathbb{Z}} f(n) 10^n[/tex]
[tex]\lim_{n \rightarrow +\infty} f(n) = 0[/tex]

and we have a theorem that every real number has a decimal expansion.


Applying the transfer principle tells us that in the nonstandard model, the hyperdecimal expansion of a hyperreal number s is nothing more than an (internal) function f:*Z->{0, ..., 9} satisfying

[tex]s = \sum_{n \in {}^\star \mathbb{Z}} f(n) 10^n[/tex]
[tex]\lim_{n \rightarrow +\infty} f(n) = 0[/tex]

and we have a theorem that every hyperreal number has a hyperdecimal expansion.