Learning: The Journey of a Lifetime
or A Cloud Chamber of the Mind
September 2005 Mathematics Learning Log

An Example of a "Learning Process" Journal
Friday September 30, 2005
Learning Log Number 18

6:40 am Lethbridge, Alberta

I am returning to a regular routine of some math in the morning. It feels good.

Mathematics      
Description
Start
End
Done
What is Mathematics? (1941) Richard Courant & Herbert Robbins      

Read chap 1 The natural numbers [p. 1 - 20]

Sept 26 Sept 29 Yes

Notes for chap 1.1 Calculation with integers [p. 1 - 9]

Sept 26 Sept 26 Yes

Notes for chap 1.2 The infinitude of the number system [p. 9 - 20]

Sept 26 Sept 26 Yes
Read chap 1 supplement The theory of numbers [p. 21 - 51] Sept 29    
Notes for chap 1 supplement 1.1 The prime numbers [p. 21 - 31] Sept 29    
Notes for chap 1 supplement 1.2 Congruences [p. 31 - 40]      
Notes for chap 1 supplement 1.3 Pythagorean numbers and Fermat's Last Theorem [p. 40 - 42]      
Notes for chap 1 supplement 1.4 The Euclidean Algorithm [p. 42 - 51]      
       
Calculus (1994) Michael Spivak      
Read chap 1 Basic properties of numbers [p. 3 - 20]      
Notes for chap 1      
Exercises for chap 1 [p. 13 - 20]      
       
Prime Obsession (2003) John Derbyshire      
Reread chap 1 - 8 [p. 1 - 136]      
       

Review

I am beginning a new chapter number theory. The first subsection is about prime numbers and overlaps with Derbyshire's "Prime Obsession".


I plan to do some off line work this morning in a coffee shop while waiting for my glasses to have new lenses put in them.

Content

Courant & Robbins Supplement to Chapter 1 "The Theory of Numbers" [p. 21 - 51]

Prime numbers are important because every number may be uniquely expressed as a product of prime numbers. This is noteworthy because prime numbers are a subset of the natural numbers and are initially defined in terms of repeated addition.

Here are a few important facts about prime numbers:

There are infinitely many primes (proof by contradiction, first due to Euclid). This leads to a recursive approach for generating prime numbers (each new prime is the product of the previous primes plus 1.)

Here are the next 5 primes, beginning with 2 and 3.

  • 7
  • 43
  • 1807
  • 3263443
  • 10650056950807

I generated these using the calculator software on my laptop. One might wonder if they are indeed all primes. For example, did I inadvertently press the wrong key at some point?

Using Mathematica, I can obtain the factors for a number. Here are the results: (web, Mathematica)

Fundamental Teorem of Arithmetic: Every integer N greater than 1 can be factored into a product of primes in only one way.

 


7:40 am

Total elapsed time: 1 hr. 0 min

I am back from a very pleasant 1 hour and 10 minutes at the Penny Coffee House making some hand written notes.

One of the tasks I identified was to construct two alternative representations of what I am about to Learn.

Here is the table format:

Description
Start
End
Quality
Number Theory Sept 30 Sept 30
1
Prime Numbers
Sept 30 Sept 30
1
Algebraic relationships
Sept 30 Sept 30
3

Distribution of prime numbers

Sept 30 Sept 30
3

The Prime Number Theorem

Sept 30 Sept 30
3
Congruences
 Sept 30 Sept 30
1

Geometric representation

 Sept 30 Sept 30
3

Rules for divisibility

Sept 30  

Fermat's Theorem

   

Quadratic residues

   
Pythagorean Numbers
   

Fermat's Last Theorem

   
Division
    

Euclidean algorithm

   

Diophantine equations

   

Euler's phi function

   
     
     
     
     

It is also possible to represent this same situation by means of a concept map.

Total elapsed time for the day: 1 hr. 0 min