Skip to main content
\(\newcommand{\vf}[1]{\mathbf{\boldsymbol{\vec{#1}}}} \renewcommand{\Hat}[1]{\mathbf{\boldsymbol{\hat{#1}}}} \let\VF=\vf \let\HAT=\Hat \newcommand{\Prime}{{}\kern0.5pt'} \newcommand{\PARTIAL}[2]{{\partial^2#1\over\partial#2^2}} \newcommand{\Partial}[2]{{\partial#1\over\partial#2}} \renewcommand{\SS}{\vf A} \newcommand{\dS}{dA} \newcommand{\dA}{dS} \newcommand{\MydA}{dA} \newcommand{\dV}{d\tau} \newcommand{\RR}{{\mathbb R}} \newcommand{\HR}{{}^*{\mathbb R}} \newcommand{\BB}{\vf B} \newcommand{\CC}{\vf C} \newcommand{\EE}{\vf E} \newcommand{\FF}{\vf F} \newcommand{\GG}{\vf G} \newcommand{\HH}{\vf H} \newcommand{\II}{\vf I} \newcommand{\JJ}{\vf J} \newcommand{\KK}{\vf K} \renewcommand{\aa}{\VF a} \newcommand{\bb}{\VF b} \newcommand{\ee}{\VF e} \newcommand{\gv}{\VF g} \newcommand{\iv}{\vf\imath} \newcommand{\rr}{\VF r} \newcommand{\rrp}{\rr\Prime} \newcommand{\uu}{\VF u} \newcommand{\vv}{\VF v} \newcommand{\ww}{\VF w} \newcommand{\grad}{\vec\nabla} \newcommand{\zero}{\kern-1pt\vec{\kern1pt\bf 0}} \newcommand{\Ihat}{\Hat I} \newcommand{\Jhat}{\Hat J} \newcommand{\ii}{\Hat\imath} \newcommand{\jj}{\Hat\jmath} \newcommand{\kk}{\Hat k} \newcommand{\nn}{\Hat n} \newcommand{\NN}{\Hat N} \newcommand{\TT}{\Hat T} \newcommand{\ihat}{\Hat\imath} \newcommand{\jhat}{\Hat\jmath} \newcommand{\khat}{\Hat k} \newcommand{\nhat}{\Hat n} \newcommand{\rhat}{\HAT r} \newcommand{\shat}{\HAT s} \newcommand{\xhat}{\Hat x} \newcommand{\yhat}{\Hat y} \newcommand{\zhat}{\Hat z} \newcommand{\that}{\Hat\theta} \newcommand{\phat}{\Hat\phi} \newcommand{\DD}[1]{D_{\hbox{$#1$}}} \newcommand{\LargeMath}[1]{\hbox{\large$#1$}} \newcommand{\INT}{\LargeMath{\int}} \newcommand{\OINT}{\LargeMath{\oint}} \newcommand{\LINT}{\mathop{\INT}\limits_C} \newcommand{\Int}{\int\limits} \newcommand{\dint}{\mathchoice{\int\!\!\!\int}{\int\!\!\int}{}{}} \newcommand{\tint}{\int\!\!\!\int\!\!\!\int} \newcommand{\DInt}[1]{\int\!\!\!\!\int\limits_{#1~~}} \newcommand{\TInt}[1]{\int\!\!\!\int\limits_{#1}\!\!\!\int} \newcommand{\Bint}{\TInt{B}} \newcommand{\Dint}{\DInt{D}} \newcommand{\Eint}{\TInt{E}} \newcommand{\Lint}{\int\limits_C} \newcommand{\Oint}{\oint\limits_C} \newcommand{\Rint}{\DInt{R}} \newcommand{\Sint}{\int\limits_S} \newcommand{\Item}{\smallskip\item{$\bullet$}} \newcommand{\LeftB}{\vector(-1,-2){25}} \newcommand{\RightB}{\vector(1,-2){25}} \newcommand{\DownB}{\vector(0,-1){60}} \newcommand{\DLeft}{\vector(-1,-1){60}} \newcommand{\DRight}{\vector(1,-1){60}} \newcommand{\Left}{\vector(-1,-1){50}} \newcommand{\Down}{\vector(0,-1){50}} \newcommand{\Right}{\vector(1,-1){50}} \newcommand{\ILeft}{\vector(1,1){50}} \newcommand{\IRight}{\vector(-1,1){50}} \newcommand{\Partials}[3] {\displaystyle{\partial^2#1\over\partial#2\,\partial#3}} \newcommand{\Jacobian}[4]{\frac{\partial(#1,#2)}{\partial(#3,#4)}} \newcommand{\JACOBIAN}[6]{\frac{\partial(#1,#2,#3)}{\partial(#4,#5,#6)}} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} \)

Section 3.2 Properties of Infinitesimals

Every non-infinite hyperreal number \(x\) is approximately equal to some real number \(r\text{,}\) formally called its standard part, in the sense that \(x-r\) is either zero or infinitesimal.  1 A better name for the standard part of a non-infinite hyperreal number would be the real part, but that would conflict with the usage of this term for complex numbers. We will write \(x\cong r\) to indicate that \(r\) is the standard part of \(x\text{.}\)  2 The most common notation for standard part is \(r=\hbox{st}(x)\text{;}\) other notations are also used.

We will often turn this notation around and write \(r\cong x\text{,}\) which should be interpreted as defining \(r\) to be the standard part of \(x\text{;}\) it should be clear from the context which of these two forms is meant. In this notation, an infinitesimal number \(\alpha\) satisfies \(\alpha\cong0\text{,}\) and any non-infinite hyperreal number is either real, or can be expressed as \(r+\alpha\) for some real number \(r\) and infinitesimal number \(\alpha\text{.}\)

More generally, we often need to compare hyperreal numbers, neither of which is real. Two nonzero hyperreal numbers \(x,y\) will be called close, written \(x\approx y\text{,}\) if \(\frac{y}{x}\cong1\text{.}\) A nonzero hyperreal number \(x\) is close to zero, written either as \(x\approx0\) or \(0\approx x\text{,}\) if \(x\) is infinitesimal. Finally, we include the special case \(0\approx0\text{,}\) so that every hyperreal number is indeed close to itself.

The sum or difference of infinitesimals is clearly infinitesimal, as is their product. The product of an infinitesimal and a (nonzero) real number is again infinitesimal. However, division of infinitesimals could be infinitesimal, real, or infinite, depending on the relative “order” of the infinitesimals. For example, if \(\alpha\) is infinitesimal, \(\frac{\alpha^2}{\alpha}=\alpha\) is infinitesimal, \(\frac{2\alpha}{\alpha}=2\) is real, and \(\frac{\alpha}{\alpha^2}=\frac{1}{\alpha}\) is infinite. Similarly, the product of an infinitesimal and an infinite number could again be infinitesimal, real, or infinite. Analogous statements hold for operations on two infinite numbers.