 Now suppose that the set \(\bbN\) of natural numbers is a member of \(U\). Then so is the set \(\Re\) of real numbers, since each real number may be identified with a set of natural numbers. \(\Re\) may be regarded as an ordered field, and the same is therefore true of its inflate \(\hat{\Re}\), since the latter has precisely the same first-order properties as \(\Re\). \(\hat{\Re}\) is called the hyperreal line, and its members hyperreals. A standard hyperreal is then just a real, to which we shall refer for emphasis as a standard real. Since \(\Re\) is infinite, nonstandard hyperreals must exist. The saturation principle implies that there must be an infinite (nonstandard)  hyperreal,[42]  that is, a hyperreal \(a\) such that \(a \gt n\) for every \(n \in \bbN\). In that case its reciprocal 1/\(a\) is infinitesimal in the sense of exceeding 0 and yet being smaller than 1/\(n+1\) for every \(n \in \bbN\). In general, we call a hyperreal a infinitesimal if its absolute value \(\lvert a\rvert\) is less than 1/\(n+1\) for every \(n \in \bbN\). In that case the set \(I\) of infinitesimals contains not just 0 but a substantial number (in fact, infinitely many) other elements. Clearly \(I\) is an additive subgroup of \(\Re\), that is, if \(a, b \in I\), then \(a - b \in I\).