Notes on Talk Slowly and Write Neatly and Big - Ideas in Mathematics | MATH 170, Study notes of Mathematics

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Lecture Notes for Math 170: Ideas in

Mathematics (Spring 2007)

by Nathanael Leedom Ackerman January 8, 2007

1

1 TALK SLOWLY AND WRITE NEATLY AND BIG!! 2

1 TALK SLOWLY AND WRITE NEATLY AND BIG!!

2 Introduction

Before Class

(1) Write my name, office and office hours on the board. (2) Write name of teaching assistants.

• Dan Pomerleano ([email protected])
• Chintan Mehta ([email protected]) (3) Write the course web address (www.math.upenn.edu/ nate/teaching/2007/spring/math 170/ind Also write my homepage address (www.math.upenn.edu/ nate/index.html) (1) Introduce myself and tell about my office hours (Wednes- days 2:15 pm - 3:15 pm). (2) Introduce the TA’s

2 INTRODUCTION 3

(3) Say that for each class I took when I was in college outside my major, if you were to ask me one year afterwards what I learned, I would be lucky to tell you a single fact.

However I am greedy. My goal in this class is for each of you, one year from not, not to remember just one fact but 3. My goal in ths class is for you to learn and remember (1) Math is a way of thinking (2) Math is beautiful (3) Math is all around us (4) Explain structure of the grading and how there are two types of grades. The main grade is made up of

• (Homework) ∗ Weekly homeworks worth 30%. These will be

2 INTRODUCTION 4

due each week at the beginning of class. There is no better way to learn math than doing problems and so these are vital for learning the material. ∗ Math is also meant to be a social activity. So you are all strongly encouraged to find a cou- ple friends in this class and to discuss the the problems and your solutions. However when it comes time to do the homework you must do it write up the solutions yourself. ∗ If not turned in to class, turned in to math office by 5 pm. No late homeworks will be accepted.

• (Presentation) ∗ Each student will also be responsible for a 10- 15 minute presentation in section on material

2 INTRODUCTION 5

related to what we are covering. This will be worth 20% of your grade ∗ You will all have to sign up for a week to present in your first section. ∗ You must also meet with me and your TA before your presentation so we can go over what we want you to discuss. This will most likely take place the Friday before but I will let you know next time for sure. ∗ You will also be responsible for typing your notes up to your presentation and e-mailing them to me so they can be posted on the web- site.

• (Midterm Paper) 20% of your grade ∗ 3-4 pages on a work of mathematical fiction. ∗ You must sign up with your TA by Monday

2 INTRODUCTION 6

January 29 (no two people with the same TA can do the same book so first come first serve) ∗ Due Friday March 2.

• (Final) To be announced. (4) Bonus points. Pure extra credit. Quizzes in class. (5) Reading related to the class can be found on the website. And when I have them I will post my lec- ture notes there as well. It is very important to stay caught up with the reading so you don’t fall behind. (6) Say that we are going to be covering a lot of material, so it is important to stay on top of it. And if at any point during the lecture something is unclear, please ask about it. Because if something doesn’t make sense to you, then it probably doesn’t make sense to others as well.

3 BIRTHDAY PROBLEM 7

3 Birthday Problem

As it is the first day, lets play a game. We all know that there are 365 days in a given year. Now our intuition tells us that if we were to have half that many people, say 182.5, in a room, there is a descent chance that two of them would have the same birthday.

What if we only have 1/10 that many, say 36 people? Do we think that it will be

• Very likely
• Somewhat likely
• As likely as unlikely
• Unlikely
• Very unlikely that they would have the same birthday?

4 DEFINITIONS 8

Take a vote and write on the board what people think.

Ask them to bet their grades (do the experiment 2-3 times depending on the number of people). Have anyone who wants to write on a piece of paper how many time they think it would happen.

Collect the papers and go through the class and figure out how many.

4 Definitions

Repeated Experiment Say we want to

assign a value between 0 and 1 to express how likely some- thing is.

Go through simpler example. Ask about flipping coins.

Observe that all the possible outcomes add to 1.

Explain that if we do the same experiment over and over

4 DEFINITIONS 9

again, the more likely an event is the more times we would

expect it to happen. Write on my calculator a progr

Take them through several hundred coin flips (start with 1, then 10, then 100, ect.)

Observe that as we do more and more coin flips we get closer and closer to 1/2.

Definition 4.0.1. We say the probability of an outcome OUTCOME for an event EVENT is equal to the limit of (NUMBER OF TIMES OUTCOME OCCURS)/(NUMBER OF TIMES EVENT OCCURS) as number of events goes to infinity.

This has two problems.

(1) In general it often isn’t practical to perform an event

5 IF EXTRA TIME 10

more than a few times. (2) While in the limit we get the right answer, along the way we might not.

5 If Extra Time

3 coins Work out the probability after flipping

three coins they all come up the same.

List options.

Observe that any one coin flip has two equally likely out- comes so each of the above is equally likely.

Explain how to use that to get the probability.

Work out how many possibilities they are for the birth- day case.

5 IF EXTRA TIME 11

Theorem 5.0.2. If the probability of event X happens is Y and the probability of event A is B and they are independent, the probability of X and A is Y * B.

Proof. Proof with marbles. Explain how to get the birthday solution we figure it out.

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