Re: Elementary Theory of Numbers
- From: Tim Smith <reply_in_group@xxxxxxxxxxxxxxxx>
- Date: Tue, 07 Oct 2008 01:27:38 -0700
In article
<89553e78-c957-4acb-814a-f104251c4439@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
conrad <conrad@xxxxxxxxxx> wrote:
I'm reading through LeVeque's book the name of
which is in the subject line. My current knowledge
of math extends to first semester calculus.
I'm reading through LeVeque's book in my spare
time. Therefore, I don't really have a means
to check my understanding of number theory
except through mediums such as this.
I am working through the first chapter
and covering proofs by induction.
The next section of chapter 1 is on
indirect proofs. I mention all this so that
the reader doesn't have to make
assumptions about my mathematical
knowledge and can therefore respond
in the most helpful way.
I have the following problem:
Show that every integer greater than 1
can be represented as a product of one
or more primes.
Now, I'm trying to think about this
problem in terms of the peano postulates
(specifically the fifth postulate and
the well-ordering axiom). My problem
seems to be on how I should represent
a general number with some number of
prime numbers.
I'm going to give you a general hint. This is a hint that I suspect
will cause most of the mathematicians reading this to furrow their brows
and cast severe virtual frowns of disapproval in my general direction.
Note also that I don't think I have LeVeque's book (damn...I really need
to get my library sorted...it never recovered from moving 16 months
ago...I genuinely have no idea whether or not I have that book!). From
what you write, though, it sounds like he follows a pattern that many
other "elementary" books follow.
That pattern is to have a first chapter that covers fundamentals, that
starts at a very low level, with axioms for the integers, some set
theory, and so on.
This first chapter is often the most difficult chapter in the book, as
it requires ignoring most of what you've learned in your prior study of
mathematics. You might think you've proved something, and then realize
you've assumed that 3 > 2 or something like that, which has not yet been
proved with the tools you have available at that point in the book. 3 >
2 might turn out to be a deep theorem at that point.
The funny thing is, most of the time when books follow this pattern
(well, other than books whose subject is foundations of mathematics or
similar things!), it turns out that you can just *skip* the first
chapter. Go right to chapter 2, or whichever chapter starts covering
what you actually think of when you think of the stated subject of the
book. There's usually nothing proved in that first, low level, chapter
that you didn't already know from prior mathematical study. You should
at some point acquire a firm grounding in the construction of the
integers, the rationals, and so on--but when you decide you want to
study that, not because some book on another subject decided to start
from the very beginning for completeness.
I'm *not* saying there is not value to be found in the first chapter.
There is great value there. It's just that you generally don't need it
for the particular subject of the book--the understanding of the
integers and rational numbers that you got from your prior calculus
course, and from any algebra you've done, is probably good enough.
Note that this is a general observation about many math books. I don't
know if it applies to the book you are reading. I'm guessing it might,
based on your mention of Peano postulates.
--
--Tim Smith
.
- References:
- Elementary Theory of Numbers
- From: conrad
- Elementary Theory of Numbers
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