Ma330 Spring 2013 (MWF 9-9:50 in CB 214)

This page is always under modification. Watch for new material here.

Useful links

Course Policy

Check out a BBC link discussing Indian Math.

Project I info      
Due date March 4!

 Final Project info       New
Due date for decisions April  2

 link to important files.

 

Many new files added! Look!

A most useful resource

Another useful resource at UTK

Special file for Quiz on 4/1

In files section!

 

 

 

 

 

·       Assignment for week of 1/9-1/11.
Read Chapter 1 of the Crest of the Peacock.
Submit a short description of what you wish to learn from this course and why. This should be your personal reflections and should be submitted in class on Friday 1/11.

·       For week of 1/14-1/19.
We will finish the discussion of number systems.
I will describe the constructions of rationals and reals.
I will describe the questions about countable and uncountable sets.  
Google up Peano axioms, Dedekind Cuts and Countable sets on Wikipedia.
By the end of the week, we shall be looking at the various number systems introduced around the world. This material can be picked up from our textbook.
Listen to the interview on BBC (link above). I will comment and expand on it next week. One of the people interviewed is the author of our textbook!

The contact information for the TA is Robert Davis davis.robert@uky.edu. Please contact him to set up your group and choose a topic. Check the above link for Project I info for more details. Robert will also resolve conflicts resulting from two groups choosing the same topic.

                                                                                                Important:
Each student should email Robert Davis by 5 pm on Wednesday, Jan. 23.
In your message, please write down who is in your group and what topic your group will be focusing on.
If you do not have a group by that time, group assignments will be made based on your topic of interest. 

·       For week of 1/23-1/26
We will formally start discussing history topics. The first one on the agenda is representations of numbers around the world. Be sure to browse through the relevant information in chapters 2, 3, 4. The chapters have several discussions, but we will concentrate on the number systems. Also, finish listening to the BBC interview. Relevant discussion of the Indian system will also be discussed.

·       New announcement: The following homework shall be due on Monday Jan. 28 in class. Write a short note explaining any one of these topics, as if you are explaining it to a friend. Be sure to write correct information without being verbose! This may be handwritten or typed as you wish. Expected length is one or two pages. Be sure to cite the source, if you consult a book or a website.

The problems are progressively more and more challenging. Try to do the hardest of these that you can do!
 

o   Explain how the natural numbers are defined by an axiom system, say the Peano axioms. Be sure to explain at least some of the integer operations, using your axioms.

o   Explain how the set of integers is constructed from the natural numbers using equivalence classes of pairs of natural numbers. Be sure to explain how the operations on integers are carried out using the equivalence classes. Pay attention to why the operations are well defined.

o   Explain how the set of rational numbers are constructed from the set of integers using equivalence classes of pairs of integers (suitably restricted). Be sure to explain the usual operations of addition and multiplication in terms of the equivalence classes. Pay attention to why the operations are well defined.

o   Assuming that the real numbers in the interval (0,1) can be represented as decimals uniquely (by omitting tails of 9's), explain the argument that they are uncountable. Give a convincing explanation.

o   Give the "Dedekind cut " definition of real numbers in detail and explain how the addition and multiplication is defined using the cuts. Explain which cuts correspond to rational numbers.

o   Prove that if a positive integer has a rational square root, then it (the square root) must be an integer.  Deduce that a product of distinct primes has no rational square root.

o   Let D be one of the numbers between 2 and 16 which is not a square. Prove that there are positive integers x,y such that x^2-Dy^2=1.

·       For week of 1/28-2/1

o   We will begin with the discussion of alphabet codes and number systems in India.

o   Discussion of calculation methods around the world. Some unusual techniques.

o   Note the link to important files on top of the page. All useful files will be linked there in the future.

·       For week of 2/11-2/15

o   Summary of BBC discussion and points to be discussed appear in the files. We will expand on these points.

o   We will also pick up on the more Modern discussion of the Chakravala discussion.

·       For week of 2/18-2/12

o   Aryabhata algorithm.

o   Divisibility test generation.

 

·       For week of 2/25-3/1

o   Infinity in India. Read in files’ section.

o   More topics from Indian Mathematics. Read from the calculus development in India in the files’ section. Especially, look at the proof of the arctan series.

o   Finish up with some combinatorics topics.

·       For week of 3/4-3/8

o   The first project is due on March 4. You must at least submit an outline (something like the first page or table of contents) on March 4. Additional time for actual submission will be allowed until the end of the week – no later than the Spring Break!

o   Expect some quiz(zes) on the various mathematical concepts learned.