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. 1.

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

2. 2.

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

3. 3.

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

4. 4.

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

5. 5.

the inverse of an infinite hyperreal is infinitesimal.

The above properties can be described more informally like:

1. 1.

finite $+$ finite $=$ finite

2. 2.

infinitesimal $+$ infinitesimal $=$ infinitesimal

3. 3.

infinite $+$ finite $=$ infinite

4. 4.

finite $\times$ finite $=$ finite

5. 5.

infinitesimal $\times$ finite $=$ infinitesimal

6. 6.

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

7. 7.

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

Title properties of hyperreals under field operations PropertiesOfHyperrealsUnderFieldOperations 2013-03-22 17:26:19 2013-03-22 17:26:19 asteroid (17536) asteroid (17536) 4 asteroid (17536) Result msc 26E35