Mathematics Chronology

5:30 am

I am not satisfied with my note making yesterday about abstract algebra. The main problem is with the proofs. I do not fully understand the logic nor the sequence of steps in them yet. I am like a dog worrying a bone. I want to feel fully in control of this. I have four books on the topic, one an undergraduate text, the other three at either an upper-level undergraduate or a graduate level. Surprisingly, I feel that the first chapter of Michael Spivak's textbook "Calculus" to provide the best introduction to the basic properties of numbers. Spivak provides a meticulous sequence of steps to show that if x + 3 = 5, then x = 2. He then goes on to say, "Naturally, such elaborate solutions are of interest only until you become convinced that they can always be supplied. In practice, it is usually just a waste of time to solve an equation by indicating the steps so explicitly..." [p. 5] Now to make some notes on "Contemporary Abstract Algebra" Second edition. by Joseph Gallian (1990).

Chapter 0 Preliminaries

• Properties of Integers
• Modular Arithmetic
• Mathematical Induction
• Equivalence Relations
• Functions (Mappings)
• Exercises

Properties of Integers [3 - 7]

Well Ordering Principle. Every nonempty set of positive integers contains a smallest member.

This is self-evident. Think of any set of positive integers. For example {3,12, 2, 99}. Reorder the numbers from smallest to largest. Then the first number in the reordered set will be the smallest member. In this example, the reorderd set is {2, 3, 12, 99} and the smallest member is 2.

The principle also applies to any finite set of integers, positive, negative or zero, but it does not apply to the infinite set of all integers since there is no end to the number of negative integers and thus this set does not have a smallest number.

Definition: A nonzero integer t is a divisor of an integer s if there is an integer u such that s = tu.

Notation: In this case we write t | s and say "t divides s"

Definition: A prime is a positive integer greater than 1 whose only positive divisors are 1 and itself.

Theorem: Division Algorithm

Let a and b be integers with b > 0. Then there exist integers q and r with the property that a = bq + r where .

Definition: q is called the quotient upon dividing a by b, and r is called the remainder.

Examples: Suppose a = 28 and b = 3. Then 28 = 9x3 + 1. That is, when 28 is divided by 3 the quotient is 9 and the remainder is 1. But one must be careful. Suppose a = -28 and b = 3. Then -28 = (-10)x3 +2. The quotient is -10 and the remainder is 2.

Definition: The greatest common divisor of two nonzero numbers a and b is the largest of all the common divisors of a and b.

Notation: gcd(a, b)

Definition: When gcd(a, b) = 1, we say a and b are relatively prime.

Theorem: GCD is a Linear Combination

For any nonzero integers a and b, there exist integers s and t such that gcd(a, b) = as + bt.

Theorem: Euclidean Algorithm (Rotman, p. 5)

Let a and b be positive integers. There is an algorithm that finds the gcd, d, and there is an algorithm that finds a pair of integers s and t with d = sa + tb.

The algorithm iterates the division algorithm. Begin with b = qa + r, where 0 <= r <=a.

Example: Suppose a = 2520 and b = 154.
2520 = 154x16 + 56
154 = 56x2 + 42
56 = 42x1 + 14
42 = 14x3

Therefore gcd(2520, 154) = 14.

GO Chronology

Notes for Learning to play the strategy game of GO.

Here are the results of my 3 games of GO++:

• Game 1: I lose by 2.5 points.
• Game 2: I lose by 3.5 points. I played defensively and was not able to overcome the 6.5 point komi.
• Game 3: I lose by 88.5 points. I got behind early and then failed to defend my area and soon was unable to form two eyes.

"Graded GO problems for Beginners Volume One Introductory Problems" Level Two

• Section 8 Ko. Problems 113 - 116. All easy.
• Section 9 How to Play in the Opening. Problems 117 - 118. Both easy.
• Section 10. Endgame. Problems 119 - 120. Both easy.

SUMMARY of the session: The first two 9x9 games were close, but I need to learn to be slightly more aggressive. The third game was a total disaster. I failed to defend my territory and soon found it impossible to form two eyes. The graded problems are still easy, which is an indication that I understand something about the game. The next set of problems is at level three. I need to set an hour aside and do some reading about the game. I still like the book "Lessons in the Fundamentals of GO" that I recall starting a few years ago. 8:00 am