The set of all hyper-real numbers is denoted by $R^*$. Every real number is a member of $R^*$, but $R^*$ has other elements too. The infinitesimals in $R^*$ are of three kinds: positive, negative and zero.
I think zero is not an extension in the set of real numbers.
Question 1: Can we call any real number a hyper-real number, too? For example, $2$ is a real number, can we say that $2$ is a hyper-real number?
Question 2: Does the set of hyper-real numbers $R^*$ include such infinitesimals say $\epsilon$, such that $-a<\epsilon<a$ for every positive real number $a$?
Addition: Is it true that if $\epsilon$ is a positive infinitesimal, then $\epsilon>0$. However, $-\epsilon$ which is a negative infinitesimal is less than zero. But $0, \epsilon$ and $-\epsilon$ are greater than any negative real number and are less than any positive real number?