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Archive-name: sci-math-faq/0to0
Last-modified: February 20, 1998
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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.
More importantly, keep in mind that the value of a function and its
limit need not be the same thing, and functions need not be continous,
if that serves a purpose (see Dirac's delta).

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.

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 Zeitschrift f|r Mathematik und
Physik. 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  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 . August Mvbius  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.
such as f(x) = e^(-1/x) and g(x) = x. (And paper  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 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.

 Anonymous and S... Bemerkungen zu den Aufsatze |berschrieben,
`Beweis der Gleichung 0^0 = 1, nach J. F. Pfaff', im zweiten Hefte
dieses Bandes, S. 134, Journal f|r die reine und angewandte
Mathematik, 12 (1834), 292--294.

 uvres Complhtes. Augustin-Louis Cauchy. Cours d'Analyse de
l'Ecole Royale Polytechnique (1821). Series 2, volume 3.

 Guillaume Libri. Mimoire sur les fonctions discontinues,
Journal f|r die reine und angewandte Mathematik, 10 (1833),
303--316.

 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.

 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.

Kahan, W. Branch Cuts for Complex Elementary Functions or Much Ado
about Nothing's Sign Bit, The State of the Art in Numerical Analysis,
editors A. Iserles and M. J. D. Powell, Clarendon Press, Oxford, pp.
165--212. \

article 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.

Crimes and Misdemeanors in the Computer Algebra Trade. Notices of the
American Mathematical Society, September 1991, volume 38, number 7,
pp.778-785

_________________________________________________________________

--
Alex Lopez-Ortiz                                         alopez-o@unb.ca
http://www.cs.unb.ca/~alopez-o                       Assistant Professor
Faculty of Computer Science                  University of New Brunswick
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