Number Theory Section Summary: 1.1-3

Some Preliminary Considerations

``The theory of numbers is concerned, at least in its elementary aspects, with properties of the integers and more particularly with the positive integers 1, 2, 3, ...(also known as the natural numbers)....'' p. 1.

1. Definitions

   Well-Ordering Principle: Every nonempty set S of non-negative integers contains a least element; that is, there is some integer a in S such that for every b belonging to S.

2. Theorems

   Archimedean property: If a and b are any positive integers, then there exists a positive integer n such that .

   (Proof is by contradiction).

   First Principle of Finite Induction: Let S be a set of positive integers with the properties that

   1. The integer 1 belongs to S, and
   2. whenever the integer k is in S, then the next integer k+1 is also in S.
   Then S is the set of all positive integers.

   Our text assumes the Well-Ordering Principle, and uses it to prove mathematical induction; alternatively we could prove the Well-Ordering Principle based on assuming Mathematical Induction. This means that the two are equivalent: We have Well-Ordering Induction.

   Second Principle of Finite Induction: Let S be a set of positive integers with the properties that

   1. The integer 1 belongs to S, and
   2. whenever the integers are in S, then the next integer k+1 is also in S.
   Then S is the set of all positive integers.

   Binomial Theorem:

   

   where

   

   (Proof is by induction).

3. Properties/Tricks/Hints/Etc.

   Pascal's rule:

   

   (this is the source of Pascal's triangle).

4. Summary

   Chapter 1 is preliminary, as the title says. We assume that you've seen this stuff before (except for the history of number theory, which I hope that you'll find interesting!).

   The story concerning the drowned disciple of Pythagoras is often told of another student, who may have revealed the existence of irrational numbers (in particular, ). Irrational numbers were not welcomed into polite Pythagorean society...!