# Week 2 - The Pythagoreans and Proof - Instructor Notes

## Housekeeping

I think Week 0 & 1 have gone well. While Week 1 didn't have much mathematics, this week we are going to jump in feet first! This week will be have more new mathematics than the typical week. Also, this week is very proof-based. This is not common for the course.

There will be a homework assignment. It's mainly math. You can do it on paper (with hand-writing!) if you wish (I would). You may type if if you wish. If you do it on paper than scan it somehow and submit it in the dropbox.

Assuming you are going to do homework on paper, you need to have an efficient (and good quality) way to scan. A number of you have done many online classes, so you might put some suggestions in the discussion area: Questions/Comments/Tips on Computers and the Internet.

A few notes on videos. I have about 6 videos below. Please let me know if they are not viewable (or poor quality). It is also important to be able to pause a video (and move forward and backward and repeat). If this isn't possible for you, let me know. Each video is less than 5 minutes long.

## Overview

The main goals for this week

1. Get introduced to Pythagoras and the Pythagorean Society (aka, the Pythagoreans).
2. Learn about proof and two primary proof methods: the direct proof and the indirect proof (aka, Proof by Contradiction).
3. Prove the Pythagorean Theorem - 3 ways, in fact!
4. Prove that there are numbers that can not be written as fractions.
5. Prove that the number of primes is infinite.
6. Generate Pythagorean Triples (whole numbers that satisfy a² + b² = c²).
(This is more math than usual for one week, but it all fits together well.)

Read 1.1 pp. 1-6 and 13.2 pp. 225-226. (BTW, each week you can decide when to do the reading. You may want to read it after reading this page of notes and viewing the videos. You decide.)

## Content

### Pythagoreans

I won't rehash the reading (you can read it). The two big contributions the Pythagorean Society gave us was (a) the importance/necessity of proofs, and (b) there are number that are "incommensurate." This (b) means that there are number that can not be written as fractions. That is there are numbers that are irrational. (BTW, whole numbers, integers, rational numbers, irrational numbers are an important part of the CCSSM.)

Here are examples of rational numbers (rational numbers are numbers that can be written as a fraction):

• 4/7
• 2 (because it is 2/1)
• 3.2 (because it is 3 1/5 which is 16/5) terminating decimals
• .6666666..... (because it is 2/3) repeating decimals
• - 4/7

Square root of 2 (and most square roots) and pi (π) are not rational (they are irrational). We will prove this later! Most of the rest of my comments are in the following videos.

### Overview of Proofs

Here's a video on proofs (in general). Click>>Video (will open in new tab/window)

### Common Factors and Being Relatively Prime

Here's a video on Common Factors and the idea of Being Relatively Prime. Click>>Video (will open in new tab/window)

### First Proof of the Pythagorean Theorem

Here's a video. Click>>Video (will open in new tab/window)

### Proof that there are infinitely many prime numbers. "The Infinitude of the Primes."

A prime number is a positive integer that has exactly two divisors (exactly two factors). Note that 1 is not prime. 1 has only 1 factor. You can google for a list of the primes. The question is, do they go on forever? YES! See the video. See pp 225-226 in the book. Euclid proved this fact. Our proof will be similar to(OK, but not exactly like) Euclid's proof. (There are other proofs than the one given in the video.) Note: This proof is quite understandable and straightforward. Bright middle school students can follow this. It is an indirect proof. (Numbers with more than two factors are composite. One is not prime and not composite.)

Here's the video. Click>>Video (will open in new tab/window)

### Proof that square root of 2 is irrational

Here's a video. Click>>Video (will open in new tab/window)

### Pythagorean Triples

Here's a video. (See pages 4-6.) Click>>Video (will open in new tab/window)

## Syllabus Assignments

The only syllabus assignment I haven't given more details about is the Idea File. As you find useful learning activities you can save them. Next week I'll show you what to include on the cover page/analysis. If I forget, someone remind me!

## Discussions

See Week 2 discussion area.

If you haven't done so already, sign up of a mathematician to provide info for myHistro (which is an assignment). (Week 1 area)

Sign up of a topic to present (probably a mathematician). (Week 1 area)

## To Do

Go to the discussion area. Participate. Everyone does not need to reply. If a good answer is given, there is no need to pile on and repeat what has been said. If you have additional info, do please add that.

## Homework

There is a homework assignment in the dropbox. Generally (not always) you get a week and 2 days to turn in HW. Due on Friday.