Let ${}^*\mathbb{R}_b$ denote the set of finite (or limited) hyperreal numbers and ${}^*\mathbb{R}_0$ the set of infinitesimal hyperreal numbers.

Proposition - 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