Standard part function

<h2 id="definition">Definition</h2>

<p>Nonstandard analysis deals primarily with the pair 
  
    
      
        
          R
        
        ⊆

∗
          
        
        
          R
        
      
    
    {\displaystyle \mathbb {R} \subseteq {}^{*}\mathbb {R} }
  
, where the <a href="/facts/Hyperreal_number/Kvf09qpo">hyperreals</a>

∗
          
        
        
          R
        
      
    
    {\displaystyle {}^{*}\mathbb {R} }
  
 are an <a href="/facts/Ordered_field/an0prSgG">ordered field</a> extension of the reals 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
, and contain infinitesimals, in addition to the reals. In the hyperreal line every real number has a collection of numbers (called a <a href="/facts/Monad_(nonstandard_analysis)/pJ3DspXx">monad</a>, or halo) of hyperreals infinitely close to it. The standard part function associates to a finite <a href="/facts/Hyperreal_number/Kvf09qpo">hyperreal</a> <i>x</i>, the unique standard real number <i>x</i>0 that is infinitely close to it. The relationship is expressed symbolically by writing
</p>

st
        ⁡
        (
        x
        )
        =
        
          x
          
            0
          
        
        .
      
    
    {\displaystyle \operatorname {st} (x)=x_{0}.}

<p>The standard part of any <a href="/facts/Infinitesimal/JKcTMIgD">infinitesimal</a> is 0.  Thus if <i>N</i> is an infinite <a href="/facts/Hypernatural/TYasI1IP">hypernatural</a>, then 1/<i>N</i> is infinitesimal, and st(1/<i>N</i>) = 0.
</p><p>If a hyperreal 
  
    
      
        u
      
    
    {\displaystyle u}
  
 is represented by a Cauchy sequence 
  
    
      
        ⟨
        
          u
          
            n
          
        
        :
        n
        ∈
        
          N
        
        ⟩
      
    
    {\displaystyle \langle u_{n}:n\in \mathbb {N} \rangle }
  
 in the <a href="/facts/Ultrapower/d4cT10hd">ultrapower</a> construction, then 
</p>

st
        ⁡
        (
        u
        )
        =
        
          lim
          
            n
            →
            ∞
          
        
        
          u
          
            n
          
        
        .
      
    
    {\displaystyle \operatorname {st} (u)=\lim _{n\to \infty }u_{n}.}

<p>More generally, each finite 
  
    
      
        u
        ∈

∗
          
        
        
          R
        
      
    
    {\displaystyle u\in {}^{*}\mathbb {R} }
  
 defines a <a href="/facts/Dedekind_cut/rqhbb24J">Dedekind cut</a> on the subset 
  
    
      
        
          R
        
        ⊆

∗
          
        
        
          R
        
      
    
    {\displaystyle \mathbb {R} \subseteq {}^{*}\mathbb {R} }
  
 (via the total order on

∗
          
        
        
          R
        
      
    
    {\displaystyle {}^{\ast }\mathbb {R} }
  
) and the corresponding real number is the standard part of <i>u</i>.
</p>
<h2 id="not-internal">Not internal</h2>
<p>The standard part function "st" is not defined by an <a href="/facts/Internal_set/IHp7mBaK">internal set</a>. There are several ways of explaining this. Perhaps the simplest is that its domain L, which is the collection of limited (i.e. finite) hyperreals, is not an internal set. Namely, since L is bounded (by any infinite hypernatural, for instance), L would have to have a least upper bound if L were internal, but L doesn't have a least upper bound. Alternatively, the range of "st" is 
  
    
      
        
          R
        
        ⊆

∗
          
        
        
          R
        
      
    
    {\displaystyle \mathbb {R} \subseteq {}^{*}\mathbb {R} }
  
, which is not internal; in fact every internal set in

∗
          
        
        
          R
        
      
    
    {\displaystyle {}^{*}\mathbb {R} }
  
 that is a subset of 
  
    
      
        
          R
        
      
    
    {\displaystyle \mathbb {R} }
  
 is necessarily <i>finite</i>.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>
<h2 id="applications">Applications</h2>
<p>All the traditional notions of calculus can be expressed in terms of the standard part function, as follows.
</p>
<h3>Derivative</h3>
<p>The standard part function is used to define the derivative of a function <i>f</i>.  If <i>f</i> is a real function, and <i>h</i> is infinitesimal, and if <i>f</i>′(<i>x</i>) exists, then
</p>

f
          ′
        
        (
        x
        )
        =
        st
        ⁡
        
          (
          
            
              
                f
                (
                x
                +
                h
                )
                −
                f
                (
                x
                )
              
              h
            
          
          )
        
        .
      
    
    {\displaystyle f'(x)=\operatorname {st} \left({\frac {f(x+h)-f(x)}{h}}\right).}

<p>Alternatively, if 
  
    
      
        y
        =
        f
        (
        x
        )
      
    
    {\displaystyle y=f(x)}
  
, one takes an infinitesimal increment 
  
    
      
        Δ
        x
      
    
    {\displaystyle \Delta x}
  
, and computes the corresponding 
  
    
      
        Δ
        y
        =
        f
        (
        x
        +
        Δ
        x
        )
        −
        f
        (
        x
        )
      
    
    {\displaystyle \Delta y=f(x+\Delta x)-f(x)}
  
.  One forms the ratio 
  
    
      
        
          
            
              Δ
              y
            
            
              Δ
              x
            
          
        
      
    
    {\textstyle {\frac {\Delta y}{\Delta x}}}
  
. The derivative is then defined as the standard part of the ratio:
</p>

d
              y
            
            
              d
              x
            
          
        
        =
        st
        ⁡
        
          (
          
            
              
                Δ
                y
              
              
                Δ
                x
              
            
          
          )
        
        .
      
    
    {\displaystyle {\frac {dy}{dx}}=\operatorname {st} \left({\frac {\Delta y}{\Delta x}}\right).}

<h3>Integral</h3>
<p>Given a function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 on 
  
    
      
        [
        a
        ,
        b
        ]
      
    
    {\displaystyle [a,b]}
  
, one defines the integral 
  
    
      
        
          ∫
          
            a
          
          
            b
          
        
        f
        (
        x
        )
        
        d
        x
      
    
    {\textstyle \int _{a}^{b}f(x)\,dx}
  
 as the standard part of an infinite Riemann sum 
  
    
      
        S
        (
        f
        ,
        a
        ,
        b
        ,
        Δ
        x
        )
      
    
    {\displaystyle S(f,a,b,\Delta x)}
  
 when the value of 
  
    
      
        Δ
        x
      
    
    {\displaystyle \Delta x}
  
 is taken to be infinitesimal, exploiting a <a href="/facts/Hyperfinite_set/algm9nlQ">hyperfinite</a> partition of the interval [<i>a</i>,<i>b</i>].
</p>
<h3>Limit</h3>
<p>Given a sequence 
  
    
      
        (
        
          u
          
            n
          
        
        )
      
    
    {\displaystyle (u_{n})}
  
, its limit is defined by 
  
    
      
        
          lim
          
            n
            →
            ∞
          
        
        
          u
          
            n
          
        
        =
        st
        ⁡
        (
        
          u
          
            H
          
        
        )
      
    
    {\textstyle \lim _{n\to \infty }u_{n}=\operatorname {st} (u_{H})}
  
 where 
  
    
      
        H
        ∈

∗
          
        
        
          N
        
        ∖
        
          N
        
      
    
    {\displaystyle H\in {}^{*}\mathbb {N} \setminus \mathbb {N} }
  
 is an infinite index.  Here the limit is said to exist if the standard part is the same regardless of the infinite index chosen.
</p>
<h3>Continuity</h3>
<p>A real function 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is continuous at a real point 
  
    
      
        x
      
    
    {\displaystyle x}
  
 if and only if the composition 
  
    
      
        st
        ∘
        f
      
    
    {\displaystyle \operatorname {st} \circ f}
  
 is <i>constant</i> on the <a href="/facts/Halo_(mathematics)/pJ3DspXx">halo</a> of 
  
    
      
        x
      
    
    {\displaystyle x}
  
. See <a href="/facts/Microcontinuity/h3fxbJHs">microcontinuity</a> for more details.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Adequality/nq5odBwa">Adequality</a></li>
<li><a href="/facts/Nonstandard_calculus/ze0BGYa6">Nonstandard calculus</a></li></ul>

<h2 id="further-reading">Further reading</h2>
<ul><li><a href="/facts/H._Jerome_Keisler/FyQhCNS3">H. Jerome Keisler</a>. <i><a href="/facts/Elementary_Calculus:_An_Infinitesimal_Approach/ATsn8xH6">Elementary Calculus: An Infinitesimal Approach</a></i>. First edition 1976; 2nd edition 1986. (This book is now out of print. The publisher has reverted the copyright to the author, who has made available the 2nd edition in .pdf format available for downloading at <a href="http://www.math.wisc.edu/~keisler/calc.html">http://www.math.wisc.edu/~keisler/calc.html</a>.)</li>
<li><a href="/facts/Abraham_Robinson/UEskqXag">Abraham Robinson</a>. Non-standard analysis. Reprint of the second (1974) edition. With a foreword by <a href="/facts/Wilhelmus_A._J._Luxemburg/yHgscRXB">Wilhelmus A. J. Luxemburg</a>. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1996. xx+293 pp. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-691-04490-2</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Katz, Karin Usadi; Katz, Mikhail G. (March 2012). "A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography". Foundations of Science. 17 (1): 51–89. arXiv:1104.0375. doi:10.1007/s10699-011-9223-1The authors refer to the Fermat-Robinson standard part.{{cite journal}}:  CS1 maint: postscript (link) <a href="https://link.springer.com/article/10.1007/s10699-011-9223-1" target="_blank">https://link.springer.com/article/10.1007/s10699-011-9223-1</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Bascelli, Tiziana; Bottazzi, Emanuele; Herzberg, Frederik; Kanovei, Vladimir; Katz, Karin U.; Katz, Mikhail G.; Nowik, Tahl; Sherry, David; Shnider, Steven (1 September 2014). "Fermat, Leibniz, Euler, and the Gang: The True History of the Concepts of Limit and Shadow" (PDF). Notices of the American Mathematical Society. 61 (8): 848. doi:10.1090/noti1149. <a href="https://community.ams.org/journals/notices/201408/rnoti-p848.pdf" target="_blank">https://community.ams.org/journals/notices/201408/rnoti-p848.pdf</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Goldblatt, Robert (1998). Lectures on the Hyperreals: An Introduction to Nonstandard Analysis. Graduate Texts in Mathematics. Vol. 188. New York: Springer. doi:10.1007/978-1-4612-0615-6. ISBN 978-0-387-98464-3. <a href="978-0-387-98464-3" target="_blank">978-0-387-98464-3</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
</ol>

Standard part function open-in-new

Standard part function