Tuesday, May 10, 2016

exponentiation - 0's Exponents are impossible?



I've had something that's been bugging me, and I tried research and asked my math teacher. None had sufficient answers. The concept of $0$ is that when $0$ goes to any exponent except for $0$, it becomes $0$. For example,



$0^3 = 0$, but $0^0 =$ undefined


However, the proof that $0^0$ is undefined is shown thus: $0^x$ ... (divided by) = $0^{(x-x)} = 0^0$ = undefined $0^x$


You can apply this to any exponent though, such as: $0^6$ ... = $0^6 = 0$ and $0^3 = 0$, so this expression is equal to $0/0$, which should be $0^3$ undefined, right?


Am I doing something wrong here? Please help! Gil


Answer



One should not say "equals undefined"; one should say "is undefined". The is the "is" of predication, not the "is" of equality.


$0^0$ is indeterminate in the sense that if $f(x)$ and $g(x)$ both approach $0$ as $x\to a$ then $f(x)^{g(x)}$ could approach any positive number or $0$ or $\infty$ depending on which functions $f$ and $g$ are.


But in some contexts it is important that $0^0$ be taken to be $1$. One of those contexts is in the identity $$ e^z = \sum_{n=0}^\infty \frac{z^n}{n!}. $$ This fails to hold when $z=0$ unless $0^0=1$, since the first term of the series is then $\dfrac{0^0}{0!}$.


In combinatorial enumeration it is also important in some contexts that $0^0$ is $1$, for the same reason $2^0$ is $1$: if you multiply by something $0$ times, it's the same as multiplying by $1$. That is also one way of looking at the reason why $0!$ is $1$.


No comments:

Post a Comment

analysis - Injection, making bijection

I have injection $f \colon A \rightarrow B$ and I want to get bijection. Can I just resting codomain to $f(A)$? I know that every function i...