Do Hyperreal numbers include infinitesimals?

Question

According to definition of Hyperreal numbers

The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + 1 + ...... + 1.[this definition has been extracted from wiki encyclopedia Hyperreal number]

According to above statement Hyperreal numbers include only infinite numbers and doesn't include their reciprocals, infinitesimals(correct me if I am wrong in saying this).

I have come across another statement mentioned in the same wiki encyclopedia which states that

The idea of the hyperreal system is to extend the real numbers R to form a system *R that includes infinitesimal and infinite numbers, but without changing any of the elementary axioms of algebra.[this statement has been extracted from wiki encyclopedia Hyperreal number/The transfer principle]

According to above statement Hyperreal numbers include both infinitesimal and infinite numbers,which contradicts the definition of Hyperreal numbers.So,what does it mean?Do Hyperreal numbers include infinitesimals?

Answer 1

Yes, they do: if $x\in{}^*\Bbb R$ is greater than any ordinary integer, then $\frac1x$ is necessarily a positive infinitesimal. There is no contradiction: the first statement doesn’t mention the infinitesimals explicitly, but the very next sentence does:

Such a number is infinite, and its reciprocal is infinitesimal.

Answer 2

Nothing in the first statement says that there are no infinitesimals, it just asserts the existence of infinite numbers. If the hyperreals are to be a field contianing infinite numbers, it must also contain their inverses, which are infinitesimals.