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

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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". http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com For more options, visit this group at http://groups.google.com/group/everything-list?hl=en -~----------~----~----~----~------~----~------~--~---