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> how about we try to make the function more continuous Look at the 3D rendering in my comment. When you look at the z=x^y surface and try to make it continuou

by etatoby 10y ago

> how about we try to make the function more continuous Look at the 3D rendering in my comment. When you look at the z=x^y surface and try to make it continuous, what comes out at 0^0 is a vertical line, aka. undefined.

dahart 10y ago
You're absolutely right. And you're also wrong in the very the same way I'm wrong. ;) We decided that 0^0=1, and by 'we', I mean the mathematicians. It's easy to show why the choice 0^0=1 is a bad choice, because it breaks some rules. It's also easy to show (as the article's author did) why it's a good choice too. But, regardless, it's still the choice that has been made, and the choice can't be unmade. It's a beautiful example, btw, thank you for contributing the visualization!
- etatoby 10y ago
  I've updated my example with this frame: https://s31.postimg.org/4kozbjqbf/x_y_lines.png https://s31.postimg.org/4kozbjqbf/x_y_lines.png that highlights 0^y=0 in red and x^0=1 in purple. Now it's even more obvious why any single answer is insufficient: 0^0 is a vertical line, not a single point!
  - dahart 10y ago
    Hey, again, I totally agree with you. You are right, no single answer is sufficient. It was obvious before, and it's still obvious. The new lines show the specific conflict between 0^x and x^0, but there is no one answer. Your re-iterating your point makes me wonder if you're actually hearing and understanding the part about choices and definitions trumping logic. If we define 0^0=1, then it doesn't matter what arguments we have, it doesn't matter that there are 3 or more right answers. The right answer can be, and in other parts of math, is what we define it to be, not what makes the most sense. 0^0 is defined as 1 (sometimes) because that has utility and consistency for some other expressions, formulas, operations, etc. It's not because it's right. It's not because it makes sense. It's because we decide to set 0^0=1. It's for convenience. Now, some sources online say that 0^0 is indeterminate, not equal to one. Other sources online say that 0^0 is 1. http://mathworld.wolfram.com/Power.html http://mathworld.wolfram.com/Power.html https://en.wikipedia.org/wiki/Indeterminate_form https://en.wikipedia.org/wiki/Indeterminate_form https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.h... https://www.google.com/search?q=0%5E0 https://www.google.com/search?q=0%5E0 https://betterexplained.com/articles/understanding-exponents-why-does-00-1/ https://betterexplained.com/articles/understanding-exponents...
  - deleted 10y ago
    [deleted]

by etatoby 10y ago

dahart 10y ago

You're absolutely right. And you're also wrong in the very the same way I'm wrong. ;) We decided that 0^0=1, and by 'we', I mean the mathematicians. It's easy to show why the choice 0^0=1 is a bad choice, because it breaks some rules. It's also easy to show (as the article's author did) why it's a good choice too. But, regardless, it's still the choice that has been made, and the choice can't be unmade. It's a beautiful example, btw, thank you for contributing the visualization!

etatoby 10y ago
I've updated my example with this frame: https://s31.postimg.org/4kozbjqbf/x_y_lines.png https://s31.postimg.org/4kozbjqbf/x_y_lines.png that highlights 0^y=0 in red and x^0=1 in purple. Now it's even more obvious why any single answer is insufficient: 0^0 is a vertical line, not a single point!
- dahart 10y ago
  Hey, again, I totally agree with you. You are right, no single answer is sufficient. It was obvious before, and it's still obvious. The new lines show the specific conflict between 0^x and x^0, but there is no one answer. Your re-iterating your point makes me wonder if you're actually hearing and understanding the part about choices and definitions trumping logic. If we define 0^0=1, then it doesn't matter what arguments we have, it doesn't matter that there are 3 or more right answers. The right answer can be, and in other parts of math, is what we define it to be, not what makes the most sense. 0^0 is defined as 1 (sometimes) because that has utility and consistency for some other expressions, formulas, operations, etc. It's not because it's right. It's not because it makes sense. It's because we decide to set 0^0=1. It's for convenience. Now, some sources online say that 0^0 is indeterminate, not equal to one. Other sources online say that 0^0 is 1. http://mathworld.wolfram.com/Power.html http://mathworld.wolfram.com/Power.html https://en.wikipedia.org/wiki/Indeterminate_form https://en.wikipedia.org/wiki/Indeterminate_form https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.html https://cs.uwaterloo.ca/~alopez-o/math-faq/mathtext/node14.h... https://www.google.com/search?q=0%5E0 https://www.google.com/search?q=0%5E0 https://betterexplained.com/articles/understanding-exponents-why-does-00-1/ https://betterexplained.com/articles/understanding-exponents...
- deleted 10y ago
  [deleted]