On 25 Jul 2009, at 15:35, m.a. wrote:

>
>              One of my fundamental problems evidently has been a  
> misconception of the use of exponents (see below in bold).



To be able to recognize a personal misconception is the key of the  
learning process.



>> >> if a is a number, usually, a^n is the result of effectuating (a
>> >> times a
>> >> times a time a ... times a), with n occurences of a. For example:
>> >> 2^3 =
>> >> 2x2x2 = 8.    I thought 2^3 meant   (2*2)* (4*2)* (8*2)= 16

You have some weird thought.

Think about this:

2*3 =  2+2+2
2^3 = 2*2*2

4*5 = 4+4+4+4+4
4^5 = 4*4*4*4*4

Take care:
a*b = b*a   (for any a and b)
a^b is different from b^a (in general. For example 5^12 = 2441140625,  
but 12^5 = 248832).


(And how could "(2*2)* (4*2)* (8*2)= 16" be true? (2*2)* (4*2)* (8*2)  
= 4*8*16 = 32*16, which is 32 times bigger than 16).

>>
>> * >>
>> >> so a^n times a^m is equal to a^(n+m)
>> >>
>> >> This extends to the rational by defining a^(-n) by 1/a^n. In that
>> >> case
>> >> a^(m-n) = a^m/a^n. In particular a^m/a^m = 1 (x/x = 1 always), and
>> >> a^m/a^m = a^(m-m) = a^0. So a^0 = 1. So in particular 2^0 = 1.
>>
>> From the above misconception you can perhaps get an idea of how  
>> utterly alien these symbols are to me. I have never run across them  
>> before in all my years (and you'd be surprised to learn how many  
>> years I'm talking about).


It just means you don't have had to compute in your life (up to now!).
If I enter 999999999999 in my pocket computer (TI Galaxy 67, a very  
old one), the machine wrote

1 E 12,

which is its way to tell me the number I enter is about 1 times 10^12.

Note that "a^n" is not the standard notation used by mathematicians,  
but it has become standard, we could say, in the electronic mails, or  
on some pocket computers. To write "2^7", which I recall is given by  
2*2*2*2*2*2*2, they will wrote "2" with "7" as little upper index.  
Engineers use often other notations.






>> When you say that I "could have found the mistakes by carefully  
>> reread the definitions"

I did not said that. I was really just asking a question. And you  
provide me now a very clear answer:



>> it's like saying that given a table of cyrillic letters I should be  
>> able to translate a passage of "Crime and Punishment".

So your answer is "no". I could have read the definition ten times, I  
could remain wrong, because of the accumulation of seemingly senseless  
symbols.

I have no problem to understand such difficulties. I can be blind  
myself on many things, not being able to find my pen on my desk,  
although it is in front of my eyes. I tend also to confuse bills and  
advertizing.
But those are only handicaps, which, if not too severe, can be  
overcome by some amounts of work. Such handicaps could be a reason to  
panick the day before the exam, but should not deter someone who  
inquires, either for fun or for personal interrogations, in an  
environment without deadline or social pression (like here).



>> A concept like  a^(-n) = 1/a^n   is like having to learn a new  
>> polysyllabic word.

Not just that. You have to understand that it is a generalization of  
a^n on the integers. You have to learn a new concept. You will not  
need to remember all such notions for the "ultimate" understanding of  
the seventh step. At some moment I will introduce "many notions" just  
with the goal to generalize. The difficulty will be more conceptual  
and related to abstraction. I face the problem of either overwhelming  
you with too much concrete examples, or abstracting too quickly. No to  
people share the same "perfect" pedagogical path, but there is a sort  
of least common path, which can take time, but there is no rush.



>> I see it and the next day I've forgotten it.

Such a problem can be overcome by work and organization. I just hope  
you have enough fun, and personal curiosity for the result, or some  
results on the path. If you forget, I can recall. I really can sum up  
each time you want. And I can sum up what we have done, or what we  
will do (it is good to remember the real goal: to understand the UDA  
"reversal", and its "constructive" aspect). The math is needed just  
for grasping what a universal dovetailer is, and why both the UD and  
its universal dovetailing are "existing" (and in which sense) in  
elementary arithmetic.



>> Having said that, let me reiterate that I do appreciate your  
>> efforts to simplify and explain every step of the way and I  
>> apologize for sometimes needing even more clarification. Your  
>> patience is saint-like and in my case, unfortunately, necessary.

I appreciate so much people who are able to say "I don't understand"  
and ask question.



>>
> Which is why when I see you make a simple mistake, I don't feel so  
> bad because I know how easy it is to do.

Ah ah! I am teaching you patience :)  Thanks.

I ask to all those who told me they were happy that I pursue this  
little teaching further the following question.  Are you ready that I  
pursue? Are you ready that I introduce a few bit of "new" material.  
Sometimes (actually most of the time) new materials can shed new light  
on what has been already seen.

What do you prefer, that I continue with the sets (with the notion of  
couples, and then of cartesian products, and then of operation,  
relation, function, etc.)
Or do you prefer I prove first that the square root of 2, you know,  
that number which multiplied by itself gives 2, is irrational (= does  
not belongs to Q, = is not a fraction,  = is not a periodic decimal). ?

I am sure many of you already know this, but this is an typical  
impossibility result, and somehow the whole machine 'theology' is a  
collection of impossibility results, so the irrationality of the  
square root of 2 is a good introduction to such type of result. Also I  
will give you a typical example of non constructive proof base on the  
square root of two. (And for those interested in the quantum  
confirmation of comp, the square root of 2 is the amplitude  
coefficient leading to the probability 1/2, which is rather important,  
if only for examples again).

It is really like most prefer. You can tell me: "do like you want",  
but I prefer to ask.

Bruno
http://iridia.ulb.ac.be/~marchal/




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