Lecture 1 21 127 Concepts of Math 08 27 2012 Syllabus and Website Open Color My Graph Features of the course website Home page contact info and links Staff page TAs and office hours use them Events page exam dates posted Homeworks posted here Documents posted on AnnotateMyPDF Clicking the link should automatically add you to the correct group Email me if you have trouble and we can fix it Folders on left documents exams extra readings homework lecture notes textbook Annotate Tutorial read it Use Hide Document List and Zoom feaures or download file as a pdf Make annotation on document Privacy settings Personal only you can see this note use this to make notes for yourself about what to read later what is helpful to study etc Admins only you and Brendan and the TAs can see this note use this to ask questions or point out errors that you aren t sure of Public everyone in the class can see this note use this to point out typos and other errors Notification settings will help you keep track of new documents and annotations Click on Remember me when you log in Open up syllabus You can read it later but here are the important parts This course will introduce us to mathematical proofs how to do them how to understand them how they work why they matter It is a writing course pretty much We care a lot about how you express your thoughts It is no longer okay to just get the right idea The ideas are the important things now Outline work on some puzzles for a while talk about induction then sets and logic Then functions and relations with some number theory then cardinality and infinite sets then just finite sets Combinatorics for a while some interesting topics If time some probability and graph theory no promises Prereqs nothing Textbook I am writing this Explain where the motivation came from and how it s going Point to chapters on the Annotate site There are 3 up now more posted as they are needed get finished Emphasize that chapter length does not correspond to time spent in class First chapter is long but not a lot of time maybe first week or so You should actually read it though 1 There are practice problems in the chapters More will be added over time so look out for that Some homework problems will come from these problems some will not Homework 30 might seem like a lot but it is important We care more about your ability to put time and effort into the problems and figure stuff out than we do your ability to work quickly under time pressure That said we still need to do exams These will be posted on CMG and Annotate one week or more in advance Due on Thursdays at beginning of recitation Returned on Tuesdays Collaboration policy It is helpful and we encourage you to work together in small groups Follow the no written record rule of thumb and make it small groups only 2 to 4 List your collaborators on your homework Go to office hours I will have some and the TAs will too I cannot emphasize enough how helpful these are Even if you aren t used to this just start going and hanging out you ll learn something Exams 3 in class exams 50 minutes 1 final 3 hours Prep Questions These are designed to alleviate the time crunch of exams You will have a week to work on these practice problems and a few will appear verbatim on the exam You shouldn t be straight up memorizing solutions rather by working through the argument you are practicing the skills we expect you to have and working with the material Really we are just helping you study Extra Credit Make annotations The more typos you catch and helpful suggestions you make the more I will like you and the more likely it is that I will bump you up a grade if you are on the border Also do a project Find an interesting topic not necessarily covered in class but you should relate to the stuff we do in class somehow and write a few pages about it Goal have a paper that a classmate can pick up and read and learn something from This can really help your grade General grade advice Yes this class is hard The rumors you ve heard are probably true However it is not prohibitively hard This is not the type of class where the smart kids all get As and that s that Far from it If you put in the time and effort that we ask of you you will absolutely do well Come to office hours Talk to me and the TAs outside of class Read the book Work on problems when they are posted and come back to them later Treat your homework like an essay write it and reread it later and edit it and keep fixing it That is DO NOT start homework two days before it s due Actually go to recitation you will get lots of examples and homework help there There is also peer tutoring available There is info on the website Behavior Come to class On time Shut up and pay attention Don t use a laptop unless you have to No need for calculators Go to recitation Work on homework diligently Don t cheat LaTeX This is an extremely helpful program and I strongly encourage you to use it It is not required though There will be info posted on the website soon about how to install and use it You will be far better off in the long run trying to use it In particular it s really easy to edit your homework We will have some seminars about how to use it soon too Stay tuned 2 Let s Do Some Math What is math The Wikipedia page on Definitions of mathematics says Mathematics has no generally accepted definition Different schools of thought particularly in philosophy have put forth radically different definitions All are controversial We will learn about what math really is over the course of the semester It will likely be rather different than what you re used to What is a proof This is also hard to characterize General idea an irrefutable logical clear and concise argument Buzzwords correctness clarity precision concision Now what is the Pythagorean Theorem What does it even say How do we write math Even yet what is a triangle A line A point What does it even mean to be true We have to agree on some axioms some fundamental facts that we just all agree are true We will prove lots of facts from those basic statements but we have to start somewhere Let s see a few proposed proofs of the Pythagorean Theorem Proof 1 Draw a square with side length a b Inside this square draw four copies of the right triangle forming a square with side length c inside the larger square The area of the larger square can be computed in two …