# 0^0 Reprise!

• Subject: [amsat-bb] 0^0 Reprise!
• From: "Stacey E. Mills, M.D." <w4sm@xxxxxxxxxx>
• Date: Mon, 16 Nov 1998 17:04:31 -0500

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OK, I admit it, I'm hooked on this minor point....but a little more reading
suggests that maybe Roy Welch and I didn't have such bad math. teachers:

The following comes from:

http://forum.swarthmore.edu/dr.math/faq/faq.0.to.0.power.html

What is 0 to the 0 power?
Questions file, which is Copyright (c) 1994 Hans de Vreught
(hdev@cp.tn.tudelft.nl).
According to some Calculus textbooks, 0^0 is an "indeterminate form." What
mathematicians mean by "indeterminate form" is that in some cases we think
about it as having one value, and in other cases we think about it as
having another.

[snip]
Other than the times when we want it to be indeterminate, 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 would otherwise require
treating 0 as a special case. Notice that 0^0 is a discontinuity of the
function f(x,y) = x^y, because no matter what number you assign to 0^0, you
can't make x^y continuous at (0,0), since the limit along the line x=0 is
0, and the limit along the line y=0 is 1.

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 f(x)^g(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 0^x
and x^0 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 of 0^0 is very old. Euler argues for 0^0 = 1 since a^0 = 1
for a not equal to 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 .

****So there you go****

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.

--
__________________________________________________________________________
Stacey E. Mills, W4SM    WWW: 	http://www.cstone.net/~w4sm/ham1.html
Charlottesville, VA	   PGP key:	http://www.cstone.net/~w4sm/key.asc
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