I have just finished reading:
Fermats Last Theorem
ISBN: 1841157910
Author: Simon Singh
I actually bought this book and had started reading it when I found
“Surely You’re Joking, Mr.Feynman!:”
but I am afraid that Mr Feynman took the lead and this book never got touched again until I had finished the Feynman book.
On the whole I enjoyed it but its a bit dry in places and meanders about the
place, or at least that was my impression. I had trouble seeing the
relevance of some of the writing to Fermats problem and this is where I got the
feeling of the book going off on tangents just to be brought back quite
sharply.
I was also a bit surprised to hear the Authors description of “divide by 0” in
one of the Appendix’s. He says that you cannot divide by zero because zero will
go into something infinitely many times.
I know this is a religious issue for some people but I would have described it
as follows.
2 x 0 = 0 : True
this looks correct and is correct. Its basic algebra. Now when transposing formula we could take the left 0 over to the right side by dividing through by
0 as follows
2 = 0/0
We can see that 2 cannot = 0/0 so division by zero is undefined. Some people
might see it better as follows.
(2 x 0)/0 = 0/0
If the left zeros cancel which they would if 0 was a normal number then we are
left we are left with the absurdity.
2 = 0/0 : Absurd
So 0 cannot be a number or at least not in any normal sense. So to say that 0
divides something infinitely many times seemed wrong to me.
If we look at it another way it might be clearer. If we look at the following
infinite sequence
1/(1/2), 1/(1/3), 1/(1/4), 1/(1/5) ………. 1/(1/n)
the we can see that as
n |–> infinty that 1/n |–> zero
so we have 1/0 which if we could say gives us infinity because 0 divides 1
infinitely many time. That sounds plausible enough but, if we look at the following
infinite sequence
1/(1/-2), 1/(1/-3), 1/(1/-4), 1/(1/-5) ………. 1/(1/-n)
then we can see that as
-n |–> negative infinty then 1/-n |–> zero
but zero is the only number that is neither negative or positive so which
infinity do we pick. Do we say that because the sequence is approaching
negative infinity that it 1/0 is -ve infinity or +ve infinity. This is another
of those absurdities that we ran into earlier when dealing with divide by zero.
Dividing by zero is indeterminable which is why no one says that it divides
something infinitely many times when in fact we have no idea what its doing.
I am sure some clever cloggs will come along after a course in complex
analysis and blow the above out of the water but this is the way I have always
thought of this question.