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0^0

askamathematician.com

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Re: 0^0

#51

Post by baddox »

dllthomas wrote: post #37
baddox wrote: post #8

Earlier quoted context omitted.

If we don't have a specific mathematical context, then saying it's undefined is intuitive to me. Without context, 0 is no more intuitive to me than 1. These two statements are equally intuitive to me, but they give different results for 0^0: "Zero raised to any power is still just zero." "Any number raised to the zeroth power is one."

For me, intuition-wise, I'd order it "undefined, 1, 0". There are 3 cases for 1 and one case for 0 that immediately spring to my mind when considering the problem: 0) Limit of 0^x, as x approaches 0 (from above). 1a) Limit of x^0 as x approaches 0 (from either direction). 1b) Limit of x^x as x approaches 0 (from above). 1c) "What did you multiply by 3 once, to get 3^1? So, multiplying 1 by zero, zero times..." Limits…

The limit in 1b is 1 from below as well, right? I'm not sure how limits work with complex numbers, but the imaginary part of x^x approaches zero as x approaches zero from below, so can we say that the limit of x^x as x approaches zero from below is also zero?

Re: 0^0

#52

Post by vlasev »

im3w1l wrote: post #16

It also plays nice with the convention of 0log 0=0, used in for example formulas for entropy.

It's perhaps more than a convention because this is also the limit that the function x log(x) naturally takes as x approaches 0 in the domain of the logarithm. Well, this is just a rehash of what the "cleverest" student does anyway.

Re: 0^0

#53

Post by analog31 »

Perhaps a related question: How should it be defined in a math library for a programming language? Should it return 1, or throw an exception?

Re: 0^0

#54

Post by haddr »

I think that the most intuitive way to get the idea of why 0^0 = 1 is to take the example from combinatorics. n^k is the number of distinct sequences for the sampling with replacement and ordering (for example ball picking from repository of n different balls and counting the number of distinct ways that k balls can be picked and ordered - with replacement). I think that there is only one way of ordering results of drawing zero balls from a set of 0 different balls :)

Re: 0^0

#55

Post by skrowyek »

While high schoolers try to prove their own intuitions about their understanding of exponents (intuition drilled into them through rote learning), mathematicians just say "we defined it that way".

It speaks to the tragedy that is the high school math curriculum.

Re: 0^0

#56

Post by masterzora »

Bahamut wrote: post #15
didgeoridoo wrote: post #11

Students: Let's come up with some crazy proofs based on our individual levels of understanding. Teachers: Let's do it by the book and come up (somehow) with conflicting answers. Mathematicians: Yeah, sorry guys. We made it all up. Pretty much captures most mathematicians I know.

Definitions in mathematics are used to make the language describing abstract concepts elegant, as explained in the entry. However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.

My favourite algebra professor always said something to the effect of "we always have to be careful about abusing notation but if we make good choices of notation we can abuse it right." Definitions are sort of the same deal.

Re: 0^0

#57

Post by baddox »

didgeoridoo wrote: post #11

Students: Let's come up with some crazy proofs based on our individual levels of understanding. Teachers: Let's do it by the book and come up (somehow) with conflicting answers. Mathematicians: Yeah, sorry guys. We made it all up. Pretty much captures most mathematicians I know.

Correct me if I'm wrong, but I infer that you're using "we made it all up" as a pejorative toward mathematicians. Of course mathematicians invented the terminology, notation, and methodology, but that's not a bad thing. It's a great thing, just like it's great that engineers "make up" bridges, chemists "make up" pharmaceuticals, writers "make up" novels, etc.

Re: 0^0

#58

Post by biot »

analog31 wrote: post #53

Perhaps a related question: How should it be defined in a math library for a programming language? Should it return 1, or throw an exception?

Why is that even a question? It should return the correct answer, of course.

Edit: At the very least, that behaviour should be a configurable option for those who desire something other than what most mathematicians accept as being the correct answer.

Re: 0^0

#59

Post by 6d0debc071 »

lovamova wrote: post #40
dinkumthinkum wrote: post #27

Earlier quoted context omitted.

I don't think math is arbitrary at all.

It is. We invented the arabic numerals because they were easy to draw and we could written any numbers with them. Just like we invented higher lever computer languages instead of using assembly. See what Fibonacci used to say in his first book Liber Abaci about using arabic numerals. http://en.wikipedia.org/wiki/Liber_Abaci

> It is [arbitrary]. We invented the arabic numerals because...

Then it's not arbitrary; chosen at random or on a fleeting whim, without reference to a reason or system. It was invented to fill a specific need based on certain limitations.

Re: 0^0

#60

Post by chunky1994 »

mason55 wrote: post #44
lovamova wrote: post #40

Earlier quoted context omitted.

It is. We invented the arabic numerals because they were easy to draw and we could written any numbers with them. Just like we invented higher lever computer languages instead of using assembly. See what Fibonacci used to say in his first book Liber Abaci about using arabic numerals. http://en.wikipedia.org/wiki/Liber_Abaci

The symbols we use to represent math are arbitrary but that doesn't mean the rules behind them are. Many concepts in math are fundamental truths.

What exactly do you mean by fundamental truths here?

Mathematics is an internally consistent (for the most part) logical framework that is extremely powerful in expressing our knowledge about the world.

However, that doesn't mean that there is some intrinsic correctness about it or its concepts.

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