# Do Hyperreal numbers include infinitesimals?

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?

• The first definition says they're an extension of the reals which include infinite numbers. Of course, if you're viewing the real numbers as a field, this extension must contain inverses for all the infinite numbers you're adding. – user82196 Oct 23 '13 at 4:55
• They definitely include infinitesimals. I don't see that the description you quote excludes infinitesimals, but it might give the impression that it does. It should therefore be corrected. – André Nicolas Oct 23 '13 at 4:56

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: