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(?).
0^0
61–70 of 256 posts
Re: 0^0
#62Re: 0^0
#63I 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…
> But how does that translate to something useful in the real world? I think this is a by-product of the way we are taught maths at the very start; that it must somehow relate to real things. We start our understanding of maths by using real world objects like apples and we show how addition works and subtraction. I think perhaps it sticks in our head that everything must somehow relate to real objects and the real w…
Square roots and integrals both relate back to the real world...
Re: 0^0
#64A 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(?).
On Android, both Chrome and Firefox still open an external tool and ruin your back button.
Re: 0^0
#65Earlier 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.
There are an infinite number of fundamental logical truths out there, but we arbitrarily picked useful ones to make a system of math.
Re: 0^0
#66A word from Knuth on the matter (warning: PDF): http://arxiv.org/pdf/math/9205211v1.pdf See page 6.
Why link to the PDF? Better to just link to the abstract: http://arxiv.org/abs/math/9205211
Edit: To be clear, this is good practice when linking to arXiv in general. From the abstract, one can easily click through to the PDF; not so the reverse. And the abstract allows one to do things like see different versions of the paper, search for other things by the same authors, etc.
Re: 0^0
#67Earlier 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.
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"?
Re: 0^0
#68Earlier 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
> 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.
But it is arbitrary; subject to individual will or judgment without restriction; contingent solely upon one's discretion.
Re: 0^0
#69I 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 space where y would be, akin to an empty set. 1 times an 'empty set' is _not_ 1, it's one empty set, which rubs me as another way of saying nothing/zero, not 1.
I know my language is imprecise and I'm probably describing empty sets and the definition of zero incorrectly. The point is that the last step of his explanation doesn't sit right with me. Just because Y exists zero times does not mean you can just throw it out of the multiplication.