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?

  • 2
    $\begingroup$ 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. $\endgroup$
    – user82196
    Oct 23, 2013 at 4:55
  • $\begingroup$ 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. $\endgroup$ Oct 23, 2013 at 4:56

2 Answers 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.


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.


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