What does 0^0 equal? Why do mathematicians and high school teachers disagree? • Hacker News Board

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#111

Post by lotharbot » Thu, Aug 25, 2011, 4:49 AM UTC

xyzzyz wrote: post #108

lotharbot wrote: post #107

Earlier quoted context omitted.

The whole point of distinguishing some forms as "indeterminate" is to work around the fact that you're trying to conduct operations on the real numbers that are not defined under the field axioms of the real numbers (or the axioms of the extended reals [-inf,inf]). That's where its essence is rooted; that's why this whole affair exists -- the fact that 0/0, 0^0, 0xinf, inf-inf, etc. are not well defined by our axioms…

Exponential function is not defined by the axioms of real numbers, as opposed to addition and multiplication, so this point is irrelevant -- you can define it in any way you want. There's no inconvenience in defining 0^0 = 1, apart from misunderstanding the concept of limits by some people. Defining 0^0 = 1 is universal convention -- people either do it like this, or do not define 0^0 at all, which I'm fighting again…

> "people either do it like this, or do not define 0^0 at all, which I'm fighting against."

I think it's silly to fight against it. There are circumstances in which leaving it undefined is good, and in which trying to define it as 1 would lead to either misunderstandings (in the case of beginners doing limits, a case you are too quick to dismiss) or actually incorrect (an equivalent problem in the hyperreals could violate the transfer principle).

It's a broad convention, but it is not universal, and it shouldn't be.

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#112

Post by parallel » Thu, Aug 25, 2011, 4:58 AM UTC

vessenes wrote: post #104

parallel wrote: post #57

> How would you explain to a 10-year old why 3^0 = 1 You draw the line 3^x. It "passes through" 1 when x = 0. So don't think about the point, think about the line. It's not rigorous but it's intuitive. http://fooplot.com/index.php?q0=3^x edit: added link and fixed typos

This is precisely the problem, though. The same 10 year old draws two lines: 0^x, and x^0. They clearly do not meet.

Yeah, I was just talking about the easier concept of x^0. Going the other way, 0^x it makes a lot less sense to me. I know that for all positive x, 0^x = 0. That it pops up to 1 at the origin means the thing is not smooth so drawing lines isn't going to help us.

Plotting z = y^x could be interesting but we're going to see asymptotic behaviour at x = y = 0 (when going in the x direction). So no easy wins there.

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#113

Post by perfunctory » Thu, Aug 25, 2011, 6:54 AM UTC

  Python: 0**0 == 1
  Javascript: Math.pow(0, 0) == 1
  Java: Math.pow(0, 0) == 1

Therefore 0^0 = 1

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#114

Post by danvet » Thu, Aug 25, 2011, 7:19 AM UTC

joe_the_user wrote: post #54

xyzzyz wrote: post #36

Earlier quoted context omitted.

What is "indeterminate form"? What does it mean for expression to "have a specific solution"? You see, 0^0 = 1, and it's obvious to a mathematician. The only problem is that the function f: [0, \infty) x R -> R, f(x, y) = x^y is discontinuous in (0, 0) and that's what causes problems -- for instance, this is the source of the whole "indeterminate form" notion. If a function f is continuous in (a, b), then for every t…

Mathematicians don't argue about what an expression "really is" (or at least, real mathematics doesn't involve this). They define functions and use axioms to prove theories about them. "No really" . Mathematics just isn't concerned with this stuff. Sometimes infinity it defined as single point making the real number compact, sometimes a "positive infinity" and a "negative infinity" are defined. Sometimes you add poin…

This.

Math is a tool (and sometimes abused for pure pleasure, 200 years later applied to make hard crypto work). If your definition doesn't make sense for the application, fix your definition and get over it.

Another example I've recently often bitched about in discussions is modern measure theory and its application to probability calculations. People just don't get the concept of theorytically possible event, but probability 0, i.e. ignore this. But without Lebesgue integration L_p function spaces are not complete and an awful lot of stuff stops to work properly. Among them essentially all of modern physics.

The sane approach is to get over the "this doesn't make intuitive sense" bitchering and just use defintions to derive useful results. And after a few years of playing around with stuff and applying the un-intuitive definition, it's becoming intuitive ;-)

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#115

Post by wlievens » Thu, Aug 25, 2011, 7:21 AM UTC

jkent wrote: post #35

kalid wrote: post #11

Technically, 0^0 is an indeterminate form and has no specific solution. Accurate but unhelpful. Practically, 0^0 highlights the issue that most of us don't have a good conceptual model for what exponents really do. How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out". I use an "expand-o-tron" analogy http://betterexplained.com/articles/understanding-exponen…

I successfully managed to explain 3^0 to 10 year olds (as a recovering high school math teacher) as: 3^2 = 9 3^1 = 3 (divide 9 by 3) 3^0 = 1 (divide 3 by 3) 3^-1 = 1/3 (divide 1 by 3) etc This can logically be explained as n^0=1 for all real numbers. Unfortunately this doesn't really handle 0^0 but fortunately 10 year olds are rarely that difficult.

That's what the parent poster meant with "it's necessary to make the algebra of powers work out".

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#116

Post by psykotic » Thu, Aug 25, 2011, 8:08 AM UTC

nova wrote: post #53

I like my high school math consistent with set theory. ø : empty set, 1 : {ø}, A : nonempty set, ~= : isomorph to. A^ø ~= 1, because there is only one function ø->A, the empty function. ø^A ~= ø, because there is no function with empty codomain and nonempty domain. ø^ø ~= 1, because there is again one function ø->ø, the empty one. So yes, 0^0 = 1.

This! The empty function is the function whose graph is the empty set. When you stop thinking of functions as symbolic expressions and make friends with the empty set, it's all clear as crystal. The empty set has perplexed a lot of people over the ages, so it's unsurprising that 0^0 prompts puzzlement. But with the benefit of modern hindsight, there's really no reason to stay confused. Set theory in the large is still a great mystery but we got the empty set well figured out by now.

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#117

Post by lloeki » Thu, Aug 25, 2011, 8:19 AM UTC

kalid wrote: post #11

Technically, 0^0 is an indeterminate form and has no specific solution. Accurate but unhelpful. Practically, 0^0 highlights the issue that most of us don't have a good conceptual model for what exponents really do. How would you explain to a 10-year old why 3^0 = 1 beyond "it's necessary to make the algebra of powers work out". I use an "expand-o-tron" analogy http://betterexplained.com/articles/understanding-exponen…

> Technically, 0^0 is an indeterminate form and has no specific solution.

Precisely, as this is the true mathematician answer: "it depends where 0^0 comes from".

As a f(x,y): RxR->R function, come from the top of the R² plane and 0^0 is 0 but come from the right side and it's 1. Limits and extension by continuity give us this easily enough for fh:x->x^0 and fv:y->0^y.

Writing this I asked myself, what if we came from some funky other path, like the diagonal, or a curve?

h: R->RxR, x->(x, 0) defines "coming from the top", and foh = fh

v: R->RxR, x->(0, y) defines "coming from the top", and fov = fv

d: R->RxR, x->(x, x) defines coming along the diagonal, where things could get interesting.

s: R->RxR, t->(e^(at)sin(t), e^(at)cos(t)) defines coming along a log spiral whose tangent at t=0 is vertical, so fos looks like fun around t=0.

Now what happens if we build a path function p: RxR->RxR, (t, z)->? that endlessly approaches v when z->0? the log spiral with z=1/a as a parameter is a possible one. With such a p function, what does lim fop(x) when x->0 (which is a function of z) look like when subsequently z->0?

Damn. It was supposed to be a two-line comment.

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#118

Post by rikthevik » Thu, Aug 25, 2011, 9:38 AM UTC

The engineer in me says, "Why don't you guys pick something, and I'll go ahead and use it in my calculations (where it's appropriate)? I don't care much about the theory behind it."

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#119

Post by nhaehnle » Thu, Aug 25, 2011, 9:46 AM UTC

jberryman wrote: post #51

Almaviva wrote: post #41

Earlier quoted context omitted.

Math is out there. I don't understand how anyone can say that the Mandelbrot set was created or is formed from arbitrary axioms. It was discovered full stop.

Your parent never says anything about arbitrary axioms. To your main point: are irrational numbers, say, "out there"? If so, where? Until a few centuries ago it was mathematical standard practice to fudge 1/2 as 25/49 when taking it's square root. But then mathematicians invented (some would argue) the notion of an irrational, because it was, well... useful. There are real metaphysical questions here; there have been…

To your main point: are irrational numbers, say, "out there"? If so, where?

I believe this to be a bad counterpoint, because you could ask the same thing about natural numbers as well. I've personally never seen a natural number. Sure, I have seen and worked with lots of representations of natural numbers, but the numbers themselves are - as far as I understand it - not physical objects. There is no qualitative difference between natural numbers and any other mathematical objects in that respect. They are all on the same "plane of existence".

At least that's a valid world view or ontology. I know that not everybody thinks like that, but besides clarifying what the parent poster probably meant I think it's not a very meaningful discussion.

Re: What does 0^0 equal? Why do mathematicians and high school teachers disagree?

#120

Post by Dn_Ab » Thu, Aug 25, 2011, 11:40 AM UTC

danvet wrote: post #114

joe_the_user wrote: post #54

Earlier quoted context omitted.

Mathematicians don't argue about what an expression "really is" (or at least, real mathematics doesn't involve this). They define functions and use axioms to prove theories about them. "No really" . Mathematics just isn't concerned with this stuff. Sometimes infinity it defined as single point making the real number compact, sometimes a "positive infinity" and a "negative infinity" are defined. Sometimes you add poin…

This. Math is a tool (and sometimes abused for pure pleasure, 200 years later applied to make hard crypto work). If your definition doesn't make sense for the application, fix your definition and get over it. Another example I've recently often bitched about in discussions is modern measure theory and its application to probability calculations. People just don't get the concept of theorytically possible event, but p…

Interesting that you say that. I've skimmed but have been meaning to properly read Nelson's: Radically Elementary probability theory http://www.math.princeton.edu/~nelson/books/rept.pdf and http://www.stat.umn.edu/geyer/nsa/. They do away with that problem all together as well as infinite constructions (replaced by hyperfinite) by replacing measure theory with non standard analysis. The gain is at least increased intuitiveness.