1.3: The Hyperreals

We will let \(\mathbb{R}\) denote the set of all real numbers. Intuitively, and historically, we think of these as the numbers sufficient to measure geometric quantities. For example, the set of all rational numbers, that is, numbers expressible as the ratios of integers, is not sufficient for this purpose since, for example, the length of the diagonal of a square with sides of length 1 is the irrational number \(\sqrt{2}\). There are numerous technical methods for defining and constructing the real numbers, but, for the purposes of this text, it is sufficient to think of them as the set of all numbers expressible as infinite decimals, repeating if the number is rational and non-repeating otherwise.

A positive infinitesimal is any number \(\epsilon\) with the property that \(\epsilon>0\) and \(\epsilon<r\) for any positive real number \(r\). The set of infinitesimals consists of the positive infinitesimals along with their additive inverses and zero. Intuitively, these are the numbers which, except for \(0,\) correspond to quantities which are too small to measure even theoretically. Again, there are technical ways to make the definition and constrution of infinitesimals explicit, but they lie beyond the scope of this text.

The multiplicative inverse of a nonzero infinitesimal is an infinite number. That is, for any infinitesimal \(\epsilon \neq 0,\) the number

\[N=\frac{1}{\epsilon}\]

is an infinite number.

The finite hyperreal numbers are numbers of the form \(r+\epsilon,\) where \(r\) is a real number and \(\epsilon\) is an infinitesimal. The hyperreal numbers, which we denote \(^{*} \mathbb{R}\), consist of the finite hyperreal numbers along with all infinite numbers.

For any finite hyperreal number \(a,\) there exists a unique real number \(r\) for which \(a=r+\epsilon\) for some infinitesimal \(\epsilon .\) In this case, we call \(r\) the shadow of \(a\) and write

\[r=\operatorname{sh}(a) .\]

Alternatively, we may call sh \((a)\) the standard part of \(a\).

We will write \(a \simeq b\) to indicate that \(a-b\) is an infinitesimal, that is, that \(a\) and \(b\) are infinitesimally close. In particular, for any finite hyperreal number \(a,\) \(a \simeq \operatorname{sh}(a)\).

It is important to note that

• if \(\epsilon\) and \(\delta\) are infinitesimals, then so is \(\epsilon+\delta\)
• if \(\epsilon\) is an infinitesimal and \(a\) is a finite hyperreal number, then \(a \epsilon\) is an infinitesimal, and
• if \(\epsilon\) is a nonzero infinitesimal and \(a\) is a hyperreal number with \(\operatorname{sh}(a) \neq 0\) (that is, \(a\) is not an infinitesimal), then \(\frac{a}{\epsilon}\) is infinite.

These are in agreement with our intuition that a finite sum of infinitely small numbers is still infinitely small and that an infinitely small nonzero number will divide into any noninfinitesimal quantity an infinite number of times.

Exercise \(\PageIndex{1}\)

Show that \(\operatorname{sh}(a+b)=\operatorname{sh}(a)+\operatorname{sh}(b)\) and \(\operatorname{sh}(a b)=\operatorname{sh}(a) \operatorname{sh}(b)\), where \(a\) and \(b\) are any hyperreal numbers.

Answer

Let \(a=r_{1}+\epsilon_{1}\) and \(b=r_{2}+\epsilon_{2},\) where \(r_{1}\) and \(r_{2}\) are real numbers and \(\epsilon_{1}\) and \(\epsilon_{2}\) are infinitesimals. Note that \(a+b=\left(r_{1}+r_{2}\right)+\left(\epsilon_{1}+\epsilon_{2}\right)\) and \(a b=r_{1} r_{2}+\left(r_{1} \epsilon_{2}+r_{2} \epsilon_{1}+\epsilon_{1} \epsilon_{2}\right)\).

Exercise \(\PageIndex{2}\)

Suppose \(a\) is a hyperreal number with \(\operatorname{sh}(a) \neq 0 .\) Show that \(\operatorname{sh}\left(\frac{1}{a}\right)=\frac{1}{\operatorname{sh}(a)}\).

Answer

Let \(a=r+\epsilon,\) where \(r \neq 0\) is a real number and \(\epsilon\) is an infinitesimal. Note that

\[\frac{1}{a}=\frac{1}{r}+\frac{\epsilon}{r(r+\epsilon)}.\]