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

askamathematician.com

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

#41

Post by Almaviva »

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…

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.

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

#43

Post by kalid »

tokenadult wrote: post #37
kalid wrote: post #31

Earlier quoted context omitted.

Yeah -- that may have been the original motivation, but repeating it as an "explanation" reinforces the notion that math is a bunch of rules (vs. models you can construct and manipulate in your head). 0 probably started as a placeholder symbol for "naught", i.e. nothing to write, and the first scribes were taught "Just write a circle when you have nothing to report". But, with greater understanding of numbers 0 evolv…

the lack of intuitive explanations for math is a major pet peeve of mine I'm a very visual thinker, and that is one reason I enjoy the new Art of Problem Solving textbook Prealgebra by Richard Rusczyk, David Patrick, and Ravi Boppana-- https://www.artofproblemsolving.com/Store/viewitem.php?item=... it is full of interesting visual "explanations" and substitute for proofs in a book intended for a young audience. That…

Thanks for the pointer! I found Needham's book awesome, I've barely made a dent in it but love the visualizations.

I don't think visualization is the only intuitive method -- you can have a general "sense", not sure how to put it more specifically -- I have a "sense" about growth of e without a specific diagram.

Agreed that not every concept can be understood... yet. There's a quote I love to rail on, in reference to Euler's formula:

"It is absolutely paradoxical; we cannot understand it, and we don't know what it means, but we have proved it, and therefore we know it must be the truth." (Benjamin Peirce, 19th-century Mathematician)

Really? Yes, it may be baffling at first, but we can _never_ understand it? Only if that's our attitude :).

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

#44

Post by xyzzyz »

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.

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 construction based on ultrafilters look like, and eventually I gave up after I understood that this space is only supposed to satisfy an universal property, and not have any intuitive shape). Deep results in algebraic methods in topology say interesting things about the models, while being completely indescribable outside of them, apart from some trivial examples (for instance, what _exactly_ are cohomology classes?).

The bottom line is, if the only things you've seen are geometry and calculus, with intuitive concepts like speed of change, area, length, then yeah, it's only making our intuitions more formal. Otherwise, it's something completely different.

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

#45

Post by Dove »

xyzzyz wrote: post #36
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…

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…

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.

0^0 is simply an expression that doesn't make sense. There isn't an answer, and there certainly isn't something we could agree to define it as. It is gibberish, nothing more, nothing less. One cannot assume just because there are mathematical symbols on paper that they make sense.

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

#46

Post by bdr »

The right convention is also "obvious" to practitioners of combinatorics. The exponentiation x^y, for integer x and y, is the number of possible strings of length y from a set of letters of cardinality x. (Hence 28 possible bytes.) How many ways are there to make a string of length 0, regardless of the alphabet size? Just one... you don't do anything.

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

#47

Post by kbutler »

I'll stick with the grade-school math approach, at least until I need to approach it differently.

  4^2 = 2 fours multiplied = 4 * 4 = 16
divide by 4 - so you take away one of the 4s by division(canceling like terms like we do in grade school fraction math):

  4^1 = 4*4/4 = 4
divide by 4 again

  4^0 = 4*4/(4*4) = 1
divide by 4 again!

  4^-1 = 4*4/(4*4*4) = 1/4
Now try it with 0:

  0^2 = two zeros multiplied = 0*0 = 0
Divide by 0. Uh-oh. Well, let's keep following grade school fraction math and cancel like terms:

  0^1 = 0*0 / 0 = 0
Divide by 0. Hmm - keep canceling like terms.

  0^0 = 0*0 / (0*0) = 1
But what's 0^-1? grin

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

#49

Post by Someone »

3am wrote: post #15

Fun digression late on a Wednesday :) The indeterminate form seems the most correct based on the analysis of the limit of f(x,y) = x^y as x approached zero from different paths. I had always thought of it more of an algebraic identity thing; x^n * x^m = x^(n+m). Obviously x^(n) = x^(n+0) = x^n * x^0 which can only be satisfied if x^0 = 1. But this article (and really, the wikipedia treatment that beej71 linked to) ma…

"Obviously x^(n) = x^(n+0) = x^n * x^0 which can only be satisfied if x^0 = 1."

...or if x^n = 0, which is true if x=0 and n != 0

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

#50

Post by xyzzyz »

qntm wrote: post #21
yaks_hairbrush wrote: post #18

Earlier quoted context omitted.

> 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". Actually, that's exactly the reason 3^0=1: it was the definition that preserved the most identities. Agreed that this explanation doesn't really help intuition.

What is true in mathematics is whatever leads to no logical contradictions.

There are cases when both a claim and its negation do not lead to contradiction, but them both being true obviously does.
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