x^0 and x^1 Power

Tokyo... the more I think about it, the more I am sure this "implicit 1" as a "explanation" of the fact that x^0 = 0 is nothing more than silly mathematical trickery. (Note: I agree multiplicative identity is important, I am only rejecting the use of an implicit one as a "proof" of anything).

I think the term "mathematical trickery" is valid, because this concept happens to work in this one isolated example-- but it doesn't provide any intellectual value outside of the narrow example it was constructed to match.

Let's look at a couple of related examples.

This concept doesn't provide any insight to any related problem. You can't use an "implicit 1" to explain what x ^ (1/2) is. Nor can you use it to explain negative exponents.

I hate the idea of mathematics as a set of rules to memorize... this is why this kind of trickery, which discourages questioning or real meaningful exploration of ideas really bugs me.

Credit should go to Douglas Adams. A very memorable line from Hitchiker's Guide when the Babblefish disproves the existence of God.

Oh... I had forgotten that. Very nice.

Well, that is certainly interesting; a user who lays claim to being an educator who finds the idea that a language having rules is less than absolutely necessary. Since mathematics is nothing more than an elaborate form of communication, if your door equals my automobile, how in the world are we supposed to communicate correctly?

This is not something I "constructed" one day. This is an elementary truth that you apparently will do anything to refute or Heaven forbid you admit you're wrong. And I never said x^0 = 0 ... is that a typo?

And I noticed a subtle shift in your argument.

The question wasn't for something that could be used to explain real or negative exponents. The question originally posed concerned an easy conceptualization for x^0 and x^1, and I more than delivered. You can call it "mathematical trickery" all you want, and you can hold all the irrational beliefs about mathematics that you want as well. It doesn't change the fact that my explanation isn't hard to grasp, if the pupils in question have an understanding of basic axioms that we all learn in elementary school.

But I'll bite, anyway. "My" implicit 1 very much so applies to real and negative exponents. For real exponents, as the exponent approaches zero (i.e. gets smaller and smaller), the result approaches 1, and for negative exponents that get smaller and smaller, the result approaches 0. In all that, the central figure is always the ever-present 1 that exists beside all values. These assertions by themselves don't completely explain the concepts, I won't argue with that, but they nevertheless provide important conceptual starting points.

If I may give an opinion about this argument, I'd say it has not gone far since my posting of my "mini theorem," if it indeed has gone anywhere at all. I'd say it's ultimately up to the original poster of this thread to decide the "best" explanation. I should only need to make a true assertion once; several times is excessive. So, I'll not make more.

Tokyo,

The mathematics as language metaphor is a powerful one. There are rules in language, but the rules don't make the language. The concepts are more important than the rules (and many of the greatest ideas have been communicated while breaking rules of language). But the point is the ideas and concepts are what is important, not any system used to communicate them.

Ironically, you are using the term "implicit one". This is not a term I have ever encountered in my formal study of mathematics (and I have studied graduate level mathematics in college).

Interestingly enough, DrewDad, I and the others here aren't attacking you for the rather unorthodox terminology you use. We are suggesting the the idea you are trying to convey does not illustrate anything about the underlying concepts. It is the concepts and ideas... not the rules of the language... that are important.

Sure, there is an "elementary truth" that we all agree upon. Multiplying a number by one results in the same number.

But this isn't a very deep elementary truth (maybe that is why you call it an "elementary" truth), and you have yet to show that it leads to any real insight about exponents.

Cute but no cigar. Concerning the Babblefishes "theory", The more rigorous you apply "its" argument, the more rigorously the poor fish would be forced to disprove his own existence. I leave it to the readers to discern why that should be the case.

Concerning the idea of axiomatics trythsm and math as a language in general. While its true that the most fundemental asioms in math are based on a "logical", common sense progression, this is only true in the most basic ideas.

A far greater percentahe of math funactions/topics/ideas are absolutely "abstract", that is not bound by notions of common sense and real world limitations. If a student is unwilling to simply accept the "truth of the -i operator he is in for a tough kearning process. I could name any number of mathematical concepts that cannot, probably never will be, based in "common sense" and real world provability. Indeed if they could be math would not be an abstract concept at all.

As a physicist, I would be sorely pressed to explian the "logic" of the Dirac function, or the square root of -1. I'd be even harder pressed to explain what a negative frequency is. I can barely characterize a positve frequency as a vacilating electric field, at the same time not ever having anything like a complete discription of the lowly electron. Imaginary numbers are simply not provable in any physical sense, but they work, and the math as it has evolved is "proven" by the results of workable science and successful material devices.

Whatever you want to say, math remains a purely abstract language. A model that seems to follow/align to some extent with out every day experiences, but is not at all required to do so. (Thankfully).

A usefull place to start unraveling the course of arguments I've seen in this thread is to understand the difference between "count" and "math". The first is based on real world experience, the latter on an entirely self referential rigorously abstract set of ideas that we have come to believe in by results.

Ironically, most mathematicians abhor the very idea that the quailty of their work be based on practical results.
Bon Appitite'.

Ok BigBang, first, I think you are conflating a couple of types of understanding. Any physics student has the experience of needing to deal with ideas that go against "common sense" (for me this happened well before GR and QM). We are trained to think and reason mathematically.

"Common Sense" and "physical understanding" are orthogonal to the ideas I am trying to express.

In my physics/math program, I was never asked to take anything on faith. (There are times in physics when there are starting assumptions, or axioms if you will, but these are clearly and honestly stated as such and they are rigorously challenged experimentally).

In my physics/math program, any new mathematical concept was carefully developed from concepts I had already studied and grasped.

You are right, that as things get more abstract, they get further detached from common experience. But you are wrong to suggests that good mathematicians or physics lose sight of the ideas behind the mathematics. The mathematics are simply tools (powerful tools, but tools nonetheless). It is the ideas expressed by the language of mathematics that drives physics and math.

Dirac (or any other great physicist) didn't receive his functions as some sort of divine law on tablets of stone. No. He had the powerful insights (based on work done before him) which he then expressed in a way that other brilliant minds-- who were also familiar with the work he was basing his ideas on-- could understand, discuss and even challenge.

"The concepts are more important than the rules (and many of the greatest ideas have been communicated while breaking rules of language). But the point is the ideas and concepts are what is important, not any system used to communicate them."

Thats like saying the car is more important than the engine. A rather elaborate attempt to make one part of a "system" more important than another, and not a terribly good one either.

If you want to get into the deeper subject of signed heuristic symbolgy - the process of communicating, then go for it. We could discuss things like "authors intent", and "readers inpressions", symbol/sign translation, ect.

But why go there. A language is only as useful as its ability to communicate useful information. Without rules that both the sender and reciever understand to have the same meanings and convey that information in the same way, we have nothing.

The progress you're alluding too is made by taking different paths in our logical thinking and predictions, thinking outside the box, not by redefining the signs we use to communicate, bevomd adding new ones.

It has yet to give you any insight, that much is apparent. But I ought not have to worry about you, seeing as you supposedly have the credentials to make pronouncements like that. I feel so humbled by your presence.

In the meantime, you are not saying anything, and continue to go off on irrelevant tangents. The task was to make a simple, teachable conceptualization of x^0 and x^1. Nothing more, nothing less. A simple mnemonic, an axiomatic truth, that may or may not be "deep" (which is a subjective term and thus has no place in an objective field as mathematics).

I don't know what to tell you anymore. Use your "imagination" that you hold in so high regard if the "triteness" of it disconcerts you so much. I know it works for me.

Big Bang, I think I agree with everything you are saying (assuming I understand correctly through the language you are using). Yes, having a system of rules for communication that both sides understand is crucial. And I am not advocating for arbitrarily changing the symbols.

But understanding the rules of grammar doesn't get us anywhere unless we have concepts and ideas (which go beyond language) for us to understand and expand. You can have perfectly meaningless conversation that follows all of the rules of grammar and syntax.

The ideas that are being communicated are the purpose of language (and of mathematics)-- that is all I am saying. This does not mean that the rules of language aren't critical.

In your car analogy, the mathematics would be the engine. What I am interested in are the passengers.

Tokyo, sarcasm doesn't add anything to the discussion.

As a student... I have never found "simple mnemonics" or "axiomatic truths" to be useful at all. As an educator, I found they often get in the way of a deep understanding of the material (and for me, understanding is the goal of education).

I guess our basic disagreement revolves around the word "teachable".

I don't disagree with you about the need for the establishment of a deeper understanding. What I disagree on is how one would go about doing so ... I never claimed my mini theorem would provide a one-stop complete revelation of exponentiation, only that it provides but a humble starting point. It's about building blocks.

Remember, we're just talking about the case of x^0 and x^1.

"in your car analogy, the mathematics would be the engine. What I am interested in are the passengers."

Well no, not to put too fine a point on it, But I would subnit that the car, traveling from point A to point B as a goal, would be the vehicle of mathematics, whereas the sybology/systemic rules, would be the engine.

But I think we are in violent agreement with each other, if not in every aspect of the critical, and sometimes problematical process, of teaching ideas, some of which are necesarily abstract, at least certainly in the importance of consistency.

I also think, based on most of what you posted, you are expressing a desire for clear concise ideas in your teaching methods to ease the understanding of concepts for the student, which I have not the least quarrel with.

In summary, I think you would agree, that if the effective learning of the passengers is your chief concern, you wouldn't get then from A to B very predicatebly with an engine that needed parts replaced every few niles.

Perhaps I should clarify for the sake of closure ...

My explanation serves as a good starting point because it provides a grounding of exponents whilst establishing a key link between exponents, multiplication, and division. It forms a "big picture," if you will, that lays the groundwork for more advanced discussion of the topic. The "implicit 1" (actually, the empty product) does not need a proof of existence because its existence is (or at least ought to be; sorry, Hume) self-evident.

Given the apparent education of my "opponents," it is perplexing that such a simple idea cannot be accepted. I guess the study about kid vs. adult thinking is mostly true; sometimes adults complicate matters where a simple explanation would suffice (cf. the riddle where the king of the jungle invites the animals to his party). Feel free to disagree with me on this, but do realize that it is a very revealing insight into the pitfalls of pedantry.

EDIT: Oh, look at all those applications for the empty product in the link ... gee, and I thought my little theorem wasn't "important" or "profound." Tsktsktsk, and they criticize me for not being inquisitive enough.

I'm beginning to understand why mathematics is such a source of anxiety for so many people. I always found great comfort in it, and still do, but if I had to have conversations like these on a regular basis I'd go into ditch digging.

lol

FreeDuck.... I don't understand why you feel this way about conversations like this (although the country does need ditch diggers, and there is nothing wrong with that).

Presenting and challenging ideas is what intelligence is all about. Perhaps it is not for everyone, but when it comes to getting any real work done in a thinking profession (such as math and science), these types of discussions are where it is at.

Oh, don't worry, FreeDuck, most of this "conversation" wasn't even necessary to begin with, so I totally sympathize with you.

Let's face it, it's not math that sucks, it's those we put in charge of it (or most of them, at any rate ... Euclid, Archimedes, Leonard, Isaac, and Albert (and anyone else who made great strides in mathematics) are my homeboys). There's too much elitism and not enough clear thinking going on.

It is false that: x^0 = 1, for all x.

0^0 = 0^(1-1) = 0^1/0^1 = 0/0 = ???

There is no unique number that is equal to 0/0, certainly not the number 1.
Indeed, 0/0 does not exist. ~Ey(0/0 = y).

Thus, (x^0 =1, for all x) is false.


[~(x=0) -> (x^0 = 1), for all x] is a theorem.