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sci.math FAQ: What is 0^0?


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Archive-Name: sci-math-faq/specialnumbers/0to0
Last-modified: April 26, 1995  
Version: 6.2

What is 0^0 

   
   According to some Calculus textbooks, 0^0 is an ``indeterminate
   form''. When evaluating a limit of the form 0^0 , then you need to
   know that limits of that form are called ``indeterminate forms'', and
   that you need to use a special technique such as L'Hopital's rule to
   evaluate them. Otherwise, 0^0 = 1 seems to be the most useful choice
   for 0^0 . This convention allows us to extend definitions in different
   areas of mathematics that otherwise would require treating 0 as a
   special case. Notice that 0^0 is a discontinuity of the function x^y .
   
   
   This means that depending on the context where 0^0 occurs, you might
   wish to substitute it with 1, indeterminate or undefined/nonexistent.
   
   Some people feel that giving a value to a function with an essential
   discontinuity at a point, such as x^y at (0,0) , is an inelegant patch
   and should not be done. Others point out correctly that in
   mathematics, usefulness and consistency are very important, and that
   under these parameters 0^0 = 1 is the natural choice.
   
   The following is a list of reasons why 0^0 should be 1.
   
   Rotando &Korn show that if f and g are real functions that vanish at
   the origin and are analytic at 0 (infinitely differentiable is not
   sufficient), then f(x)^g(x) approaches 1 as x approaches 0 from the
   right.
   
   From Concrete Mathematics p.162 (R. Graham, D. Knuth, O. Patashnik):
   
     Some textbooks leave the quantity 0^0 undefined, because the
     functions x^0 and 0^x have different limiting values when x
     decreases to 0. But this is a mistake. We must define x^0=1 for all
     x , if the binomial theorem is to be valid when x = 0 , y = 0 ,
     and/or x = -y . The theorem is too important to be arbitrarily
     restricted! By contrast, the function 0^x is quite unimportant.
     
   Published by Addison-Wesley, 2nd printing Dec, 1988.
   
   As a rule of thumb, one can say that 0^0 = 1 , but 0.0^(0.0) is
   undefined, meaning that when approaching from a different direction
   there is no clearly predetermined value to assign to 0.0^(0.0) ; but
   Kahan has argued that 0.0^(0.0) should be 1, because if f(x), g(x) -->
   0 as x approaches some limit, and f(x) and g(x) are analytic
   functions, then f(x)^g(x) --> 1 .
   
   The discussion on 0^0 is very old, Euler argues for 0^0 = 1 since a^0
   = 1 for a != 0 . The controversy raged throughout the nineteenth
   century, but was mainly conducted in the pages of the lesser journals:
   Grunert's Archiv and Schlomilch's Zeitshrift. Consensus has recently
   been built around setting the value of 0^0 = 1 .
   
   On a discussion of the use of the function 0^(0^x) by an Italian
   mathematician named Guglielmo Libri.
   
     [T]he paper [33] did produce several ripples in mathematical waters
     when it originally appeared, because it stirred up a controversy
     about whether 0^0 is defined. Most mathematicians agreed that 0^0 =
     1 , but Cauchy [5, page 70] had listed 0^0 together with other
     expressions like 0/0 and oo - oo in a table of undefined forms.
     Libri's justification for the equation 0^0 = 1 was far from
     convincing, and a commentator who signed his name simply ``S'' rose
     to the attack [45]. August Mvbius [36] defended Libri, by presenting
     his former professor's reason for believing that 0^0 = 1 (basically
     a proof that lim_(x --> 0+) x^x = 1 ). Mvbius also went further and
     presented a supposed proof that lim_(x --> 0+) f(x)^(g(x)) whenever
     lim_(x --> 0+) f(x) = lim_(x --> 0+) g(x) = 0 . Of course ``S'' then
     asked [3] whether Mvbius knew about functions such as f(x) =
     e^(-1/x) and g(x) = x . (And paper [36] was quietly omitted from the
     historical record when the collected words of Mvbius were ultimately
     published.) The debate stopped there, apparently with the conclusion
     that 0^0 should be undefined.
     
     But no, no, ten thousand times no! Anybody who wants the binomial
     theorem (x + y)^n = sum_(k = 0)^n (n\choose k) x^k y^(n - k) to hold
     for at least one nonnegative integer n must believe that 0^0 = 1 ,
     for we can plug in x = 0 and y = 1 to get 1 on the left and 0^0 on
     the right.
     
     The number of mappings from the empty set to the empty set is 0^0 .
     It has to be 1.
     
     On the other hand, Cauchy had good reason to consider 0^0 as an
     undefined limiting form, in the sense that the limiting value of
     f(x)^(g(x)) is not known a priori when f(x) and g(x) approach 0
     independently. In this much stronger sense, the value of 0^0 is less
     defined than, say, the value of 0 + 0 . Both Cauchy and Libri were
     right, but Libri and his defenders did not understand why truth was
     on their side.
     
     [3] Anonymous and S ... Bemerkungen zu den Aufsatze |berschrieben,
     `Beweis der Gleichung ... , nach J. F. Pfaff', im zweiten Hefte
     dieses Bandes, S. 134, Journal f|r die reine und angewandte
     Mathematik, 12 (1834), 292-294.
     
     
     
     [5] Oe uvres Complhtes. Augustin-Louis Cauchy. Cours d'Analyse de
     l'Ecole Royale Polytechnique (1821). Series 2, volume 3.
     
     
     
     [33] Guillaume Libri. Mimoire sur les fonctions discontinues,
     Journal f|r die reine und angewandte Mathematik, 10 (1833),
     303-316.
     
     
     
     [36] A. F. Mvbius. Beweis der Gleichung 0^0 = 1 , nach J. F. Pfaff.
     Journal f|r die reine und angewandte Mathematik,
     
     
     
     12 (1834), 134-136.
     
     [45] S ... Sur la valeur de 0^0 . Journal f|r die reine und
     angewandte Mathematik 11, (1834), 272-273.
     
     
     
   
   
   
   
   References
   
   Knuth. Two notes on notation. (AMM 99 no. 5 (May 1992), 403-422).
   
   
   
   H. E. Vaughan. The expression ' 0^0 '. Mathematics Teacher 63 (1970),
   pp.111-112.
   
   
   
   Louis M. Rotando and Henry Korn. The Indeterminate Form 0^0 .
   Mathematics Magazine, Vol. 50, No. 1 (January 1977), pp. 41-42.
   
   
   
   L. J. Paige,. A note on indeterminate forms. American Mathematical
   Monthly, 61 (1954), 189-190; reprinted in the Mathematical
   Association of America's 1969 volume, Selected Papers on Calculus, pp.
   210-211.
   
   
   
   Baxley &Hayashi. A note on indeterminate forms. American Mathematical
   Monthly, 85 (1978), pp. 484-486.
   
   
   
   
     _________________________________________________________________
   
   
   
    alopez-o@barrow.uwaterloo.ca
    Tue Apr 04 17:26:57 EDT 1995

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