Re: [math-fun] Proof that 0^0 = 1
Neil Sloane wrote: ----- Let's not start that discussion again. 0^0 is undefined. End of story. ----- Agreed. Without meaning to extend the discussion, I'd say that for certain applications, especially in combinatorics like counting the number of functions between two finite sets, there might be some value in defining 0^0 = 1 for that purpose. For situations that are more gooey than crunchy*, not so much (e.g., the value of x^y as (x,y) approaches (0,0) along some curve in the half-plane x > 0). So there's no point in trying to say what its only correct value ought to be. —Dan ——————— * Back around 1970 my grad-student friends and I used to classify various fields of mathematics as either gooey (continuous) or crunchy (discrete). Mathematicians, too.
"I don't know what you mean by '0^0,' " Alice said. Humpty Dumpty smiled contemptuously. "Of course you don't-—till I tell you. I meant 'there's a nice knock-down argument for you!'" "But '0^0' doesn't mean 'a nice knock-down argument'," Alice objected. "When I use a word," Humpty Dumpty said, in rather a scornful tone, "it means just what I choose it to mean—neither more nor less." "The question is," said Alice, "whether you can make words mean so many different things." "The question is," said Humpty Dumpty, "which is to be master-—that's all." (with apologies to the artful Dodgson.)
For me, the most common application of 0^0=1 is the fact that 0 log 0 = 0. That is, a coin whose probability of coming up heads is 0 or 1 has zero entropy. - Cris
On Jun 13, 2020, at 12:45 PM, Dan Asimov <dasimov@earthlink.net> wrote:
Neil Sloane wrote: ----- Let's not start that discussion again. 0^0 is undefined. End of story. -----
Agreed. Without meaning to extend the discussion, I'd say that for certain applications, especially in combinatorics like counting the number of functions between two finite sets, there might be some value in defining 0^0 = 1 for that purpose. For situations that are more gooey than crunchy*, not so much (e.g., the value of x^y as (x,y) approaches (0,0) along some curve in the half-plane x > 0). So there's no point in trying to say what its only correct value ought to be.
—Dan ——————— * Back around 1970 my grad-student friends and I used to classify various fields of mathematics as either gooey (continuous) or crunchy (discrete). Mathematicians, too.
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participants (3)
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Cris Moore -
Dan Asimov -
Marc LeBrun