Simply from the title of the course called "Fundamentals" I wouldn't know what material would be covered. The course description says it is about number theory and combinatorics. While these two topics are not necessarily interrelated, they do have some things in common. The most interesting is that both topics do not require very much background in order to encounter very difficult problems.

What we find early on in this course that there are still things that are not known about the integers. For instance, while we know for sure that there are an infinite number of primes, we do not know if there are an infinite number of pairs of prime numbers of the form p and p+2 (e.g. (3,5), (5,7), (11, 13), (17,19), (29, 31), ...). Primes themselves are very hard to pin down and describe. It is still very surprising to me that there doesn't seem to exist a formula for an infinite sequence of primes.

One of the main applications of number theory is that of cryptography (sending coded messages so that an interceptor is unable to read them). One of the commonly used codes in modern cryptography is based theorems about integers that were discovered hundreds of years before the computer age. It still amazes me that the pieces of number theory that were discovered by mathematicians that could not dream of a modern computer became so relevant to making sure that your credit card number is not stolen when you send it over the internet. Although we only briefly touch on this application of number theory (the textbook was written before its invention!), most of what we learn is relevant for further study in cryptography.

Combinatorics is similar to number theory in that most people are familiar with the basic ideas of counting. However, it is easy to describe a set of things which is hard to count. For instance, how many ways are there of seating 10 couples (or more generally, n couples) around a table if no person is allowed to sit next to his or her spouse? To learn some of the basic rules of counting we build up arguments out of simple rules which readers accept and agree upon.

What makes these two topics "fundamental?" The main thing is that they provide a perfect setting for practicing a skill which clearly is fundamental to all of mathematics, writing and expressing yourself clearly and basing your arguments on principles that your reader accepts. This course takes some fundamental ideas (that explanation, justification, logic, proof and reason are behind mathematics) and ties them into some classical mathematics. Number theory is at its base addition, multiplication and division of integers. Combinatorics is about enumerating sets of objects. Both subjects involve familiar concepts to students and are taught at all levels. We will go deeper into these subjects, but what will make them 'fundamental' is we will also practice explaining why one applies basic rules to build up clear arguments.

You can look back at past web pages for this course to see what happened in past years. I've taught this course 3 times already but I tend to adjust it slightly each time.

## Monday, August 24, 2009

### About MATH 5410 Analysis for teachers

MATH 5410 Analysis for Teachers

In mathematics the concept of a real number comprises a rigorous treatment of the intuitive notion of `quantity'. The intuitive idea of a `relation' between quantities is encapsulated in rigorous mathematical terms by the

concept of a function of a real variable. The study of real numbers and functions of a real variable, together with higher concepts built upon them---such as measure and integration, metric spaces, families of functions, and so forth---is collectively referred to as analysis. Complex numbers and functions of a complex variable are closely related to real numbers and functions of a real variable, and study of the former is also referred to as analysis, or, more precisely , complex analysis.

MATH 5410 aims to convey a flavor of analysis as seen from a more sophisticated mathematical perspective than that of the Ontario high school curriculum, in the hopes that teachers will gain a new, enriched understanding that allows them to view the high school curriculum as part of a bigger picture.

Another objective of MATH 5410, is to convey a sense of the place of mathematics within the wider world. After all, why should math be taught as a basic subject from primary through secondary school? It turns out that the information age which is now upon us is largely based on the fact that a mathematically rigorous treatment of `quantity' is applicable to just

about anything: sounds, images, language, music, motion. The particular branch of mathematics that treats the `quantification' of sounds, pictures and communication is known as information theory, and it is the basis for digital communication as manifest in the world wide web, with its myriad applications such as youtube and iTunes, as well as cellular telephones and other wireless (and wired) media. But in order to have anything beyond a superficial understanding of information theory, one has to have a mastery of basic analysis. Any technical discussion of mp4 or coding theory, for

example, makes use of concepts from analysis. It is thus hoped that teachers who take MATH 5410 will be better equipped to convey to their students the place of mathematics in the broader world.

The format of the course will be weekly lectures and worksheets, with evaluation based on regular assignments, two in-class

tests, and a presentation.

In mathematics the concept of a real number comprises a rigorous treatment of the intuitive notion of `quantity'. The intuitive idea of a `relation' between quantities is encapsulated in rigorous mathematical terms by the

concept of a function of a real variable. The study of real numbers and functions of a real variable, together with higher concepts built upon them---such as measure and integration, metric spaces, families of functions, and so forth---is collectively referred to as analysis. Complex numbers and functions of a complex variable are closely related to real numbers and functions of a real variable, and study of the former is also referred to as analysis, or, more precisely , complex analysis.

MATH 5410 aims to convey a flavor of analysis as seen from a more sophisticated mathematical perspective than that of the Ontario high school curriculum, in the hopes that teachers will gain a new, enriched understanding that allows them to view the high school curriculum as part of a bigger picture.

Another objective of MATH 5410, is to convey a sense of the place of mathematics within the wider world. After all, why should math be taught as a basic subject from primary through secondary school? It turns out that the information age which is now upon us is largely based on the fact that a mathematically rigorous treatment of `quantity' is applicable to just

about anything: sounds, images, language, music, motion. The particular branch of mathematics that treats the `quantification' of sounds, pictures and communication is known as information theory, and it is the basis for digital communication as manifest in the world wide web, with its myriad applications such as youtube and iTunes, as well as cellular telephones and other wireless (and wired) media. But in order to have anything beyond a superficial understanding of information theory, one has to have a mastery of basic analysis. Any technical discussion of mp4 or coding theory, for

example, makes use of concepts from analysis. It is thus hoped that teachers who take MATH 5410 will be better equipped to convey to their students the place of mathematics in the broader world.

The format of the course will be weekly lectures and worksheets, with evaluation based on regular assignments, two in-class

tests, and a presentation.

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