This course ended May 11, 2014.

## Math 110: Discovering in Mathematics

```   Department of Mathematics and Computer Science
Hobart and William Smith Colleges

Spring 2014.

Instructor:  David J. Eck  (eck@hws.edu)

Course Handout:  http://math.hws.edu/eck/courses/math110_s14.html

Monday, Wednesday, Friday, 3:00 -- 3:55 PM
Room Stern 304.

```

### First Week: January 22 and 24

Welcome to the course!

This week will be an introduction to the course. I will give a short "low-stakes" assignment on Wednesday that will be due at the beginning of Friday's class. On Friday, you should expect to do some group discussion, with an assignment based on that group work due Monday. I will add the assginments to this post after each class.

There is no reading from the textbook for this week.

Here is a link to the YouTube video we viewed on the first day of class:

Is Math Created or Discovered?

The first assignment is to write a paragraph to answer each of the following two questions. This assignment is due in class on Friday. It is a "low stakes" assignment; that is, it will be graded as checked minus, checked, or checked plus.

1. Do integers such as 17 exist?
• If you answered yes, what sort of things are they? What is the nature of their existence?
• If you answered no, I don't know, or anything else, how can we make sense of statements such as 17 = 5 + 12 if the numbers don't exist?
2. Do you believe that there are infinitely many prime numbers? The collection of primes can be written 2, 3, 4, 7, 11, 13, 17, ... Do you believe that this infinite collection of numbers exists?

Here is the simulation of the Erdos theorem that we saw in class:

A Simulation of a Theorem of Erdos

### Second Week: January 27, 29, and 30

You can start reading Chapter 2 of the textbook. We will cover material from Sections 1, 2, and 3 this week and might move on to Section 4 if there is time. We will not cover everything from these sections (although it wouldn't hurt you to read them in their entirety). Here are the parts that you should pay attention to:

• Section 2.1, pages 44–46 (the pigeonhole principle)
• Section 2.2, pages 53–58 (Fibonacci numbers)
• Section 2.3, pages 68–80 (prime numbers and the infinity of primes)

Here is the solution to the 9-coin problem that we looked at in class on Monday:

9coins.pdf

Here is the worksheet for the homework that is due on Wednesday:

Homework due Wednesday, January 29

Here are vhart's Fibonacci YouTube videos. We saw Part 1 in class. You might look at parts 2 and 3 for some discussion of why Fibonacci numbers come up (or don't come up) in nature:

Here is the first graded homework assignment, which is :

Homework, due next Wednesday, February 5

Here is the web page where you can make Fibonacci-like sequences:

NmberSeq.html

### Third Week: February 3, 5, and 7

The homework that was supposed to be due on Wednesday has been postponed until Friday. We worked on it for a while in class on Monday, and decided that more time was needed. The whole class period Wednesday was spent on group work on the the assignment.

The reading assignment for the week is Chapter 2, Sections 6 and 7, covering rational and irrational numbers and decimal representation of numbers. We will spend Friday's class on this material.

Homework 1 Solutions

### Fourth Week: February 10, 12, and 14

There is a test this week on Friday, February 14. Here is a study guide, including some sample questions:

Test 1 Information

We will be working on Chapter 2, Sections 3.1 through 3.3 this week. We should more or less finish 3.1 and 3.2 on Monday. On Wednesday, we will move on to 3.3 if there is time. Note that 3.3 is not on the test. We will devote as much of the class on Wednesday as necessary to going over test material and working on sample questions from the study guide.

### Fifth Week: February 17, 19, and 21

On Monday, I will finish up my discussion of infinity. I will cover Cantor's proof that the cardinality of the real numbers is strictly greater than the cardinality of the positive integers, from Section 3.3. I would also like to look at the "Ping Pong Ball Conundrum" from Section 3.2. After that, we will be moving on to Chapter 8, which is about probability theory. The reading for the week is Section 3.3, Section 8.1, and Section 8.2.

Note: I expect to start assigning homework problems and in-class exercises from the textbook. You should have a copy—or know someone who does.

Here is a link to the New York Times article that I mentioned in class, about the possibility that we are living in a computer simulation and the idea that that might explain the nature of our mathematics:

Is the Universe a Simulation?

Now that we are starting in on a more quantitative part of the course, it is possible that you might be able to get some help from the Quantitative Fellows in the Center for Teaching and Learning. Here are their hours, all held in the CTL:

```        Sunday 7-9
Monday 6-9
Tuesday 2-4
Wednesday 8-10
Thursday 6-8
```

Here is the worksheet from Wednesday, Feb. 19, in-class work.

### Sixth Week: February 24, 26, and 28

We will continue with Chapter 8, probability for the entire week. The reading for the week is Sections 8.3 and 8.4. We might get to 8.5 by Friday.

The following graded homework is due on Wednesday:

Section 8.2, Exercises 4, 10, 15, 18, 19, 28, 38, 39, 40

Some of these questions are straightforward, some of them require thoughtful analysis. In all cases, you should explain your reasoning, even when the problem requires only a simple numerical answer. Remember that you can work together with other students on the homework problems, but you should always write up your answers in your own words (and mathematical symbols).

### Seventh Week: March 3, 5, and 7

We will continue with Chapter 8. We will just be getting to 8.4 on Monday, and we will finish with Chapter 8 after finishing that section. Next, I would like to spend some time in Chapter 10.

The following graded homework is due on Friday, March 7:

```           Section 8.3,  # 11, 15, 23, 34
Section 8.4,  # 5, 6, 8, 11, 13, 24, 31
```

Don't forget that all answers must be justified!

Here are sample answers to this homework.

### Eighth Week: March 10, 12, and 14

There is a test on Wednesday of this week. A study guide is available and was handed out in class on Friday.

We will spend most of the day Monday going over material for the test. In particular, we need to spend a little more time on expected value. On Friday, we will discuss Newcomb's Paradox, which is in Section 10.1 of the book. Next week is Spring Break. As homework over break, I would like you to tell a couple of friends or family members about Newcomb's Paradox and get their reactions. You should write up a short report about your interactions. What do they think about the paradox? What would they do? What would you do, and why? The report should be about a page long and will be graded on a check/check+/check- basis.

### Ninth Week: March 24, 26, and 28

We start on Monday with a discussion of the assignment on Newcomb's paradox. After that, we are going take a break from numbers for a while. The next section of the course will be geometric. To start, we will work on a view of infinity in geometry, Section 3.5. After that, we will look at symmetry and symmetry groups. This topic is not really covered in the textbook, except for a bit in Section 4.4 (where it mostly talks about unsymmetric patterns).

Here is the Assignment Worksheet from Monday's class
(and here are my sample answers.)

Here is Graded Homework 4, and
the first sheet and second sheet of patterns
that were used in class on Friday.

### Tenth Week: March 31; April 2 and 4

On Monday, the class meets in the computer lab Gulick 208 to work on the second part of the homework. Here are links to the three web pages that you will use in that lab:

On Wednesday, we will continue our discussion of symmetry. We will look at what it means to say that we have "symmetry groups." On Friday, we will move on to fractals, Chapter 7 in the textbook. You can look at the pictures in Section 7.1, and you should read Section 7.2. Next Monday, we will have another computer lab where you will work with the Chaos Game that is discussed in Section 7.2 and with the Mandelbrot set.

Here is the worksheet from Wednesday's class

.

### Eleventh Week: April 7, 9, and 11

Once again, we meet in the Gulick 208 on Monday. In the lab, you will work with the Chaos Game and the Mandelbrot set. The assignment for the lab work is due this Friday, April 11.

We will continue our look at fractals this week. We will finish Section 7.2 and do Section 7.3, which covers fractal dimension.

Here are two links for use in the lab on Monday:

Here are the complicated commands that you can copy-and-paste into a DOS window in the lab to run the programs (quotation marks must be included):

```   "C:\Program Files (x86)\Java\jre7\bin\java"  -jar  "I:\Software Install Kits\Public\Math-110\ChaosGame.jar"

"C:\Program Files (x86)\Java\jre7\bin\java"  -jar  "I:\Software Install Kits\Public\Math-110\xMandelbrot.jar"
```

### Twelfth Week: April 14, 16, 18

There is a test on Monday, April 14. Here is the study guide.

After the test, we will be spending some time on "The Fourth Dimension" and the question of dimensions in general. The reading from the textbook is Section 4.7.

Here is the in-class worksheet from Wednesday.

### Thirteenth Week: April 21, 23, and 25

We will start the week with some final discussion of the fourth dimension and dimension in general. After that, we will move on to Chapter 6, Sections 6.1 and 6.2. These sections are about "graph theory", where the term graph here refers to a collection of vertices (points) and edges (line or curve segments that have vertices as their endpoints).

Here is the homework that is due next Monday.
and here are my Sample Answers to this Homework

XPlorMath-3d (jar file / Java program for Friday's class)

### Fourteenth Week: April 28 and 30; May 2

We have just a bit more about graphs to cover. After that, we will be discussing voting. Voting is covered in the textbook in Section 10.4. I recommend reading only pages 811 to 818 and "Arrow's Impossibility Theorem" at the bottom of page 821.

There will be a final graded homework assignment, which will be due next Monday, May 5.

and here are my Sample Answers to this Homework

### End of Classes and Final Exam: May 5 and 11.

The last day of class in Monday, May 5. We will spend the day wrapping up the course and reviewing for the final exam. The final exam is at the officially scheduled time: Sunday, May 11, at 1:30 PM. It will be given in our regular classroom.

Study guide for the final exam.

```   End-of-semester office Hours in Lansing 313 (but also look for me in Lansing 310):

Tuesday, May 6:   12:00 – 1:20 PM
Thursday, May 8:   2:00 – 4:00 PM
Friday, May 9:     1:00 – 3:00 PM
Saturday, May 10: 11:00 AM – 2:00 PM
Sunday, May 11:   12:00 – 1:15 PM and 6:00 – 6:45 PM
```