Lecture 1: Unique Factorization
- Sections 1.1-1.2 of the text.
- Maple Getting Started
(also look for "Getting Started" in the Table of Contents in Maple Help)
- Using Help - Go to the "Help" menu and select "Table of Contents"
and look for the "Using Help" folder. Also see "Using Help" under the
"Help Menu" within "Help System Menu Bars" in the "Menus" folder.
- Worksheet Management - Go to the "Help" menu and select "Table of
Contents" and look for Overview in the "Worksheet Management" folder.
- Overview of the course.
- Grading policy and expectations
- Introduction to Maple
- Maple worksheet interface
- Maple help facility
- Basic capabilities of Maple
- Variables, Values, and Names
- Symbolic vs. Numeric Computation
- Watch Out
- End with a semi-colon
- Names vs. Values and Evaluation
- Equal vs. Assignment
- Dynamic Scope
- Case Sensitivity
- Answers you don't understand
- Maple's assumptions might be different than yours
- Maple is not perfect
- Theorem: p|ab => p|a or p|b, p a prime number
- Proof that sqrt(2) is irrational.
- Investigation of Pythagorean triples.
- Unique Factorization (the fundamental theorem of arithmetic) - the
existence of the factorization of an integer into primes follows from
the correctness proof of a simple algorithm to compute the prime
factorization of an integer, and the uniqueness of the factorization follows
from the previous theorem - see
ufd.mw for worksheet containing the algorithm.
- Example where unique factorization does not hold: Z[sqrt(10)].
Maple worksheets and documentation
This assignment is a practice assignment not intended to be handed in.
Created: mar. 31, 2008 by
"jjohnson AT cs DOT drexel DOT edu
- Install Maple (go to IRT's software site and download - follow installation
- Load and run the examples from the worksheet from this lecture.
- Create a simple Maple worksheet and perform several computations
- Use evalf to approximate sqrt(3) to 100 decimal places.
- Use the plot command to plot x^2-3 over the interval [0,2]
- Use the solve command to solve the equation x^2-3=0.
- Use the fsolve command to solve the equation x^2-3=0.
- Use Maple to determine the number of primes less than 100.
- How many primes are there less than 1000?
- Create a title and some text explaining your computations.
- Add a section to your worksheet with the title "Bisection"
- Write a Maple procedure to approximate sqrt(3). Your procedure should take as
input a number e and should return an approximation s to sqrt(3) such that
|s - sqrt(3)| &le e. Use bisection to obtain your approximation. I.E. start with
the interval [0,1] and repeatedly compute the midpoint choosing either the left
or right subinterval, whichever contains sqrt(3). Continue this until the width
of the interval containing sqrt(3) is less than or equal to e. Finally return the
midpoint of the final interval.
- Generalize the previous procedure to approximate an arbitrary square root.