PlanetMath: structure of finite hyperreal numbers

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structure of finite hyperreal numbers

(Theorem)

Theorem - Every finite (or limited) hyperreal number $x \in {}^*\mathbb{R}$ admits a unique decomposition of the form

$\displaystyle x = a + \epsilon$

where $a \in \mathbb{R}$ and $\epsilon$ is infinitesimal.

Remark : This theorem just says that every finite hyperreal number has a real part and an infinitesimal part (just like real and imaginary parts in complex numbers).

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26E35 (Real functions :: Miscellaneous topics :: Nonstandard analysis)

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