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0^0

askamathematician.com

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Re: 0^0

#101

Post by vlasev »

marcosdumay wrote: post #67
vlasev wrote: post #48

Earlier quoted context omitted.

I had an idea for a couple of years now of creating a wikipedia-style mathematics textbook that will be crowd-sourced, standardized and cover all of math in a way that's accessible to learn from on your own. It would have a kind of a zoom function where you can expand details on explanations and calculations to a depth that you prefer. Ideally this kind of thing would start off with basic math and get progressively f…

Your "basic" in basic Math means "foundational", or "what we teach to children"?

That's a good question. In the spirit of making learning easier it would perhaps "start" with what we teach children. However, that might be ultimately misguided because that would just mimic the curriculum and force the whole thing into a box. But then again the math knowledge itself forces a structure on any such textbook because to know A you need to know B, C, D, etc.

This is further complicated by the fact that most subjects in math can be approached from multiple equivalent angles. For example, complex analysis can be taught from the complex derivative perspective or from the power series perspective equally well. Ideally the format of the whole thing will allow the student to naturally learn things from the angle they understand better.

Re: 0^0

#102

Post by scythe »

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.

0^0, like any indeterminate form, can be made to equal anything via sufficient cleverness. Consider the limit:

y = lim_[x->0] x^[a / log(x)]

We have log y = lim_[x->0] (a / log(x)) log(x) = a. So 0^0 equals any number at all! Of course, nobody in their right mind would define exponents this way, but the indeterminacy is inherent in the definition of the symbols.

Re: 0^0

#103

Post by clebio »

my_username_is_ wrote: post #98
blackrabbit wrote: post #93

Can anyone explain why this is a valid operation? http://wp.com/latex.php?latex=\lim_{x%20\to%200^{+}}%20x^{x}...

it's using log to mean log base e, also known as the natural log or ln. you can do that because they're inverse operations (I think that's the right term for it, it's been a while since a formal math class) so they undo each other

Yes. It's an identity relationship. x = exp(log(x))

http://en.wikipedia.org/wiki/List_of_logarithmic_identities#...

Re: 0^0

#104

Post by dragonwriter »

davyjones wrote: post #61
yaks_hairbrush wrote: post #29

A word from Knuth on the matter (warning: PDF): http://arxiv.org/pdf/math/9205211v1.pdf See page 6.

Are PDF warnings relevant anymore? Of late Chromium and Firefox display PDF natively, sandboxed(?).

There are browsers that are not the desktop versions of Chrom(e/ium) or Firefox, and which are widely used.

Re: 0^0

#105

Post by rntz »

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…

In a certain sense, "1+1 is 2" is also merely a definition, in the same sense that "0^0 is 1" is a definition. Addition can be formally defined in mathematics; we habitually omit this definition because it is tedious, and because addition is such an intuitive operation that we do not require a definition in order to reason about it.

Much as the question "what if the parallel axiom didn't hold?" leads to alternative geometries, the question "what if addition didn't work in the same way?" (or "how can we generalize addition?") leads to some basic notions in abstract algebra.

Re: 0^0

#106

Post by noonespecial »

Eleutheria wrote: post #18

Whenever you try to cross-breed an apple tree with a donkey, you get nonsense. But for convenience, whatever the offspring is, we may call it a donkapple. That's they beauty of math.

The utility of math is that you can then do a bunch of calculations with donkapples and at the end, get a meaningful real-world answer.

Re: 0^0

#107

Post by Cacti »

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…

Invented, obviously. ;)

Re: 0^0

#108

Post by olalonde »

Are there any examples where the x^0=1 definition turns out to make other definitions more complicated to write down? For example, you'd have a general definition and need a special definition for when you get an exponent that equals 0.

Re: 0^0

#109

Post by dvanduzer »

baddox wrote: post #91
dvanduzer wrote: post #90

Earlier quoted context omitted.

Mathematicians are engaged in a dramatically different kind of project than almost any other human discipline. Mathematical details emerge from definitions, but they appear exactly the same way for everyone else using the same definitions . And what's really surprising is how robustly those purely rational results compare to messy empirical reality. There is no good reason to believe that this should be the case! In…

I don't think the fifth axiom is considered particularly mysterious anymore. The traditional fifth axiom clearly isn't a logical result of the first four, since it can be replaced with other parallel postulates to yield non-Euclidean geometries which are themselves perfectly workable and consistent. In fact, that section of the Wikipedia article notes that Beltrami proved the independence of the parallel postulate.

> clearly

It's mysterious because it is only clear in retrospect. The fact that you can negate the parallel postulate and get a system that is still self-consistent is incredibly mysterious. This is what sets mathematics apart. We "make it up" like other human accomplishments, and yet we can't actually just make it up. If we arbitrarily defined 0^0 = π, we'd just be speaking nonsense.

Re: 0^0

#110

Post by jacobolus »

yaks_hairbrush wrote: post #29

A word from Knuth on the matter (warning: PDF): http://arxiv.org/pdf/math/9205211v1.pdf See page 6.

I really like that Knuth paper: the Iverson bracket in particular is some really handy notation. Mathematicians would do well to spend more effort on notation. Currently it doesn’t seem nearly as valued as theorem proving, but in my opinion it’s just as important, because it defines how we think about the structures we’re working with.

I think computer programming would actually be quite excellent exercise for a mathematician, because it involves such heavy intimate experience with the problems of naming things, working with notation, and defining the boundaries of various abstractions. From kindergarten up through the end of an undergraduate degree, mathematics students mostly take existing notation and definitions for granted, and don’t get much hands-on experience with the problems which result from inventing bad notation or bad names. As a result, they have a less visceral understanding of the importance of good notation and good names.

[I also think programming students should spend at least a bit of time working with as many different abstraction styles and notations as they can, as well as e.g. trying to implement new toy programming languages with new semantics.]

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