5:40 am Edmonton, Alberta
I purchased 6 books at the University of Alberta Bookstore yesterday. Three of these were on mathematics and one was on Learning:
 The Calculus Tutoring Book (1986) Carol Ash & Robert B. Ash.
 The Adventure of Numbers (2004) Gilles Godefroy
 The Mathematics of Oz (2002) Clifford Pickover
 Content Area Reading (2005) Richard & Jo Anne Vacca, Deborah Begoray
The first looks like a promising supplement to my other books on calculus. I must return to this topic! It should be scheduled for alternate days. The Godefrey book appears as a classic description of the evolution of the concepts that deal with the various ideas about numbers. Pickover's book is another of his delightful compediums of recreational mathematics. These are an important component of Learning math  where the emphasis is on divergent thinking, problem solving in the true sense of the word, and having fun. I must remind myself to collect together both Pickover's books as well as many I have by Martin Gardner.
Thus my topics are now:
 Calculus
 Number Theory
 Recreational Mathematics
This feels right to me: a nice balance between formalism and fun.
Now to rearrange the following table to incorporate these books.
Mathematics 



Description 
Start 
End 
Done 
Calculus 
Calculus (1994) Michael Spivak 







What is Mathematics? (1941) Richard Courant & Herbert Robbins 







The Calculus Tutoring Book (1986) Carol & Robert Ash 







Number Theory 
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 
Oct 01 
Yes 
Notes for chap 1 supplement 1.2 Congruences [p. 31  40] 
Oct 03 


Notes for chap 1 supplement 1.3 Pythagorean numbers and Fermat's Last Theorem [p. 40  42] 
Oct 04 


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] 







In Code (2001) Sarah Flannery 



Part I Background [p. 1  40] 
Oct 07 
Oct 08 
Yes 
Part II Mathematical Excursions [p. 41  186] 
Oct 08 






Prime Obsession (2003) John Derbyshire 



Reread chap 1  8 [p. 1  136] 







The Adventure of Numbers (2004) Gilles Godefroy 







Recreational Mathematics 
The Mathematics of Oz (2002) Clifford Pickover 







Review
I reread the first 90 pages of Sarah Flannery's book, "In Code". It is a nice blend of autobiography and mathematics, a bit like this web site. Now to make a few notes on the first part (chapters 1  3)

Here are a few quotes from "In Code" that I enjoyed: 
Quotation 
Comment 
"Ever since I can remember, my father has given us little problems and puzzles. ... These problems challenged us and encouraged our curiousity ... they taught us how to reason and think for ourselves. [p. 8] 
This reinforces my decision to buy the Pickover book, and to set aside a few others that I have purchased over the years.
I am now going to set aside specific times where I play with these (1/3 calculus, 1/3 number theory, and 1/3 recreational math).
Such puzzles, which may require time to explore, should also occupy 1/3 of the school curriculum. 
"This variation on a theme lets you learn the very valuable lesson that sometimes problems may not have solutions of the type you seek." [p. 10] 
The idea that some problems may not have a solution is one that needs reminding. 
"Once you see and solve (or just hear the solution to) a problem of this type, you learn to be careful and to watch out for little wrinkles." [p. 12] 
Another useful reminder. 
"When you finally crack a problem, whether by yourself or with some help, you experience a great feeling of selfsatisfaction and pride." [p. 13] 
I agree, yet I wonder how many people have experienced this feeling. 
"For years I had the advantage ... of hearing mathematics spoken in a natural way between my father and a physicist friend of his in our kitchen. ... I realized with some relief that the people I supposed could solve every math problem were, in fact, often struggling." [p. 31] 
I recall a similar event when I was an undergraduate and asked a professor for help on a statistics problem that was outside of our regular work. 
"But those projects were really no more than gathering and marshalling information in an attractive way (in fact, organizing  which I love; I'm not quite good as keeping organized, though ...) [p. 33] 
Two points:
1) many activities are superficial and require little or no original thought,
2) organizing is a major theme in this web site, and keeping organized is a challenge. 
" 'It'll take time to assimilate the concepts, to let them mature and become clear in your mind. Anyway, you are almost sure to run into all sorts of difficulties at the programming end when you try to implement the mathematics. Projects don't have to be hectic affairs; better if you give yourself plenty of time and that you take your time.' " [p. 36] 
Once again, this applies to this web site, and to my goal of Learning calculus.
But I am very comfortable with the time scale. This may take a few years, but they will be enjoyable years with a "great feeling of selfsatisfaction". 
"He often advises, 'Speak clearly and simply. Tell a person what you know and come clean when you don't know something. Don't bluff.' " [p. 37] 
This has been my mantra since I first worked as a computer programmer in Montreal in the 1960's. 
"The project required that I learn number theory (really learn it), the necessary cryptographic ideas and how to program in Mathematica so I could implement and illustrate the cryptographic schemes." [p. 38] 
Three points:
1) there is a huge distinction between learning and "really learning". Really learning must be selfimposed.
2) I have yet to tackle programming in Mathematica. It is time to have a look at this!
3) The idea of simulating a situation in Mathematica is one that I should begin to play with. In my own words, "If you think you really understand something, try building a model to represent your understanding." 
"One of Dad's favorite books, Excursions in Calculus by Robert M. Young." [p. 44] 
I have checked this at amazon.ca and it is available but 4 6 week delay. It is not available at chapters.ca. There is not much information on it, 
"I would come to appreciate that it wasn't good enough to know any old way of doing something  you had to find methods or techniques that implement the theory in highly efficient ways." [p. 59] 
This is in contrast with the view of many mathematicians, who remain in the realm of the abstract. 


The book is full of useful asides. 
Content
In Code (2001) by Sarah Flannery
Hilbert numbers: 4n + 1
 closed under multiplication
 Hilbert prime: no other Hilbert numbers as factors
 Prime factorization is not unique.
 this does not appear to be mainstream mathematics
Fermat numbers:
Euler numbers:
 the first 40 values of n yield prime numbers
Mersenne numbers:

Description 
Start 
End 
Quality 
Number Theory (Courant & Robbins) 
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 
Oct 03 
2 
Fermat's Theorem

Oct 03 
Oct 04 
2 
Quadratic residues

Oct 04 


Pythagorean Numbers 



Fermat's Last Theorem




Division 



Euclidean algorithm




Diophantine equations




Euler's phi function








The next chapter in "In Code" is called The Arithmetic of Cryptography [p. 71  112]. That, and the following chapter, Sums With a Difference provide the detail of the mathematics underlying cryptography. Now to do some slow reading, some yellow highlighting, and some hand written notes. This has been a good morning.
8:00 am Total elapsed time for the day: 2 hr. 20 min 