properties of hyperreals under field operations

Let ${}^{*}ℝ_b$ denote the set of finite (or limited) hyperreal numbers and ${}^{*}ℝ_0$ the set of infinitesimal hyperreal numbers.

- We have that

1. ${}^{*}ℝ_b$ and ${}^{*}ℝ_0$ are subrings of ${}^{*}ℝ$.

2. ${}^{*}ℝ_0$ is an ideal of ${}^{*}ℝ_b$.

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