properties of hyperreals under field operations

Let ${}^{*}\mathbb{R}_{b}$ denote the set of finite (or limited) hyperreal numbers and ${}^{*}\mathbb{R}_{0}$ the set of infinitesimal hyperreal numbers.

- We have that

1.

${}^{*}\mathbb{R}_{b}$ and ${}^{*}\mathbb{R}_{0}$ are subrings of ${}^{*}\mathbb{R}$ .
2.

${}^{*}\mathbb{R}_{0}$ is an ideal of ${}^{*}\mathbb{R}_{b}$ .
3.

the sum of an infinite hyperreal with a finite hyperreal is infinite.
4.

the inverse of a non-zero infinitesimal hyperreal is infinite.
5.

the inverse of an infinite hyperreal is infinitesimal.

The above properties can be described more informally like:

1.

finite $+$ finite $=$ finite
2.

infinitesimal $+$ infinitesimal $=$ infinitesimal
3.

infinite $+$ finite $=$ infinite
4.

finite $\times$ finite $=$ finite
5.

infinitesimal $\times$ finite $=$ infinitesimal
6.

infinitesimal ${}^{-1}$ $=$ infinite
7.

infinite ${}^{-1}$ $=$ infinitesimal

Title	properties of hyperreals under field operations
Canonical name	PropertiesOfHyperrealsUnderFieldOperations
Date of creation	2013-03-22 17:26:19
Last modified on	2013-03-22 17:26:19
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	4
Author	asteroid (17536)
Entry type	Result
Classification	msc 26E35