Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

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

askamathematician.com

Post reply on HN
Print view View original

61–70 of 136 posts

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

#61

Post by Almaviva »

xyzzyz wrote: post #44
sliverstorm wrote: post #30

Earlier quoted context omitted.

Although my mathematician friends would probably yell at me if they heard this, as far as I see things it's hardly any different from physics. Just a model we create to describe observed phenomenon.

Oh, math only works this way in intuitive fields, like elementary calculus, elementary probability and staticstis, graph theory or Euclidean geometry. However, there are fields in math that are different -- for instance, there are topological spaces that exhibit phenomenons unseen anywhere else and that are very hard to grasp intuitively (I had very hard time trying to imagine what Cech-Stone compactification constru…

I'm exactly the intuitive thinker you describe, with only an undergrad degree in math. I have not a clue what a Cech-Stone compactification construction based on ultrafilters is.

But... I think that whatever it is, you decide on some fairly simple properties you want to satisfy, and then go off discovering what they lead to and what the consequences are. Sometimes (mostly all the time?) you get something trivial or that reduces to being isomorphic to something else, but sometimes you get out a lot more than you put in, in surprising ways. I call that a lot more like "discovery" than "invention" though both are strained as analogies.

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

#62

Post by RodgerTheGreat »

sliverstorm wrote: post #30
jpadkins wrote: post #22

Earlier quoted context omitted.

wait.. you mean math is simply made up? It's not the language of the universe? well that's just depressing.

Although my mathematician friends would probably yell at me if they heard this, as far as I see things it's hardly any different from physics. Just a model we create to describe observed phenomenon.

Bingo. Mathematics is a consistent model which can be applied to do useful things and make useful predictions, but like any logical system it springs from axioms that must simply be taken as given. Most of these axioms (Modus Ponens[1], for example) are so simple and obvious that it seems bizarre to construct a way of thinking without them, but that does not intrinsically make them "true".

[1] http://en.wikipedia.org/wiki/Modus_ponens

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

#63

Post by jberryman »

Almaviva wrote: post #58
jberryman wrote: post #51

Earlier quoted context omitted.

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…

I don't think the word "invented" is much better an analogy for real numbers than "position" is for "where" the Mandelbrot Set is. This goes beyond what is strictly math, but I don't think it's reasonable to say that properties of the real numbers (say, roots, pi, e, and so forth) are invented. In some sense they seem like the simplest thing that fits a few properties (and not that many). Similarly with Euclidean Geo…

> I realize this puts me quite firmly into the category of people being belittled here!

I hope I'm not participating in a discussion where people are being belittled! I think all the viewpoints here are fascinating.

Yours is certainly interesting and valid, I'm simply offering that the "full stop" at the end of your previous comment doesn't reflect the kind of deeper metaphysical questions that underly the whole progression of the history of math.

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

#64

Post by ubasu »

dxbydt wrote: post #60
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…

>most of us don't have a good conceptual model for what exponents really do Instead of matching math to real world objects (1= one banana, 2 = two bananas, 1+2 = 3 bananas etc. ) and building up to exponentiation, multiplication etc. thereby introducing all sorts of paradoxes, group theory dodges all that and treats the whole thing as a very consistent rule-based system. Things fall into place quickly once the rules…

Group theory is formed by abstracting out the observed properties of number systems. If you want to show that some collection is a group, you will have to do the computations to show that they follow the rules, in which case, it helps to have an understanding of the mechanics of the computation.

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

#65

Post by dataduck »

yaks_hairbrush wrote: post #17

The high school teacher in the link is a B.S. in math education. They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essentially a series of completed works, with little history attached to it. This teacher probably hasn't thought critically about why, say, 1/x^2 = x^(-2). By contrast, a mathematician has a Ph.D. in m…

Not entirely fair. While there is a lot of choice about how to define stuff around the edges of mathematics, after the axiomatic choices have been made, there's quite a lot to discover in their structure. Particularly if the relevant axioms relate to something outside of mathematics (the examples are too many to even scratch but start out thinking of the definitions of natural and rational numbers) then you really are discovering things in the same way as a physicist - in fact, this is how many physicists go about discovering things, for better or worse.

In most situations it makes sense to define exponentiation as repeated multiplication, and a^0 as the absence of multiplication by a, hence a^0 = 1 as the multiplicative identity. I wouldn't introduce the idea of anything else to a student unless they specifically asked me about one of the problems which can arise in choosing 0^0 = 1.

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

#66

Post by sid0 »

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?

On the sheet of paper in front of me, as the hypotenuse of the isosceles right-angled triangle with unit length I drew a moment ago. Irrational numbers are probably a bad example of what you're talking about.

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

#67

Post by pshapiro »

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.

Thanks, I was gonna say the same thing. :)

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

#68

Post by lotharbot »

xyzzyz wrote: post #55
Dove wrote: post #45

Earlier quoted context omitted.

What is "indeterminate form"? http://en.wikipedia.org/wiki/Indeterminate_form You see, 0^0 = 1, and it's obvious to a mathematician . . . we define 0^0 = 1, to be consistent with exponentiation rules Well, you're going to be inconsistent with them no matter how you define it, since, as you point out, x^y should be zero if you approach (0,0) along the x=0 axis, and it should be one if you approach along the y=0 axis.…

By "exponentiation rules" I mean algebraic equalities, like a^x * a^y = a^(x+y). Most of them work no matter if you define 0^0 = 1 or 0, but some of them are cleaner with 0^0 = 1. It's also consistent with cardinal and ordinal exponentiation (look it up). "Approaching along x axis" is not algebraic notion, it's analytic one. 0^0 makes no less sense than, say, -e^(i pi). They're both 1 because we define them like this…

I haven't seen any reason why -e^(i pi)=1 could be considered incorrect. It's consistent with the Taylor Series expansion of e^x. It's consistent with the view of complex exponentiation as rotation. I don't know of any particular problems that arise from taking -e^(i pi)=1.

This is not the case with 0^0=1, which is inconsistent with many limits. That's why 0^0=1 is an agreed-upon convention sometimes. http://en.wikipedia.org/wiki/Exponentiation#Zero_to_the_zero... has a fairly nice summary of the issues involved in defining it.

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

#69

Post by MetaMan »

xyzzyz wrote: post #56
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…

Are you sure you're replying to right post? I ask, because nothing I see in yours can be seen as reply to mine -- it actually repeats my point.

joe's reply clarified things somewhat - at least for me. Are replies always supposed to be rebuttals?

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

#70

Post by cperciva »

jpadkins wrote: post #22
yaks_hairbrush wrote: post #17

The high school teacher in the link is a B.S. in math education. They're usually reflexive Platonists, believing that math is out there, and we merely discover it. This is a result of the teaching of undergrad math as essentially a series of completed works, with little history attached to it. This teacher probably hasn't thought critically about why, say, 1/x^2 = x^(-2). By contrast, a mathematician has a Ph.D. in m…

wait.. you mean math is simply made up? It's not the language of the universe? well that's just depressing.

God created the integers. All else is the work of man.
Post reply on HN