Posts Tagged ‘modular arithmetic’

For those of you who missed last week’s math circle, Dr. Page has supplied us with his lecture notes and slides.  Lecture Slides and Lecture Notes.

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Many talented young problem solvers can find their first math circle visits challenging because they simply haven’t been exposed to some of the concepts and tools of discrete mathematics.  In the US many of these concepts are reserved for “university level” mathematics but it has been proven that these ideas can be well understood and applied by very young students.  The students who add these concepts to their toolbox will have much more success in math circles and math contests as well as a great advantage going on to university.

To help fill this void, the good people at IMACS, creators of the popular eIMACS courses, have launched a series of short and focused online courses under the title of Elements of Mathematics: Foundations.  The first course in this series, Operational Systems, focuses on the very important concept of modular arithmetic and is free for students who register before January 1, 2013.

This course covers modular arithmetic using secret codes and online games. Learn about operational systems and their properties (commutativity, associativity, neutral elements, invertibility) by building interactive machines and evaluating non-numeric operations. Get a solid introduction to the concepts of least common multiple and greatest common divisor, as well as to the geometric notions of midpoint and reflection.

IMACS has provided the following description of the origins of this series and curriculum.

The Elements of Mathematics: Foundations online courses are based on Book 0: Intuitive Background of the Elements of Mathematics (EM) series of textbooks. EM is the result of a collaborative effort of an international team of eminent mathematicians and mathematics educators.

Through more than a decade of research and development, these scholars created an original curriculum that is fun and engaging for talented middle and high school students while maintaining a level of mathematical rigor found only at the university level. Aimed solely at talented students and unconstrained by the need to follow a standards-based curriculum, EM focused on providing precocious students with a deep understanding of mathematical structure.

Along the way, Book 0: Intuitive Background covered all of middle and high school mathematics except calculus before the end of middle school. After completing the Intuitive Background series, students would begin the EM formal logic series which covered a significant portion of a college undergraduate mathematics degree by the end of high school. The formal logic segment of the EM curriculum is already available online at www.eimacs.com as part of the Advanced Mathematical Logic track.

Don’t allow yourself to forget over the holidays, sign up for the first course in the series today.


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We look forward to seeing old and new friends today at the September 17  session of Metroplex Math Circle.  Next week we look forward to a lecture from Joshua Nichols-Barrer.   Dr. Nichols-Barrer earned his PhD at MIT and is the AwesomeMath Academic Director, a two time IMO silver medalist and a multiple winner of the USAMO.

Here is a description of the session in Dr. Nichols-Barrer’s own words:

Modular arithmetic is an essential tool for properly treating number theory problems in contest mathematics.  While there is far more to talk about than we have time for today, we will extensively cover the foundations of arithmetic mod an integer $m$, looking to differences between mod $m$ arithmetic and that which we are all familiar with, as well as those things which distinguish arithmetics mod $m$ for distinct values of $m$.  We will also begin to think about algebra mod $m$ should we have the time.

Modular arithmetic is one of the many fields ignored by standard math curricula but critical for success in math competitions or a career in mathematics or sciences.

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The second lecture of the season provided lessons in both applied mathematics and number theory. Dr. Bennette Harris began with an easy to follow introduction to binary and computer operations including XOR. He then defined the terms “encoding” and “encryption” and gave several examples of early encryption methods.

Dr. Harris described the difference between “strong” and “weak” encryption methods. This led to a deep dive into the popular RSA encryption method which depends on extremely large prime numbers. Dr. Harris used this as an opportunity to give the students a sense for mind-boggling large numbers and some of the very creative and efficient ways of determining whether or not a given number is a prime.

In this part of the discussion, Dr. Harris taught the students various techniques in modular arithmetic. Modular arithmetic is a tool that is not just useful in encryption but is critical for success in math contests and other pursuits.

Dr. Harris used several techniques to engage students including a series of 10 problems that he worked into the lecture for the students to solve. He also demonstrated the UBASIC program which very quickly found tremendously large prime numbers.

Dr. Harris has provided his slides from the lecture and answers to the problems. Members of the Metroplex Math Circle e-mail group can download these files from the group site. To join the e-mail group simply click below.

Click to join MetroplexMathCircle

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