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Monday October 23, 2006 6:05 am Lethbridge Sunrise 8:05 Sunset 18:26 Hours of daylight: 10:21
A. Morning Musings
6:05 am It is +10 C at the moment. The forecast is for a high of + 9 C. We should have no trouble making this.
My weight is down 1 to 181. My original goal of 180 is definitely within sight.
First activity this morning is coffee with friends. We leave for Calgary early this afternoon for an overnight stay. This will give me an opportunity to visit the train store in Calgary and discuss a few questions that I have about DCC.
The early morning goal is to make some notes for chapter 5 of "Fearless Symmetry".
B. Plan
Immediate Health Walk & exercise 1 hr Model Trains Assembly of CN 5930, an SD40-2 with a NAFTA logo 1 hr Mathematics Make notes for "Fearless Symmetry" chap 5 1 hr Begin reading "The Equation that Couldn't be Solved" 1 hr Literature Continue reading "Standing Stones" by John Metcalf 2 hr Later Chores Investigate water softeners for home Technology Read manual for cell phone Make notes for chap. 4 of "Switching to the Mac" Begin reading "iPhoto" digital photography - learn about using the various manual settings
Literature Read "The Art of Living" by Epictetus Read "The Song of Roland" Mathematics Larson "Calculus" Read "The Computational Beauty of Nature" Chap 3 Gardner "The Colossal Book of Short Puzzles" History Continue reading "Citizens" Watson "Ideas" Model Trains Build oil refinery diorama: add ground cover Assemble second oil platform kit Puzzles The Orange Puzzle Cube: puzzle #9
C. Actual/Notes
6:20 am
The first step for each session should be a review of the previous session's notes. The first few paragraphs of my previous session deserve repeating.
The two major ideas so far are those of a group and those of a field.
A group is a set of elements and a binary operation (often called a composition) that describes how to obtain an element given two other elements. The set is closed under this operation (i.e. given any two elements in the set and the operation, one always obtains another element in the set). There is an identity element (usually represented by the letter e) such that for any element in the group, xe = ex = x. Every element has an inverse (i.e. xy = yx = e). Finally, the associative law holds for the operation (i.e. for any three elements, (xy)z = x(yz) ).
A field is a special type of group that has a second binary operation (the two operations are usually called addition and multiplication) where element has a multiplicative inverse. One example of a field is the set of integers 0, 1, 2, ... (p - 1) where p is a prime integer and the two operations of addition and multiplication modulo p.
Now to make some notes for chapter 5. Making these notes also acts as a form of review.
Fearless Symmetry Chapter 5 Complex Numbers [p. 42 - 48]
- "Complex numbers are an extension of the real numbers useful for solving equations." [p. 42]
- "The set of complex numbers is another example of a field." [p. 42]
- "It [complex numbers] is handy because every polynomial in one variable with integer coefficients can be factored into linear factors if we use complex numbers. Equivalently, every such polynomial has a complex root. This gives us a standard place to keep track of the solutions to polynomial equations." [p. 42]
- "We also introduce an important subset of the complex numbers, namely, the set of all 'algebraic numbers' - those numbers that are the roots of polynomial equations with integer coefficients. This set is also a field, and will be important when we study the structure of solutions of polynomial equations." [p. 42]
Notations:
- a + bi
Terms
- algebraic numbers
- real number
- complex number
- real part
- imaginary part
- complex conjugate
- theorem
- lemma
- corollary
- algebraically closed
Theorems
5.2 Let f(x) be a polynomial whose coefficients are any complex numbers. (For example, f(x0 might have integer coefficients.) Then the equation f(x) = 0 has solutions in C.
The only real new concept for me is that of an algebraic number. Although I must admit that the term complex conjugate is one that I had forgotten. However the entire emphasis of this book (i.e. solving polynomial equations) is new to me.
I was not aware that complex numbers were useful in understanding polynomial equations.
- "Multiplication by i rotates the complex plane by 90° counterclockwise as we look down upon it, while -i rotates it the same amount clockwise." [p. 43]
I have just learned how to type the symbol for degrees ( ° ). It is the keystroke alt-shift-8. I googled "symbol for degrees" to discover this. Simple, when you know how.
There is reference to the book "Imagining Numbers" by Mazur, which I happen to have. I must have another look at this book!
- Definition: A real number is any number that can be expressed as a decimal. [p. 43]
- The set of all real numbers is usually denoted by the symbol R. [p. 43]
- If one tries to solve a cubic equation using a formula like the one for solving quadratic equations, then one unavoidably encounters the square root of a negative number. This was the motivating force behind the creation of complex numbers. [p. 44]
- Definition: A complex number is a number of the form a + bi, where a and b are real numbers.
- Addition and subtraction of complex numbers is simple: just add (or subtract) the real and imaginary parts separately. That is:
(a + bi) + (c + di) = (a + c) + (b + d)i
- Multiplication of complex numbers is defined as follows: (a + bi)(c + di) = ac + adi + bci + bdii = (ac - bd) + (ad + bc)i
- Division by complex numbers is defined as multiplication by the reciprocal. The reciprocal of a complex number turns out to be a fairly complex expression (no pun):
This was a basic chapter. But it whets the appetite for the next chapter where this book begins to take off: the solutions of equations.
Now to begin reading chapter 6 Equations and Varieties [p. 49 - 66]
I also want to begin reading "The Equation that Couldn't be Solved". This appears to give a historical perspective on the origins of group theory and the language of symmetry.
D. Reflection