Mathematics Chronology

7:30 am I will begin with exercises 1.3

Algebra: Abstract and Concrete. Edition 2.5 (online) Frederick Goodman

Chapter 1. Algebraic Themes

There are 12 sections to this chapter. The main themes appear to be symmetry (first four sections), permutations, divisibility, integers, group, rings, and fields.

1.1 What is Symmetry?

See Mathematics 12 notebook.

1.2 Symmetries of the Rectangle and the Square

See Mathematics 13 notebook.

1.3 Multiplication Tables

See Mathematics 13 notebook.

Exercises 1.3 1.3.1 List the symmetries of an equilateral triangle (there are six) and work out the mutiplication table for the symmetries. Refering to Figure 1.3.6 There are 3 symmetries which involve flipping the triangle through each of its perpendicular bisectors (a, b, c) plus there are 2 rotations of 120 degrees (r and ) plus one nonmotion symmetry, e. e r a b c e e r a b c r r e . . . e r . . . a a . . e . . b b . . . e . c c . . . . e   1.3.2 Consider the symmetries of the square card. (a) Show that any positive power of r must be one of . First work out some examples, say through . Show that for any natural number k, , where m is the nonnegative remainder after division of k by 4.   (b) Observe that is the same symmetry as the rotation by about the axis through the centroid of the faces of the square, in the clockwise sense, looking from the top of the square, that is, is the opposite motion to r, so . Define for any positive integer k. Show that , where m is the unique element of {0, 1, 2, 3} such that m + k is divisible by 4. 1.3.3 Here is another way to list the symmetries of the square card that makes it easy to compute the products of symmetries quickly. (a) Verify that the four symmetries a, b, c, and d that exchange the top and bottom faces of the card are a, ra, , and , in some order. Which is which? Thus a complete list of the symmetries is . (b) Verify that . (c) Conclude that for all integers k. (d) Show that these relations suffice to compute any product.

Literature Notes

Part I Book Three: The Sensualists

Chap. 1 In the Servants' Quarters

See Literature 23 .

Chap. 2 Stinking Lazaveta

See Literature 23 .

Chap. 3 The Confession of an Ardent Heart. In Verse.

See Literature 23 .

Chap. 4 The Confession of an Ardent Heart. In Anecdotes.

See Literature 23 .

Chap. 5 The Confession of an Ardent Heart. "Heels Up".

See Literature 23 .

Chap. 6 Smerdyakov

Alyosha arrives at His father's house to see Fyodor with Ivan and Smerdyakov (the servant). Fyodor and Ivan are drinking and Fyodor has already had a bit too much. Fyodor relates an anecdote about Smerdyakov that shows how honest he is when he returned some money that Fyodor had lost. However Smerdyakov just stands in the background and refuses to participate in the conversation.

Chap. 7 Disputation

Fyodor baits Smerdyakov into discussing religion and we gain some additional insights into his personality. He is very taciturn and private, but has some interesting issues involving religion that are very logical and straightforward, yet show the illogic of some beliefs.

Chap. 8 Over the Cognac

By now Fyodor has clearly had too much to drink as he now tries to draw both Ivan and Alyosha into a discussion of religion but both sons are treating what he says lightly. Near the end of the conversation Alyosha suddenly has an attack of sudden trembling and silent tears, just like Fyodor remembers his wife (Alyosha's mother) having. At the same moment Dmitri rushes into the house and Fyodor rushes to Ivan asking him to save him as he thinks Dmitri will kill him.

Chap. 9 The Sensualists

Dmitri has overpowered Grigory and Smerdyakov and yells "Where is she?". Fyodor immediately realizes Dmitri is not after him but Grushenka, and begins to chase after Dmitri, thinking that Grushenka must be in the house as well. Fyodor breaks away from Ivan and rushes at Dmitri. Dmitri knocks him down and kicks him in the head. Alexei convinces Dmitri that Grushenka is not here and Dmitri leaves in a rush. Fyodor is put to bed but he asks Alexei to find Grushenko and find out if she prefers Dmitri or Fyodor.

Chap. 10 The Two Together

Early that evening Alyosha goes to visit Katerina and discovers that Grushenka and Katerina have been visiting. They appear to have shared stories about each other and Dmitri and Katerina says the Grushenka has agreed to tell Dmitri that she does not love him and that he should return to Dmitri. But then Grushenka laughs and says that she agreed to no such thing. Katerina is devastated at his betrayal by Grushenka. As Alyosha is leaving he is given a note from one of Katerina's aunts.

Chap. 11 One More Ruined Reputation

Dmitri sees Alyosha leaving Katerina's house and asks him what transpired. Alyosha tells him everything that occurred and Dmitri decides to continue with his plan to leave Katerina. Alyosha returns to the monastery where he read the note from Lise, the daughter of Madame Khoklakov, where she admits her undying love for Alexei, and asks to see him tomorrow.

SUMMARY of the session: This is a very fast paced novel, with developments every few pages. The intrigues and cross-currents are everywhere. It is difficult to imagine a more tangled situation between a father, his three sons, and two women.