0 Number Symbol

In recent times, 0 number symbol has become increasingly relevant in various contexts. factorial - Why does 0! - Mathematics Stack Exchange. The product of 0 and anything is $0$, and seems like it would be reasonable to assume that $0! I'm perplexed as to why I have to account for this condition in my factorial function (Trying to learn Haskell).

It is possible to interpret such expressions in many ways that can make sense. The question is, what properties do we want such an interpretation to have? $0^i = 0$ is a good choice, and maybe the only choice that makes concrete sense, since it follows the convention $0^x = 0$. On the other hand, $0^ {-1} = 0$ is clearly false (well, almost —see the discussion on goblin's answer), and $0^0=0 ...

algebra precalculus - Zero to the zero power – is $0^0=1 .... This perspective suggests that, @Arturo: I heartily disagree with your first sentence. Here's why: There's the binomial theorem (which you find too weak), and there's power series and polynomials (see also Gadi's answer). For all this, $0^0=1$ is extremely convenient, and I wouldn't know how to do without it.

In my lectures, I always tell my students that whatever their teachers said in school about $0^0$ being undefined, we ... Is $0$ a natural number? Inclusion of $0$ in the natural numbers is a definition for them that first occurred in the 19th century. In this context, the Peano Axioms for natural numbers take $0$ to be one though, so if you are working with these axioms (and a lot of natural number theory does) then you take $0$ to be a natural number.

In this context, is $0^\infty$ indeterminate? Is a constant raised to the power of infinity indeterminate? In relation to this, say, for instance, is $0^\\infty$ indeterminate? Or is it only 1 raised to the infinity that is?

Equally important, i have learned that 1/0 is infinity, why isn't it minus infinity?. 93 The other comments are correct: 1 0 1 0 is undefined. Similarly, the limit of 1 x 1 x as x x approaches 0 0 is also undefined. However, if you take the limit of 1 x 1 x as x x approaches zero from the left or from the right, you get negative and positive infinity respectively. What is the integral of 0?

So there is a sense in which the only "really geometrically meaningful" integral of $0$ is $0$ itself. But your friend is still wrong, since the term "integral" in this context means antiderivative and not area function. Justifying why 0/0 is indeterminate and 1/0 is undefined. Similarly, in the context of limits, $0/0$ is an indeterminate form (limit could be anything) while $1/0$ is not (limit either doesn't exist or is $\pm\infty$). This is a pretty reasonable way to think about why it is that $0/0$ is indeterminate and $1/0$ is not.