Welcome!
Here you will see info about what we did in class, upcoming exams, solutions for past exams, etc.,
CHECK AT LEAST ONCE A WEEK!!
COURSE PAGE WITH SYLLABUS, GRADING SCHEME ETC

Started 8/17/2018




FINAL EXAM DEC 7 FRIDAY 2 TO 4PM ROOM G-06 (USUAL CLASSROOM).
Tuesday, 12/4/18
Please read Final exam review problem solutions.
Please also study class notes, homework and test solutions and anything else that is on the update page in addition to the review problems.


Monday,12/3/18.
Please study class notes, homework and test solutions and anything else that is on the update page in addition to the review problems.


Friday, 11/30/18.
Please try Final exam review problems.

PLEASE TRY THEM BY YOURSELF IF YOU WANT TO DO WELL IN FINAL.
Will post solutions next week, likely by tuesday night.

ALL OUTSTANDING ISSUES MUST BE ADDRESSED BEFORE FINAL EXAM.
NO MAKE-UPS ETC AFTER FINAL EXAM.
The Fall 2018 COAS Course Evaluations extended through Tuesday, December 4, 2018:
Please check your email for link to evaluation.


IMPORTANT: From here onwards, each class will be a hw session, worth 10 points.
No more turning in of homeworks.
I will write down a few problems at the beginning and you will have to do it in class.
I will guide you through it but it will not be done on the board.

Presentations week after thanksgiving.
Task 1: 30 points (penalty 10 points /week for late submission); Task 2 Due by 11/26 Mon: 30 points; presentation: 40 points.
Please finish project and preparation for presentation and for answering questions before coming back to school after the break.

Monday, Nov 19, 2018.
Today we continued with geometric proof of the formula to generate Pythagorean triples.
Please study these Notes on Pythagorean triplets.
You might need to reload your browser.
We broke down the main proof into pieces, and problems similar to these pieces might be in the final exam.
I will be in office 1 to 3 pm tuesday (tomorrow) and 10 to 12 pm wednesday.


Tuesday, Nov 13, 2018
Congrats to all of you for submitting the task 1 on time.
There were just one or two projects with omissions and I have mentioned that in the next task assignment. You can finish those as part of task 2.
I have given 30 points to everyone for task 1.

Just posted task 2 (final task) for the projects for each group in Blackboard.
Please look under group tasks in the group with your name in it, after you log in and select this course (Proofs).
Please use the group wiki as well as file exchange to put up your project related work.
All work must be submitted in Blackboard in your group page.
Please try to finish the current task before leaving town for thanksgiving.
You will have to present your project to the class the week after thanksgiving.
Each group member may be required to give a presentation, so every group member should know and understand all the work being done by the group.


PROJECT: Task 1 due by friday Nov 9.
Task 2 to be posted later this week, due monday Nov 26.
Wednesday, 11/7.
I have updated the notes from monday (see below).
Please reload it to see latest version.


Tuesday, Nov 6, 2018.
Yesterday we did three problems as part of step by step proof of the formula to generate Pythagorean triples.
They are on page 2.
Notes for monday's class Pythagorean triplets
You might need to reload your browser.


Thursday Nov 1.
IMPORTANT: From here onwards, each class will be a hw session, worth 10 points.
No more turning in of homeworks.
I will write down a few problems at the beginning and you will have to do it in class.
I will guide you through it but it will not be done on the board.

PROJECT: Task 1 due by next friday Nov 9.
Task 2 to be posted next week, due monday Nov 26.
Presentations week after thanksgiving.
Task 1: 30 points (penalty 10 points /week for late submission); Task 2: 30 points; presentation: 40 points.

Notes for wednesday's class Pythagorean triplets

Solutions for hw5


Friday, Oct 26,2018.
I have posted projects for each group in Blackboard.
Please look under group tasks in the group with your name in it, after you log in and select this course (Proofs).
You can use the group wiki (right now it just has the instructions for the next week) as well as file uploads to put up your project related work.
All work must be submitted in Blackboard in your group page.
Please try to finish the current task by the end of next week.
Each group member may be required to give a presentation, so every group member should know and understand all the work being done by the group.


Monday, Oct 22, 2018.
Notes for today's class Two examples for strong induction (to help with HW5)
HW5 due wednesday, 10/24 by 3pm.
Please try the problems everyday, and by yourself as much as possible.


Friday, Oct 19, 2018.
Notes for wednesday's class Proof by strong induction (INCLUDES HW5)
You may need to refresh your browser.
HW5 due wednesday, 10/24 by 3pm.
Please try the problems everyday, and by yourself as much as possible.


Tuesday, 10/16/18.
Please read Solutions for Second Midterm Exam.
Wednesday we will continue discussion of strong induction
Midterm Exam Problems.


Thursday, 10/11/18.
Please read Review Problems with Solutions for Midterm Exam.
Monday we will have the make-up midterm.
Please study these review problems in addition to all things you studied for midterm exam.
Make sure to try each problem at least for half an hour before reading solution.


Monday, 10/8/18.
We will have a make-up midterm exam on monday, 10/15.
This wednesday (10/10) we will do a review for the same.
I will ask you to do problems in class.
Today we went over Strong Induction.
Please read notes.
We will continue this after monday's exam.

.
Thursday, 10/4.
Please read Solutions for Midterm Exam.
Monday we will talk about strong induction.
Midterm exam problems.



MIDTERM EXAM WEDNESDAY OCT 3.
Need to get it done by Oct 5, when midterm grades are due.
Review on monday, Oct 1.
NO CALCULATORS OR CELLPHONES ALLOWED.
Please study all hw problems and class notes.

Tuesday, 10-2-18.
I have uploaded the review set (below)with solutions to all problems except last one.
Last problem is a challenge problem and optional.
Please reload the page after opening the review set, in order to get the latest version.
Note that in problem 10 the question should read:
Find irrational numbers a,b such that a^b is irrational. (NOT "ab is rational").
Also, NO CALCULATORS OR CELLPHONES ALLOWED.


Monday, 10-1-18.
I have uploaded the revised review problem set with solutions to some of the problems.
Please refresh your browser.
Will put in solutions to rest of problems asap.


Friday, 9-28-2018.
Please try these Review problems.

We will do some of them monday.

Tuesday, Sep 25, 2018.
Please do as many problems as possible in the exercises following section 1.2 of Introductory Number Theory Textbook.


Monday, Sep 24, 2018.
Updated Notes for today's class Proof by induction and using Pigeonhole principle (Revised!)
includes SOLUTIONS FOR HW4. Please note that problem 9 has been revised. Formula should have 2a, not just a.
You may need to refresh browser.
I showed a geometric way to think of the formula for first n odd natural numbers.


Wednesday, Sep 19, 2018.
Updated Notes for today's class Proof by induction and using Pigeonhole principle (Revised!)
includes HW4 DUE NEXT MONDAY. Please note that problem 9 has been revised. Formula should have 2a, not just a.
You may need to refresh browser.
Today we covered arithmetic and geometric sequences, identifying patterns and using that to find formulae for sums of sequences.
I also showed a geometric way to think of the formula for first n natural numbers.
See if you can come up with similar formula for sum of first n odd numbers.


Monday, Sep 17, 2018.
Notes for today's class Proof by induction and using Pigeonhole principle
includes HW4 DUE NEXT MONDAY.
Please read notes and try all problems before coming to class wednesday.
Please also read 1.2.3, all of 1.6, 2.1, and 2.2 and 2.3.1 from the Introductory Number Theory Textbook.


Thursday, Sep 13, 2018.
Yesterday we went over some hw3 problems and talked about GCD.
Monday we will talk about Pigeonhole principle and Principle of Induction.
Please read 1.2.3, all of 1.6, 2.1, and 2.2 and 2.3.1 in preparation, from the Introductory Number Theory Textbook.


Wednesday, Sep 12, 2018.
Revised Notes for today's class (Some examples from basic number theory)
includes SOLUTIONS for hw3.
You may need to refresh browser.


Monday, Sep 10, 2018.
Revised Notes for today's class (Some examples from basic number theory)
includes hw3 with HINTS for same !
Will do problems 3,4,5,8,9 on wednesday.
Problem 4 uses result of problem 6 and it can also be done using 3 (unique factorization) although that will be silly since we use result of problem 4 to prove that in first place.

Solutions for Homework 2


Friday, Sep 7, 2018.
Here is an Introductory Number Theory Textbook that you can download.
For now, please read sections 1.1,1.2.1,1.2.2, all of 1.3 and all of 1.5.


Wednesday, Sep 5, 2018.
HW2 can be turned in tomorrow by 3pm in office or mailbox.
HW3 (see below) is due next Wednesday (9/12) by 3pm. No more postponement

I will be in office tomorrow from 1 to 3 if you need help.

Notes for today's class (includes hw3):
Will do problems 4,6,7 on monday.
Problem 4 uses result of problem 6 and it can also be done using 3 (unique factorization) although that will be silly since we use result of problem 4 to prove that in first place.
Some examples from basic number theory
I have revised Homework 2 (Problem 2 should say "positive real numbers."
You may need to refresh your browser


Thursday, Aug 30,2018.
HW2 (see below) is due Wednesday by 3pm.
I will be in office tomorrow from 10 to 2 and tuesday from 1 to 4 if you need help.

Notes for wednesday's class:
Notes about converse, contrapositive, proof by contradiction.
Please study the notes and do the Homework 2.
HW 2 is worth 16 points.

Homework 1 Solutions. .


Friday, Aug 24,2018.
On Monday NO CLASS but I will be on campus from 9 to 11.
Please come to office at that time if you need help for homework.


Extra Work and Reading for wednesday's class:
Notes about Euclid's fifth postulate with an extra credit problem.
Please study the notes and answer the questions in the last page.
The problem will be for extra credit and you can, if you wish, show me the answer by wednesday.

Additional reading material: Euclid's first four postulates (mostly about the fourth but also talks about first three).
Some geometry facts that follow from Euclid's first four postulates.

The homework due monday is separate from the above.
Note that this week's problems were only a demonstration of how in math we try to prove everything starting from first principles.
Starting next week we will study methods of proofs systematically.
The problems will be much simpler.


Wednesday, 8/22/2018
NO CLASS ON MONDAY
Office hours from 9 to 11 am.
You can get help for HW1 (see below) and submit during that time.
You can do the same tomorrow or friday as well.
Call 806-6832 if you want to make sure I am in my office.
HW1 will consist of only the 2 questions mentioned below in yesterday's notes.
For the second question you can assume that the triangles in Picture 1 (on the LHS) are all congruent
and that the gray area in the middle is a square of length c on all sides,
since we proved those two things for the first problem.


Tuesday, Aug 21.
Notes for class yesterday
Please study the notes and answer the questions in the last page.
If you attended class you should be able to answer them.
They are part of the homework due next monday.