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

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

#31

Post by kalid »

yaks_hairbrush wrote: post #18
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…

> 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.

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 evolved into its own entity and we saw numbers on a "line", a powerful mental model (why not 2d numbers? N-dimensional numbers? etc.)

Re-teaching that 3^0 = 1 "because the math is convenient" doesn't help us build a mental model of what exponents could be (I know you don't agree with this, just stating it again because the lack of intuitive explanations for math is a major pet peeve of mine).

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

#32

Post by jackpirate »

dxbydt wrote: post #26

Try these for size ( actual interview questions ) PITA interviewer > What's bigger, e^pi or pi^e ? If you get past that one, PITA interviewer > What's i^1 ? Clever student> Its just 1 unit on the imaginary axis. PITA interviewer >Good! So then, whats i^i ? Clever student > Probably a few more units on the imaginary axis! PITA interviewer >Then why does google say 0.207 ( http://www.google.com/search?q=i^i ) Clever st…

What were you interviewing for?

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

#33

Post by yaks_hairbrush »

noahl wrote: post #23
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…

> 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. It's not entirely clear what you mean by this, but in the most obvious interpretation this idea is correct. Here's what I mean by that: math deals with pure logic. All logical deductions (…

> Here's what I mean by that: math deals with pure logic. All logical deductions (derivations from axioms to conclusions in formal systems) are in a sense "out there", waiting to be discovered. Certainly they have always been true, before people knew about them, and there are deductions that are true even though no one knows about them yet.

From one viewpoint, math deals with pure logic. It's something of a poor viewpoint from my perspective: Newton couldn't back up his calculus with logic, Euler and Riemann made numerous unfounded assumptions when looking at the zeta function, Heaviside built a telegraph across the Atlantic despite lacking a proper logical foundation.

My personal viewpoint is that mathematics is a series of shortcuts for understanding and manipulating a wide variety of phenomena, and mathematical research is the development of further shortcuts. Often logic comes in, but it's usually after you get the result.

Nothing wrong with logic (I sure do like computers, for example!), it's just not the panacea for anything mathematical.

> But either way, I think you are wrong that high school teachers don't think critically about why 1/x^2 = x^(-2). Some may be bad teachers, but I have known a lot of incredibly good high school math teachers, and I suspect they have thought about this. In fact, when I talk to mathematicians about high school education, they usually agree that people doing research know more parts of mathematics than high school teachers, but good high school teachers have a much deeper understanding of elementary math than people doing research, because they have had to approach it from many different angles in order to teach different students.

You're absolutely right about this, and I was specifically thinking of two very bad high school teachers when I wrote my post. The fact is that I know very few HS math teachers, and so my opinion is quite clouded by these two.

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

#34

Post by foysavas »

Only after seeing how many comments on the article attempted to genuinely refute the article did I get a sense of how few people grasp the foundation of mathematics (that is, the composition of arbitrary assumptions to agreeable statements).

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

#35

Post by jkent »

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.

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

#36

Post by xyzzyz »

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 two sequences a_n, b_n, such that lim a_n = a, lim b_n = b, we have lim f(a_n, b_n) = f(a, b). That's why lim (a_n)^(b_n) = a^b if (a, b) != (0, 0), and this is "determinate form". But if (a, b) = (0, 0), then no matter how we define 0^0, it does not follow that lim (a^n)^(b^n) = a^b = 0^0, because in this case, lim (a_n)^(b_n) can be every positive value, and so mathematicians used to call it "indeterminate form" (it's not common today, though). So, since this problem is unsolvable in a consistent (continuous) way, we define 0^0 = 1, to be consistent with exponentiation rules, at least.

I've never seen a need for an "intuitive" explanation of exponentiation -- the usual definition is as intuitive as one can get. The thing is, most people do not know, _why_ expressions like pi^e are supposed to make sense -- they just take exponentiation as given. Only then they need to make up some explanation why "exponentiation rules" are like this, and what exponentiation is about. Hell, people don't even know what real numbers are! How are they supposed to make sense of exponentiation with exponent other than natural number?

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

#37

Post by tokenadult »

kalid wrote: post #31
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.

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 said, I finally realized that I was limiting my mathematical development by insisting that every mathematical idea must appeal to my visual intuition. Some mathematical ideas are proven even if they don't appeal to visual intuition. In the words attributed to John von Neumann, "in mathematics you don't understand things. You just get used to them."

http://en.wikiquote.org/wiki/John_von_Neumann

That point of view makes a lot of sense to many of the best mathematicians.

One more example of really interesting visual explanations of mathematical concepts is Visual Complex Analysis

http://usf.usfca.edu/vca/

by Tristan Needham. The book is delightful, and well reviewed, but it is not the sole path toward getting used to complex analysis.

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

#38

Post by imminentdomain »

Doesn't the first argument have a logic error, in that x^(1-1) = x^1x^(-1) has a caveat for x not equal to 0, so it shouldn't prove anything for x = 0. In other words, it just says x^0 = 1 for any x =/= 0.

Using Abstract Algebra, I think 0^0 = 1 is completely accurate. The power function (y^x) could be defined to be the amount you times (x times) you apply the operation between the y on the identity element. In our usual numbers that looks like y(y(y...(y1)...)). When x is negative y becomes the multiplicative inverse of y and everything else remains the same.

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

#39

Post by Almaviva »

yaks_hairbrush wrote: post #18
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…

> 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.

You add a bunch of stuff and you get a total. You add nothing and 0 is the total because it's the identity and starting point for addition. You multiply a bunch of stuff and you get a product. You multiply nothing and you get 1 because that's the identity for multiplication.

(That's the same as saying "the rules of algebra work out" but there's maybe something intuitive about multiplying nothing and getting back the thing that doesn't change the result of multiplication?)

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

#40

Post by webspiderus »

_delirium wrote: post #24
webspiderus wrote: post #20

Earlier quoted context omitted.

This is completely off-topic, but did you mean to call it l'hospital as opposed to L'Hôpital? I remember even my calculus book had similar errors, and I always wondered if it was just because the two looked so similar (or if there was any more reasoning behind it). didn't mean to nitpick, your comment just triggered a repressed train of thought :)

His original name was actually Guillaume de l'Hospital; French spelling reforms later did away with a number of cases of silent 's' (which had been silent for a long time already), replacing it with a circumflex over the preceding vowel. See: http://en.wikipedia.org/wiki/Use_of_the_circumflex_in_French... and: http://en.wikipedia.org/wiki/Guillaume_de_l%27H%C3%B4pital#c...

ah, thanks - I had a feeling it was something like that! I figured a Calculus book would probably get it right :)
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