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0^0

askamathematician.com

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Re: 0^0

#111

Post by scythe »

[deleted] wrote: post #74
forrestthewoods wrote: post #31

Earlier quoted context omitted.

I love Wikipedia. It's amazing. It makes the world a better place. I'm a pretty decent programmer. I do video games so I do lots of 3d math. I'd say I'm decent at that as well. I hate Wikipedia for math. Absolutely hate it. Unless you are a mathematician by trade Wikipedia is damn near useless for learning new math concepts. I don't even bother checking it anymore.

[deleted]

I have also found Wikipedia to be a convenient math reference, but it is not always the "best" that is easily available on the internet. If I'm looking at a topic, it's usually fruitful to Google "introduction to X theory" and go through a few possibilities to find a literature review. Often there is a review which is a good "companion" to the Wikipedia article in terms of explanation, since the style will be different and you get some breadth of perspective.

>that that's attributable to math being hard

(an easily underestimated property of mathematics)

Re: 0^0

#112

Post by krilnon »

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?

Mathematica (or do we call it Wolfram Language now?) evaluates 0^0 as Indeterminate and warns: `Power::indet: "Indeterminate expression 0^0 encountered."`

Re: 0^0

#113

Post by djur »

jomtung wrote: post #34
ryannevius wrote: post #17

I understand the "math"...the numbers...the work on paper. But how does that translate to something useful in the real world? That, after all, is what useful math helps us do...solve problems for the real, tangible world. Saying that 0^0 = 1 is a cool math game; but translate 0 into something in the real world (i.e. nothing, none, etc.)...and trying to make something out of it other than 0 or "indeterminate" starts t…

From the article's example the simplicity of the binomial formula is extremely useful compared to a formula that would have to account for the case where k=0. Another commenter pointed out the useful elegance of 0log0=0 for physicists. These are the real world applications for mathematicians choosing definitions directly. Saying that an idea may be defined in many ways is correct, but choosing a working definition fo…

Supporting your final paragraph: the concepts of nothingness, emptiness, absence, etc. are all much older and more universal than mathematical zero.

Re: 0^0

#114

Post by mathgrad »

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…

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.

Re: 0^0

#115

Post by alecbenzer »

mathgrad wrote: post #114
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…

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 this definition is somewhat non-standard for laypeople. Most people think of x^y to be defined as "x multiplied by itself y times" or something similar. By this definition, 0^0 is a convention/shorthand, not a truth.

I think the "functions from b to a" definition is useful to note too, though

Re: 0^0

#116

Post by blackrabbit »

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

Thanks!

Re: 0^0

#117

Post by forrestthewoods »

[deleted] wrote: post #74
forrestthewoods wrote: post #31

Earlier quoted context omitted.

I love Wikipedia. It's amazing. It makes the world a better place. I'm a pretty decent programmer. I do video games so I do lots of 3d math. I'd say I'm decent at that as well. I hate Wikipedia for math. Absolutely hate it. Unless you are a mathematician by trade Wikipedia is damn near useless for learning new math concepts. I don't even bother checking it anymore.

[deleted]

I disagree. There are math concepts that I've found impossible to pick up from wikipedia but can easily be learned in 5 minutes by having a conversation with someone who already understands them.

Honestly this article in question is a perfect example. I'd wager than almost everyone who reads Hacker News can read the blog post and understand every single step from start to finish. The same can not be said for the Wikipedia article.

Re: 0^0

#118

Post by olalonde »

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?

Re: 0^0

#119

Post by j2kun »

mathgrad wrote: post #114
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…

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.

You're just pushing the arbitrariness of defining things one step further to the definition of a number. It makes it no less arbitrary that you've defined it and force us to accept the definition to get to your conclusion.

Re: 0^0

#120

Post by Twisol »

mathgrad wrote: post #114
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…

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.
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