11–20 of 256 posts
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.
That's a rather long text to say "it's an arbitrary -- and conveniently chosen -- definition of a special case of the power function similar to how 1 is not prime". I also think the presentation was chosen poorly: lots of wrong information before the correct approach is presented. Wikipedia is probably a better source here: https://en.wikipedia.org/wiki/0%5E0#Zero_to_the_power_of_zer...
His presentation was much better because it leads you to the answer instead of dropping it on you from the sky. (The difference between learning by rote vs learning by understanding.)
This is ridiculous. Doesn't this seem counter-intuitive for it to be anything else besides 0 or undefined?
Defining 0^0 as anything but 1 is weird since x^0 is 1 given the other definitions we have adopted, i.e. x^n = x * x * ... * x (n times) for positive integers, (x^n)^(-1) being defined as the unique number such that x^n * (x^n)^(-1) = 1, (x^n)^(-1) * x^n = 1 for non-zero numbers x, and the simple result that (x^n)^(-1) = (x^(-1))^n from proof by induction & the uniqueness condition of the multiplicative inverse, we then get the natural formula that 1 = x^n * x^(-n) = x^(n - n) = x^0 for non-zero numbers. Defining 0^0 = 0 or undefined doesn't agree with the formula given for all non-zero real numbers, and goes against the limit of x^x as x approaches 0 (as detailed in the article), making x^x a discontinuous function at x = 0 if it is defined as another value.
Missing Q and A: But if mathematicians insist it is 1, why do high school teachers act like they know more than the mathematicians do? A: They don't. The statement that mathematicians uniformly say it is 1 is simply false. My high school teacher had a PhD in math, I think it's fair to say she was a mathematician. And yes, she said it was undefined.
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.
However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.
But for convenience, whatever the offspring is, we may call it a donkapple.
That's they beauty of math.
Missing Q and A: But if mathematicians insist it is 1, why do high school teachers act like they know more than the mathematicians do? A: They don't. The statement that mathematicians uniformly say it is 1 is simply false. My high school teacher had a PhD in math, I think it's fair to say she was a mathematician. And yes, she said it was undefined.
I could come up with whatever crazy definitions in math I wanted to, and sometimes even get useful results. However, this view is not... convenient for teaching, so they pretend that you are learning "math", as opposed to "a particular math".
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.
Definitions in mathematics are used to make the language describing abstract concepts elegant, as explained in the entry. However, definitions aren't chosen all willy-nilly - there are good arguments why definitions are adopted, as should have been seen in the article.