What is 0^0? - Guglielmo Libri and Augustin Cauchy

Michael Huber and V. Frederick Rickey


According to Knuth, Libri's 1833 paper [8] "did produce several ripples in mathematical waters when it originally appeared, because it stirred up a controversy about whether 00 is defined." Most mathematicians at the time agreed that 00 = 1, even though Augustin-Louis Cauchy had listed 00 in a table of undefined forms in his book entitled Cours D'Analyse (1821) [2]. Evidently, Libri's argument was not convincing, so August Möbius came to his defense. Möbius tried to defend Libri by presenting a supposed proof of 00 = 1 (in essence, a proof that \(\lim_{x\rightarrow {0^+}} x^x=1\)). After confrontations from another mathematician resulted, the paper "was quietly omitted from the historical record when the collected works of Möbius were ultimately published." Knuth goes on to write that the debate ended with the result that 00 should be undefined, and then he states,


"No, no, ten thousand times no!"

Perhaps Cauchy was developing the notion of 00 as an undefined limiting form. Then the limiting value of [f(x)]g(x) is not known a priori when each of f(x) and g(x) approach 0 independently. According to Knuth, "the value of 00 is less defined than, say, the value of 0 + 0." He reminds us to recall the binomial theorem:

\[(x + y)^n = \sum_{k=0}^n {n \choose k} x^k y^{n-k}.\]

If this theorem is to hold for at least one nonnegative integer, then mathematicians "must believe that 00 = 1," for we can plug in x = 0 and y = 1 to get 1 on the left and 00 on the right.


Editor's note: This article was published in March of 2008.