What does 0^0 equal? Why do mathematicians and high school teachers disagree?
askamathematician.com
askamathematician.com
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-exponents-...
to wrap my head around what exponents are really doing: some amount of growth (base) for some amount of time (power). This gives you a "multiplier effect". So, 3^0 means "3x growth for 0 seconds" which, being 0 seconds, changes nothing -- the multiplier is 1. "0x growth for 0 seconds" is also 1, since it was never applied. "0x growth for .00001 seconds" is 0, since a miniscule amount of obliteration still obliterates you.
This can even be extended to understand, intuitively, why i^i is a real number (http://betterexplained.com/articles/intuitive-understanding-...).
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.
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).
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
by Tristan Needham. The book is delightful, and well reviewed, but it is not the sole path toward getting used to complex analysis.
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 :).
(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?)
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
Unfortunately this doesn't really handle 0^0 but fortunately 10 year olds are rarely that difficult.
3^3 = 3*3*3*1 = 27
3^2 = 3*3*1 = 9
3^1 = 3*1 = 3
3^0 = 1 = 1
0^3 = 0*0*0*1 = 0
0^2 = 0*0*1 = 0
0^1 = 0*1 = 0
0^0 = 1 = 1
What I'd intended was that the value of 1 was reached by applying the same algorithm that was applied to arrive at the other values: start with 1, multiply by the base once per instance of the exponent. No division involved.
It's still incorrect if we want to be strict, of course. That algorithm is not quite the definition of exponentiation, because that algorithm can't really be extended to work outside rational exponents. Exponentiation is defined across complex numbers (ignoring 0^0 for the moment). I think this is acceptable because I'm only shooting for an explanation, which doesn't need to be strict.
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?
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.
0^0 makes no less sense than, say, -e^(i pi). They're both 1 because we define them like this. If you think that -e^(i pi) makes more sense than 0^0, please, explain me why.
Also, mathematicians agree in this, seriously. Go and ask one.
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.
So what? It's only a problem if you want the exponentation function to be continuous, so you escape the problem by leaving it undefined. You could place similar unbased requirements on complex exponentiation to make it seem incorrect. For instance, real exponentiation always gives a positive value for positive base, while complex does not, so e^(i pi) = -1 is wrong. I agree that this is ridiculous requirement, but leaving 0^0 undefined because the math is not as we want it to be (e.g. exponentation is not continuous) looks just as ridiculous and silly to me.
On the other hand, putting 0^0 = 1 makes it consistent many combinatoric formulas, and is also consistent with cardinal exponentation, where nobody objects to 0^0 = 1, when you look at 0 as the cardinal number.
Even the original article noted that we don't choose the 0^0 convention because it's "correct", but because it's "nice" -- which is why we define it that way in the contexts where it makes sense to define it that way. If you're working with combinatorics, 0^0=1. If you're working with cardinal exponentiation, 0^0=1. If you're taking limits, or working in the hyperreals, or in certain other contexts, the convention doesn't apply. In some circumstances, 0^0 isn't even a valid statement -- like if you're working directly with the field axioms of R.
Recognize what context you're working in, and what assumptions or conventions apply in that context. That's just good mathematics.
But the whole point of distinguishing some "forms" as "indeterminate" is to work around the discontinuity of elementary functions! My absolutely first sentence in this thread is asking, what exactly the "indeterminate form" is. I'm asking this question, because this not a formal notion and you will not find any formal definition of it. Its existence is rooted in the fact that for no reason other than the tradition (and convenience) we use special notation for some functions. Instead of +: R x R -> R, +(2, 3) = 5, we write 2 + 3 = 5. The same goes for ^: [0, \infty] x R -> R. The only reason we have all those fancy limit evaluating laws is because these function are continuous most of the time. For instance, + is continuous everywhere, so lim +(a_n, b_n) = +(lim a_n, lim b_n), if both lim a_n and lim b_n make sense. Similarly, /: R x R - {0} -> R is also continuous everywhere, so lim /(a_n, b_n) = /(lim a_n, lim b_n), if the right hand expression makes sense. If it does not make sense, for instance when both lim a_n and lim b_n are equal to zero, we need cannot approach this problem in such a simple way. Now, some people would call /(0, 0) an "indeterminate form", which makes for me no more sense than calling f(0, 0) an indeterminate form, where f(x, y) = log_(1/x) (y) -- while f is continuous everywhere where defined, you cannot extend its domain to contain (0, 0) for it to stay continuous, just like you cannot do it with / function.
As I repeated many times, the whole affair is because ^ seem to be more familiar than beta function (we have a special notation for it, for instance), people want it to behave nicely, so that for instance it conforms to some arbitrary limit evaluating laws, missing the whole underlying concept of continuity.
This actually relates to all three examples I've presented where the 0^0=1 convention fails. It should be treated as an indeterminate form in limits because it's not well-defined by the axioms of the real numbers; it's also not well-defined by the axioms of the hyperreals, but division of infinitesimals is well-defined in the hyperreals, which gives us an alternate method of computing limits that avoids the "indeterminate form" entirely.
Let me reiterate: 0^0 is not defined under the field axioms of the real numbers. The choice to define it as 1 is a convention which makes certain math easier, in certain areas of mathematics. It is by no means a universal convention; it is by no means the one and only correct definition of 0^0. You continue to argue for the convention, but miss the larger point that it is a convention which is chosen for convenience, and which is not always appropriate.
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
I am not. Are you? Let me reiterate: the whole concept of "indeterminate forms" (which, I repeat, is not formal at all) stems from misunderstanding the process of taking limits.
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.
Mathematician here; we do not. See http://math.stackexchange.com/questions/11150/zero-to-zero-p....
More precisely, as Arturo Magidin points out at http://math.stackexchange.com/questions/11150/zero-to-zero-p..., if we view exponentiation in the 'discrete setting', then $0^0$ must be $1$; whereas, if we view it in the continuous setting, there is simply no good answer—unlike $e^{i\pi}$, which also lives in the continuous setting, but has a perfectly good, unambiguous answer. (lotharbot gives a nice explanation below of the ways that this is consistent with existing mathematics; but it can also be derived from the definition of the exponential function, with no further arbitrary conventions needed.)
Mathematician here; we do not.
It seems I was a little too bold with my claim. All the mathematicians I know (and I'm a mathematician as well) agree with 0^0 = 1. It's a folklore specific thing, I guess.
As am I, by training if not by profession. As is lotharbot. You're in a thread full of mathematicians. :)
Which is what I would expect on this site, actually. I'm always timid making technical claims here unless I'm sure I'm correct; it seems to be a place frequented by arbitrarily large fish.
To answer your question, if 0^0 = 1 is _the_ answer in discrete setting, and it's _an_ answer in continuous setting, why don't we just agree that 0^0 = 1 and stop creating confusing situation where sometimes it's defined and sometimes it's not.
I'm not persuaded it is always the answer. I think the fact that it is an indeterminate form in limits is a forceful enough demonstration of that. It all depends on context. If I came across a 0^0 in, say, an engineering context, my first instinct would be to check whether the formula was defined in that case, not to just assume that 1 would work.
I mean, it's like 1/0. If you're working in R, that's simply illegal. If you're working in R*, it's the infinite point. If you're taking a limit, it means "unbounded". If you're working in my favorite field, the hyperreals, it could be any number of flavors of infinity depending on the flavor of zero it was.
It would be foolhardy to try to define the symbol; without a context to supply some sort of sense, it is nonsense. And that is how I feel about 0^0 as well.
From this point of view, the whole notion of "indeterminate form" makes just as little sense as distinguishing some arbitrary class of functions and calling them "elementary". Why are some points of discontinuity of some functions more special than other points of discontinuity of other functions? Why sin is more elementary than gamma? Historical heritage of confusion, I guess.
Similarly, if you're evaluating a limit and you get 0^0 you need to do more work. You can't just stop and say "oh, that's 1". It depends on what function you used to get there -- x^x will give you a different answer from ( e^(-1/x) )^x. Again, it has nothing to do with the continuity of exponentiation. The issue is whether the convention of 0^0=1 is correct in the specific part of mathematics you're working in.
The same argument can be made if you're working in the hyperreals, or if you're working with field axioms -- the convention 0^0 doesn't work in that context.
Please, by all means, use the convention 0^0=1 when it's appropriate. But understand that it's not always appropriate. Not every mathematician works in the particular subschool that you do; not every mathematician is going to find your convention appropriate.
What do you mean by "getting 0/0" in the process of evaluating limits?
The issue is not the continuity of x/x; the issue is whether setting a convention for 0/0 would give you the right value for a limit.
Please, tell me - what is the relation between lim f(a_n) and f(lim a_n) ?
Because sometimes it's better not to. Sometimes it's inconsistent with our definitions.
Just like sometimes we agree that you can't divide by zero, and sometimes we agree that you can. Sometimes infinity is an actual value (say, in the extended reals), and sometimes it's just a symbol for "unbounded". Sometimes we agree that you can't take the square root of a negative number, and sometimes you can. Sometimes we use the axiom of choice, and sometimes we don't (and you can have an awful lot of fun either way!)
Mathematics is contextual. How various operations behave depends on which axioms and conventions are being used.
I'd love to see even one example of 0^0=1 being inconsistent with a definition. The closest I've ever seen is that it bothers people that for reasons of their own had their hearts set on (x,y) -> x^y having no discontinuities...
Perhaps it's more precise to say "Because sometimes it's better not to. Sometimes there is no canonical choice that follows from our definitions, and it doesn't help to assign an arbitrary value that doesn't help solve any related problems."
What is "x" equal to? In general, I mean, not in the context of any equation like "x+1=2". You could say "x=7 in the study of free variables over integers when no other constraints are given", and that is completely consistent with the rest of mathematics, and yet would not be particularly useful and introduces an ugly (philosophical weasel word, yes) asymmetry in the theory (I'd say it introduces a gauge invariance (https://secure.wikimedia.org/wikipedia/en/wiki/Gauge_theory), but I'm really not qualified to discuss that in a rigorous way.)
Someone like Scott Aaronson could put this claim on more solid footing, but I would state that, intuitively, "assigning a value to an indeterminate form leads to a more complex definition of a mathematical system" in some formal complexity-theory sense.
Thinking back on my math education, part of the difference in viewpoint may be the first time I rigorously met the continuous-domain exponential.
This was in real analysis. Exponentiation is defined first for positive integer exponents, and then for rational exponents. All elementary. Then it's extended to real-valued exponents by taking the limits of rational numbers, and appealing to continuity.
I just looked, this is exercise 6 in chapter 1 of baby Rudin.
So, because the notion of limit and continuity is embedded in this definition of the exponential function, it's natural to "approach" (groan) 0^0 as a special case, because the conditions of this definition (continuity) don't hold.
Technically, 0^y is 0 only if you approach it from the right: y>0. To the left of 0 it is indeterminate or infinity depending on how you look at it. x^0, however, makes sense for all x and is always 1
"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 points to a given function to make it more tractable and sometimes you don't. But none of this "means" anything. The real number line can be embedded in a number of topological spaces. At least two division rings and various things (the complex numbers are most common). The way you extend a given function (say e^x) is going to vary depending on what space you're looking at as well as what topic you're interested in.
Math works with definition systems and get theorems out of them. If you want to know what something "really is", consult philosophy or something.
It turns out to be so. If a reply is not a rebuttal, it is usually preceded with something like "To clarify, ..." or "I wanted to add, that ...". I just got confused without it.
>If you want to know what something "really is", consult philosophy or something.
You mean like the "foundations of mathematics"? http://en.wikipedia.org/wiki/Foundations_of_mathematics
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 ;-)
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 are laid out explicitly.
Consider: finite abelian group with only 3 elements a,b,c. Given a+b=c, a+c=a, what's b+b ? Hmmm...okay, if a plus c is a, then c is acting like zero. So b+c must be b. since addition is commutative (abelian gp), b+a must be a+b which you said was c. So now we know b+a=c, b+c=b, so b+b better be a !
Students are easily convinced because you've laid out the rules very explicitly. In fact, they'll try to convince you that b plus b better be a because that's the only way to make the cayley table work out! (http://en.wikipedia.org/wiki/Cayley_table)
There are several books that argue that the teaching of Abstract Algebra must precede Calculus for this very reason. With Calculus, the mapping of math to real-world objects leads to all sorts of messy realities. With group theory, you dodge that mess by simply stating rules upfront.
My claim is the exact opposite. I claim you don't need to understand the mechanics ( just blindly abide by the rules of the group or abelian group or finite simple group or whatever), which is why the approach is better. If you show a monkey red means stop and green means go and reinforce these rules by rewarding with a banana, eventually the monkey will stop when he sees the red. Not because he understands the mechanics of traffic management. Simply because he is abiding by the rules. Similarly, large portions of math can be approached by either the definitional route ( ie. rules ie. define propositions & theorems that logically follow if those props held ) or via trying to understand actual mechanics by mapping everything to real world phenomena ( x = distance, dx/dt = velocity, d/dt(dx/dt) = acceleration etc. ) which are problematic because the mapping breaks down due to the nature of physical reality ( like friction etc. )
How would one explain say Hilbert's 7th problem via the actual mechanics ?
If a is algebraic and b is irrational show a^b is transcendental.
What does that even mean when you map them to the real world ? Instead, the solution is to build upon theorems that logically follow from the axioms you start out with. Problem: http://en.wikipedia.org/wiki/Hilbert%27s_seventh_problem Solution: http://terrytao.wordpress.com/2011/08/21/hilberts-seventh-pr...
(Warning, ascii math is confusing and ambiguous to read. Sorr.)
Exponentiation of group "multiplication" does not immediately seem amenable to the reals, sure. But real exponentation does form a group, as shown here:
Define x_g(r) = the function that raises a Real/{0} (non-zero real) number r to the exponent x (in the sense of of some reasonable definition of exponentiaton of continuous functions). Define X = the set x_g() functions corresponding to all reals (including 0)
Define x_g y_g as composition: y_g(x_g(r)) = (r^x)^y = r^ (xy). Then we have 0_g x_g = (r^0)^y = r ^ (0 y) = 1 = r ^ (y * 0) = (r^y)^0) = y_g 0_g -> identity
y_g (1/y)_g = r^y ^ (1/y) = r^0 = 0_g -> inverse
(x_g y_g) (z_g) = ((r^x)^y)^z = (r^(xy))^z = r^((xy)z) = r^(x(yz) = (r^x)^(yz) = x_g (r_g r_g) -> Associativity
That makes a group.
Now, I explicitly left out the 0^x case. Can we fit it back in?
Not particularly cleanly, as thoroughly discussed in this thread.
(Prologue: I encountered group theory first by drawing pictures of pegboards and strings to illustrate permutations (before I knew the word!), not by reading the rules up front.)
You are describing a schism between pure/formal and applied mathematics (and between formalism and intuition, to some extent). It's completely cool for you to have an interest in pure math completely separated from applied math, real-world physics, programming etc. It is also cool for someone to pursue applied math,physics, etc without any pure math, but that is sad because a lot of beautiful symmetry and cross-disciplinary value would be last. (Goodbye, encryption!)
I personally strive to connect pure and applied mathematics. After getting burned (in an emotional/psychological way) by chasing pure math study beyond my ability to intuit and apply it, I now commit myself to learning theory and application in tandem. (I'll certainly appreciate the fact that pure theorists such as your ideal have gone several steps ahead and I can study their results without trying to discover them from scratch.)
In fact, my most recent flight of fancy / big dream is to write math/CS tutorials that provide such a tight integration of theory and application, abstract and concrete, general and specific. And I want to use modern web tools (hyperlink, animation, multi-dimensional page layout) to do so.
I'd love to talk to anyone interested in working with me on that :-)
X to the 0.5 is square root, X to the 0.3 is cube root, X to the 0.25 is fourth root, etc
Therefore X^0 is what you get if you're taking the 'infiniteth' root
= 1
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.
7^2=7 * 7=49
7^2/7^2=7 * 7/ 7 * 7=49/49=7^(2-2)=7^0=1
But 0^0 was never intuitive to me.
By contrast, a mathematician has a Ph.D. in math, and has had to do original research and look at some history of topics. Thus, the mathematician knows that math is not a finished product, but is under constant refinement.
Now, let's talk about negative exponents. Students are taught that 1/x^2 = x^(-2). High school teachers often don't understand why. It is so ingrained that even asking "Why?" seems almost grammatically incorrect.
The reason why is that we know the following things about exponents: 1) x^n = x* x* ...* x for n a positive integer 2) As a consequence of 1, x^m* x^n = x^(m+n) for m,n positive integers
Now, the question is not "what is x^n if n is negative?" (which is what a high school teacher might ask). Rather, the question is "Can we define (!) x^n for negative n in a way consistent with the item two above?" (mathematician's framing). And, of course, we can. If x^n = 1/x^(-n) for n negative, then item two works.
So, a high school teacher most likely thinks that the negative exponent rule is simply a rule, handed down from the Gods of math. A mathematician recognizes that it is a convention, and such a smooth convention that there is simply no better choice.
Now, about 0^0: The HS teacher asks "What is 0^0?" and is therefore under the impression that 0^0 is undefined because according to certain reasonings it could be 0 or it could be 1. Textbooks (not written by mathematicians) wouldn't correct this. The TI-86 gives a domain error when 0^0 is input. The mathematician asks "What value of 0^0 makes my preferred formulae continue working?" and thereby defines 0^0=1.
well that's just depressing.
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.
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.
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 (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.
The other interpretation, which you probably meant, was the question of mathematical style and interest: which axioms do we study, and why do we care about them, and why are some theorems important and others not important? Certainly the definition that 1/x^2 = x^(-2) was made for consistency, which is an aspect of style. You could make some other definition and do formal reasoning just as well (although you'd have to change some other things - maybe the definition of exponentiation, maybe your interpretation of symbols on paper).
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.
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.
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 since the greeks started reaching the limits of a purely geometric (read physical) understanding of math.
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 Geometry, and the natural numbers (with primes and their structure, etc.) I really actually think something akin to Occam's Razor applies in math!
You can of course tweak your starting points and get really interesting things too, say non-standard analysis or non-Euclidean geometries. And you can derive (I would say "discover") some rich and surprising properties and patterns in them.
But I don't think any of these things are accidental. The real numbers (and pi, e, etc) would surely be "invented" in the same (modulo shifts in convention) ways by other advanced civilizations, I think. Could you imagine this not being so?
This is also extra-mathematical, and I admit this may be hooey, but I believe (as a non-mathematician) that math gives hints at large patterns and tells you when things fit well or are funny. Like, I can argue whether Pi or 2*Pi is the more "natural" constant, and it's not a discussion completely devoid of content! Also, I think math tells me something is funny about 0^0 (because of the limit 0^x) but that 1 is the more natural fit.
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.
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.
Doesn't the quantised nature of matter mean that 2^½ exists as a real measure of a material object only as much as a perfect circle actually is existent in our universe?
No, they're the perfect example and you're observation about the unit square is glib.
Trying to reason about the diagonal of the unit square basically destroyed the Pythagorean world view of integers as the fundamental building blocks of reality.
If mathematical entities are simply discovered and that-is-that, then why did it take nearly two millenia after this observation for western math to accept irrationals as numbers?
From M. Stifel, 1544:
"Now, that cannot be called a true number which is of such a nature that it lacks precision. Therefore, just as an infinite number is not a number, so an irrational number is not a true number, but lies hidden in a kind of cloud of infinity"
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.
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.
Depending on the context (are you working in set theory? are you making a new definitions for exponentiation?) you might have a different definition. But such operations are often defined recursively (e.g. in set theory, roughly, where S(x) = x+1 (or successor of x) Exp(x, 0) = 1, and Exp(x, S(y)) = x * Exp(x, y). Here you'll have 0^0 = 1, clearly.
For high school, 0^0 should be 1. It's necessary for problems high school students might encounter in calculous, and is the way it is defined in almost any field you'd be working in before graduate school.
High school teachers who insist that 0^0 != 1 likely don't understanding that it's a definition.
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 student> hmmm...ohhh...aaahhh....WTF...I hate math I don't want this stupid stupid job lemme go back to coding monads in Haskell for my ubercool startup.
PITA interviewer >Don't let the door hit you on your way out.
4^2 = 2 fours multiplied = 4 * 4 = 16
4^1 = 4*4/4 = 4
4^0 = 4*4/(4*4) = 1
4^-1 = 4*4/(4*4*4) = 1/4
0^2 = two zeros multiplied = 0*0 = 0
0^1 = 0*0 / 0 = 0
0^0 = 0*0 / (0*0) = 1
x^(-1) doesn't have an upper bound as x approaches zero.
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) made me think more about it.
...or if x^n = 0, which is true if x=0 and n != 0
http://en.wikipedia.org/wiki/Exponent#Zero_to_the_zero_power
Here's the version I found in the Google cache:
http://webcache.googleusercontent.com/search?q=cache:Jxv6vFH...
javascript:window.open("http://webcache.googleusercontent.com/search?q=cache:"+encodeURIComponent(location.href))Otherwise, very handy, thanks.
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.
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...
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.
I'm about 1/2 way through. It's a real gift.
http://en.wikipedia.org/wiki/Transreal_arithmetic#Transreal_...
We had this guy as a lecturer. Whether nullity exists or not (though James's argument is that it's as valid as j), and I have to admit his arithmetic does make some things simpler.
Python: 0**0 == 1
Javascript: Math.pow(0, 0) == 1
Java: Math.pow(0, 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
The same 10 year old draws two lines: 0^x, and x^0.
They clearly do not meet.
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.
Sometimes you have to use another language to make sense of something.
Error establishing a database connection
that is kind of funny... sorta