One of my fundamental problems evidently has been a misconception 
of the use of exponents (see below in bold).
  ----- Original Message ----- 
  From: Bruno Marchal 
  Sent: Thursday, July 23, 2009 6:37 AM
  Subject: Re: Seven Step Series

  On 23 Jul 2009, at 05:44, m.a. wrote:

    >> 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   
    * >>
    >> 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). When 
you say that I "could have found the mistakes by carefully reread the 
definitions" it's like saying that given a table of cyrillic letters I should 
be able to translate a passage of "Crime and Punishment". A concept like  
a^(-n) = 1/a^n   is like having to learn a new polysyllabic word. I see it and 
the next day I've forgotten it. 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.   m.a.

  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.

  We use the fact that multiplication is associative a*(b*c) = (a*b)*c = a*b*c. 
No need for parenthesis.

  The verification without computation gives an idea how we can convince 
ourself of the truth of the general statement:

  a^n times a^m is equal to a^(n+m)

  a^n = a*a*a* ... *a with n occurences of "a".
  a^n = a*a*a* ... *a with m occurences of "a".           


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