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0^0

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Re: 0^0

#131

Post by j2kun »

gameshot911 wrote: post #69

I have a problem with this part of the explanation: >However, this definition extends quite naturally from the positive integers to the non-negative integers, so that when x is zero, y is repeated zero times, giving y^{0} = 1, which holds for any y. Hence, when y is zero, we have 0^0 = 1. When y is zero, don't we have 0^0 = 1 x [y zero times]? Maybe I'm conceptualizing it incorrectly, but I'm envisioning an empty spa…

Another convention is that the empty product is 1, and this is precisely what is being said by putting the 1 out front.

Re: 0^0

#132

Post by chacham15 »

mathgrad wrote: post #129
chacham15 wrote: post #127

Earlier quoted context omitted.

Actually this gave me the idea: a^b is the number of mappings from set a to set b. Now, if set a is empty and set b is too, it is natural to believe that there are no mappings because there is nothing to map from or to.

A function is a relation for elements in a to elements in b such that for every element in a there is a unique element in b. This is vacuously true of the empty relation when a is empty.

I didnt discuss a function, I discussed a mapping. They are different constructs. Imagine the question like this: you have two groups of people, students and teachers. How many possible ways are there to assign students to teachers? To put it another way, how many arrows would it take pointing from the student to the teacher to illustrate every possible assignment? Now, if there are no students and no teachers, you have no arrows, not one.

Re: 0^0

#133

Post by mathgrad »

impendia wrote: post #130
mathgrad wrote: post #114

Earlier quoted context omitted.

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.

I think it's a little bit different. There are many natural ways of defining the expression a^b, which coincide when a and b are (say) positive integers, but there is basically only one way of defining 2 (namely, 1 + 1).

Look up Peano arithmetic to see how one proves 1+1 = 2 within the standard formal system.

Re: 0^0

#134

Post by j2kun »

olalonde wrote: post #118
im3w1l wrote: post #16

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

Why? With the x^0=1 definition, wouldn't log 0 be undefined?

0 log 0 = log (0^0) = log (1) = 0

Re: 0^0

#135

Post by mason55 »

chunky1994 wrote: post #60
mason55 wrote: post #44

Earlier quoted context omitted.

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.

If I have three objects and you give me two more then I'll always have five objects. You can call it cinco or 五 but there are still five of them.

Likewise, you'll always be able to determine the length of the hypotenuse of a right triangle by its two legs. No matter what system you set up, if you're cutting three boards to build a triangle the length of the big one is absolutely defined by the length of the other two, assuming Euclidian geometry. The symbols (a, b, =, c, +, superscript 2) are all totally arbitrary but if you're working with three boards there is absolutely an intrinsic correctness to the Pythagorean theorem.

There are branches of math which are just exploring the internal consistency of the system we've set up but much of physics is spent describing the real world and applying our math symbols to the universe. The amazingly cool thing is that our system is so good that we can use our abstract symbols to make predictions about physical laws and they actually come out to be true! I would argue that all of physics is basically "math that's a fundamental truth of the universe".

Re: 0^0

#136

Post by didgeoridoo »

baddox wrote: post #57
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.

Not at all pejorative. In my experience, the odd thing about mathematics is that the people who seem to treat it the least seriously are mathematicians. Everyone else thinks of "math" as a set of immutable laws. Mathematicians are the only ones who can see & treat math for what it is: a set of human languages designed to interpret and communicate ideas.

Re: 0^0

#137

Post by mathgrad »

chacham15 wrote: post #132
mathgrad wrote: post #129

Earlier quoted context omitted.

A function is a relation for elements in a to elements in b such that for every element in a there is a unique element in b. This is vacuously true of the empty relation when a is empty.

I didnt discuss a function, I discussed a mapping. They are different constructs. Imagine the question like this: you have two groups of people, students and teachers. How many possible ways are there to assign students to teachers? To put it another way, how many arrows would it take pointing from the student to the teacher to illustrate every possible assignment? Now, if there are no students and no teachers, you h…

I am a mathematician I don't need examples. What is your formal definition of a mapping.

In mathematic it is normally used as a synonym for a morphism in a given category and in the category of sets this would be a function. So in our context a mapping is a function.

Re: 0^0

#138

Post by [deleted] »

chacham15 wrote: post #121
rickhanlonii wrote: post #86

It's very important to note here that 0^0=1 is a shorthand and not a truth . Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 is 1" in the sense that they say "1+1 is 2" or "0.999... is 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why…

> If you do not accept it as true (an explicitly accept it as false), then you can prove all of Hyperbolic Geometry What is the equivalent in this analogy if you do not accept that 0^0=1 (i.e. accept that 0^0=0)?

[deleted]

Re: 0^0

#139

Post by einhverfr »

rickhanlonii wrote: post #86

It's very important to note here that 0^0=1 is a shorthand and not a truth . Mathematicians are absolutely not stating that they have proven, or that it is true, that 0^0=1. It is a definition, not a claim of equality. They're not saying "0^0 is 1" in the sense that they say "1+1 is 2" or "0.999... is 1". They're saying "we define 0^0 to be 1". The difference is more than just pedantry, it strikes at the core of why…

I see it as somewhat complex but in the end I sort of agree that it is a definition. Let's start with a problem:

0/0

The problem with this is not that the equation itself is meaningless. the limit of n/x as x->0 is infinity for any positive real number, and negative infinity for any negative real number. In essence 0/0 ends up reducing to 0 * infinity, which isn't very helpful.

I would argue that discontinuity in a form like this can never be unique to a single point in an otherwise continuous function, if the limit approached from both sides is the same, unless that point as a specific, defined value. This is because a point is arbitrarily small. The limit of 1/x as x->0 is discontinuous because from the negative side and the positive side, the limits are different, as anyone can see when you plot it on a graph.

However here we have x/x, and the limit stays at 1 from both sides. It may be formally undefined at the exact point of 0/0, but the limit from both sides is the same, and it remains constant as you get arbitrarily close. Therefore it is simpler, system-wise, to treat it as 1, than to make an exception for it.

> In the case here, you're free to reject the convention that defines 0^0 as 1 and reason with the result;

But you add complexity which is neither helpful nor desirable.

Re: 0^0

#140

Post by einhverfr »

Twisol wrote: post #120
mathgrad wrote: post #114

Earlier quoted context omitted.

This is not true. A very natural way to define a^b is as the number of functions for a set with "b" elements to a set with "a" elements. In this case there is exactly one function from the empty set to the empty set, and we have proven 0^0 = 1. This is no different than defining addition and then proving 1+1 = 2.

I prefer thinking about 0^0 = 1 as an empty product ( https://en.wikipedia.org/wiki/Empty_product ), since it generalizes nicely to any operation with an identity element. That is, if you apply any operation zero times, the result is that operation's identity. It's interesting that the analogous empty sum, 0*1 = 0, is a complete non-issue.

But here's your problem:

infinity * n = infinity, right?

0 * n = 0, right?

0 * infinity = ?

Ok, this is relevant here particularly because:

Lim 1/x as x -> 0 from the positive side is infinity, right?

So 0 * that is.....

lim 1/x as x -> from the negative side is negative infinity, right?

So 0 * that is.....

That's why 0/0 doesn't work as such. You don't know how 0 is derived or what it means. If we have x^2/x, and take the limit as x -> 0, we get 0. If we take x/x^2 and take the limit as x -> 0 we get +/- infinity.

But that doesn't mean that x/x or x^2/x are not continuous functions, any more than 1 or x are not continuous functions.

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