I did not encounter this convention while working on my math degree. I am surprised that none of the characters in the article said "0^0 is nothing, but limits of the form 0^0 can be any nonnegative real number or infinite".
0^0
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Re: 0^0
#72I don't get what the big deal is. I put 0^0 in Windows Calculator and it gives me 1 ... problem solved!
Re: 0^0
#73Perhaps a related question: How should it be defined in a math library for a programming language? Should it return 1, or throw an exception?
In JavaScript, Math.pow(0, 0) == 1.
In Python and Ruby, 0 asterisk asterisk 0 == 1.
In Windows Calculator, 0 [x^y] 0 == 1.
So I guess the consensus is that you should return 1.
Re: 0^0
#74That'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...
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.
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Re: 0^0
#75Another good reminder on how math itself is arbitrary and made up by humans (often for what's simplest/easiest), and not handed down to us by God. Luckily it's an extremely useful and extendable made up system. I see this all the time with AI/machine learning. Most algorithms are based on assumptions that make the math work out better rather than being aligned with some "fundamental truth." The world is not linear, b…
I don't know that I accept this argument. You're saying that because there are slight inconsistencies that we have to reconcile, math can't possibly given to us from God? Has God never handed anything to humans that had slight inconsistencies in it?
I... okay, but you're not really responding to the intent of the message which was saying that math isn't a bastion of purity and fundamentally and wholly right and perfectly designed.
And if math had been handed down it would all be in an old holy book, which it clearly isn't.
Re: 0^0
#76I feel like in discrete mathematics (especially combinatorics), since we don't use continuous functions, it's useful to say 0^0 is 1, along with 0! = 1, and so on. Makes a lot of things around Binomial theorem and the like easier. I'm not so sure if it's safe to use that when doing and calculus proofs or anything along those lines, but there, you have more useful tools for dealing with limits that might approach 0^0.
0! really is 1 in a much more reasonable sense. The empty product is the multiplicative identity. The continuous notion of ! also agrees: http://en.wikipedia.org/wiki/Gamma_function
Re: 0^0
#77[deleted]
Re: 0^0
#78The real problem here is that x^y is a single shorthand which refers to a few fundamentally different mathematical concepts (which happen to have significant overlap with each other).
First, it refers to a function f:C x N --> C, defined in terms of repeated multiplication. f(x,0) is 1 for all x != 0, and so we adopt the convention that f(0,0) is also 1.
But it also refers to a function g:C x C --> C, defined as g(x,y) = exp(y log(x)), which has a branch cut on the negative real axis of the first argument and an essential singularity at (0,0), and so g(0,0) is necessarily undefined.
The value of 0^0 depends entirely on what sort of mathematics one is doing at the time, and therefore which function one is referring to.
Re: 0^0
#79- "0^0. Why? Because mathematicians said so. No really, it’s true." - [Detailed explanation of the tradeoffs involved in choosing different definitions of exponentiation.] So, it's not "because mathematicians said so", it's because of a deep review of the tradeoffs of defining how exponentiation generalizes, the kind of thing that mathematicians happen to study more than other identifiable groups.
That's why we trust mathematicians' say-so on the topic. :)
(Well, and because they can justify it, of course.)
Re: 0^0
#80Had a set theory professor who taught us that for the non-negative integers, m^n was just the number of unique mappings from a set of cardinality n to one of cardinality m. Ergo, for all sets A such that |A| = k, k^0 is just all mappings from Ø, which is necessarily the one with empty image and pre-image. So 0^0 = 1.