Math. 301, Sec. 001, Fall 2017
Homework and Announcements
All handouts for the class will be posted as pdf files
after a bullet sign. All these handouts are copyright
material and are intended for the students of this class only.
IN PARTICULAR, DO NOT POST THESE HANDOUTS ON ANY OTHER
WESITE OR INTO AN ARCHIVE DATABASE!
GMU Undergraduate Math competitions Activities
Two upcoming competitions are:
(i) Virginia Tech Math Competition Saturday,
October 21, 2017, 9 -- 11:30 am.
(ii) Putnam Math Competition Saturday,
December 2, 2017, 10 am -- 1 pm (Part A) and 3 pm -- 6 pm (Part B).
Those who are interested should contact either
Prof. Dan Anderson (danders1"the-at-sign"gmu-dot-edu), or
Prof. Jim Lawrence (lawrence"the-at-sign"gmu-dot-edu) for
more info. -- Recruitment is underway, and practice for
the above competitions will be:
Friday Putnam Practice (aka "Chill Math")
most Friday's at 4:45 -- 6 pm at Math Dept. Room 4307 (open collaboration room)
For general information regarding this course, please
read the Syllabus here below, it should answer all
practical questions, like when tests are, what
the grading policies are etc.
Homework and Solutions
Here below the homework (HW) assignments and selected solutions
will be posted. Many of the problems will be exercises
from our textbook. -- Please be sure to read the solutions.
If you have questions or comments feel free to come
and talk to me.
There is no HW for Labor Day Monday, since there
is no class.
- Homework for Monday, Sept. 11. :
Be sure that you know how inductive proofs
work. Every form of induction (both weak and
strong form) is a consequence of the
Well Ordering Property (WOP) of the
natural numbers: Every nonempty set of natural numbers
has a least element. When defining a sequence
inductively, we call it recursion .
To prove some property for a recursively
defined sequence it is very often good to use
- Homework for Monday, Sept. 18
(I assigned some problems too eagerly, sorry.!)
Countable sets are precisely
those sets where we can list their elements
in the same manner as the natural numbers:
the first element, the second element and so on.
Many familiar sets are countable, but not all.
(We will see that the set of real numbers
is, in fact, not countable.) --
At this point (the end of the "Cook's Tour") we know
that there are many number sets out there. Some are countable,
meaning that we can make a list of them number 1, 2, 3 and so on,
that will cover the whole set (just like the natural numbers),
and some sets of numbers are uncountable, like the set of real
numbers and the complex numbers. This means that there is
no way one can write a computer code to list all the real
numbers say. -- The integers are interesting (and also difficult
for that matter!) since we cannot always divide one number with
another nonzero number and obtain an integer. When solving
equations involving integers we must proceed with much
more care than when solving for real or rational numbers.
However, there are some basic things that we can do. The
Division Algorithm (DA) is the procedure that
best resembles general division. EDA yields a remainder r
which is either zero or otherwise positive, but still stricly
less than the divisor. This is the best we can hope for when
working with integers. DA also implies that every integer
is uniquely represented in a given base b. --
One of the most important and interesting number sequence
is that of the prime numbers. Primes are the building blocks
of the integers, as we will see soon. They are the "atoms"
of the natural numbers. Many simply stated questions about
prime numbers are still unanswered, despite tremendous efforts.
An active area of research is to understand the distribution
of the primes on the natural number line. The
Prime Number Theorem states that the number of primes
less than a given number x is approximately given
by x/log(x), and that this approximation gets
better and better the larger x is.
- Homework for Monday, Sept. 25. :
The Euclidean Algorithm (EA) is an efficient algorithm
to compute the greatest common divisor of two positive integers.
This is one of the most important and nicest algorithms ever
invented. By back-tracking the EA, we can actually solve
equations of the form ax + by = c for
any integers a,b,c. This again can
be used recursively to solve such linear equations
in three or more variables (instead of just two!)
However, that will become increasingly messier as
the number of variables is increased. -- The
Fundamental Theorem of Arithmetic (FTA) states
precisely why the prime numbers are the building
blocks of the integers. That each positive integer
is a unique product of primes is good for many
divisibility problems. Provided that we have the
factorization of given integer, then we can spot
immediately whether or not one divides the other.
However, finding the factorization of a given integer
is computationally very hard. In fact, determining
whether or not it is a prime is hard enough, let alone
finding the complete factorization.