Useful links
Project I info |
Final Project info New |
Many new files added! Look! |
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Special file for Quiz on 4/1 In files section! |
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· 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.