The past two classes have dealt with the "division algorithm" and a variety of concepts surrounding it. For convenience I'll restate it: given any natural numbers m and n, there exist unique integers q and r with
0 ≤ r < n
and
m = nq + r.
The integer q is called the quotient of m by n, and r is called the remainder. This expression can be put into quotient form, by simply diving both sides by n:
m ⁄ n = q + r ⁄ n .
We also went over a few definitions that you are responsible for. If the remainder of m by n is zero, we say that n divides m, written
n | m.
Using this definition of "divides," you should be able to prove the following two facts: if m is any natural number, then
m | m and 1 | m.
That is, any number is divisible by itself and 1. If a number m has no divisors except these two, then it is called a prime number.
All of this is very ancient. The formal proof of the division algorithm can be found in Euclid (circa 300 BC), but the proof likely goes back to the Pythagoreans of two centuries prior, if not earlier. The practical use, but not the formal understanding, of the division algorithm is far, far older still.
Indeed it is present in one form or another at the very beginnings of essentially all civilizations---the practicum of mathematics appears to be a requirement for cities and trade, and the equivalent of the division algorithm is found in actuarial tables left in Sumerian cuneiform tablets, the intricate calendrics of the Maya, and taxation records of the Shang dynasty. What motivates its development, apparently, is the needs of tandem developments of economic systems (trade and taxation), legal systems (arbitration, settlement of land/goods disputes), the measurement of time, and the military.
We may take a simple example from the military. Suppose you live in a city-state whose smallest military unit is the phalanx, and that training is based around blocks of soldiers 8 deep and 12 wide (96 individuals). Conscription records indicate you have 5,012 recruits. To plan a campaign you need to arrange, ahead of time, supplies, logistics, and battlefield organization, and you need to know at any given time how many phalanges you'll have. Now, in the days before calculators or even before efficient number systems, this is actually quite serious problem! A city's administrative core would require a scribe or scholar who could answer (or show methods for how to answer) myriads of similar questions. In this case the scribe would tell you that enough soldiers are present for 52 phalanges, with 20 soldiers left over (maybe they can serve as scouts). That is, 5,012 by 96 has a quotient of 52 with a remainder of 20.
Now most cultures did not develop math beyond these practical uses, possibly because most cultures' needs extend no further, possibly because many cultures didn't have time for that development (many civilizations in the archeological past lasted 1,000 years or less), and possibly because of internal cultural proclivities---sometimes people are not interested in math for its own sake. Go figure!
I want to say something about this. The simple use of mathematics in solving other kinds of problems is what I call "in the box" thinking with regards to math. To relate this to our political system, it is a framework that you can function within, and attempt to use, as a tool, to accomplish your goals. If politics is used this way, you choose a side that suits you (left or right, Repubs or Dems), and you lobby, vote, speak, or participate in other ways. Or you can jump out of the box and think about the system, for instance you can think about what kind of system we should have instead, whether or not the current paradigm is just what we have to live with, and how we could change the system.
Likewise math can be used just to compute, calculate, and problem solve: that is, the system (math in this case) can be used. In some places and times in human history, people have left the system's confines and thought about the system. In mathematics, this originally occurred (to my knowledge) in two human cultures: during either the late Zhou dynasty or early Warring States period (records are unclear owing to the destruction of that period), and during the classical Greek enlightenment. Both periods were in the rough time frame of 500-200BC. In the Greek world, the concept of formal mathematical proof---a crowning jewel of that culture's intellectual achievement---was developed. Through the ebb and flow of cultural history (empire, fall, feudalism, enlightenment, global trade, technology), the concept of proof persisted, and remains with us, virtually unchanged, in the present day. The concept of proof is the main tool we use to understand math, as opposed to just using math.
So we want to ask about numbers. We have these special numbers, the primes, that have no divisors except the two that any number must have. In a sense, primes are natural phenomena: they have been observed by many cultures in disparate times and places. We have some examples of primes (2, 3, 5, 7, etc), but in total, how many primes are there? What is the proportion of all numbers that are prime? The second question was asked long ago but not answered with any degree of precision until the Prime Number Theorem was proved in 1896:
The number of primes between 1 and n is approximately n / ln(n).
The answer to the first question was provided more than two millennia earlier by the Greeks themselves: there are infinitely many prime numbers. The proof is by contradiction. To begin, we assume there are just finitely many primes, and, using this assumption, derive a contradiction.
So let's start. Assume just finitely many primes exist, and that p1, p2, . . . , pN is the full list of all prime numbers. Then consider the number
T = p1 p2 . . . pN + 1
formed by multiplying all the primes together, and adding 1. From here it is easy to show that the remainder of T by p1 is 1, the remainder of T by p2 is also 1, etc. That is, the quotient of T by any of the primes is 1. Therefore T is not divisible by any prime, and it must therefore be a new prime number. But we assumed that that p1, p2, . . . , pN was a list of ALL primes!
This impossibility means that it could not have been the case that p1, p2, . . . , pN was a list of all primes. This means is that ALL primes cannot be put into any finite list---the quantity of primes, then, must be infinite.
I mentioned that time-keeping was one of the earliest uses for math. Coming up in class, we will discuss special kinds of mathematical systems designed for exactly this kind of application.
Wednesday, September 11, 2013
Wednesday, September 4, 2013
Formal reasoning and the real world
This week we started on the course material proper. At the beginning of the book, the authors get a little silly with their "problems," for instance how a knight could cross a perfectly square 20ft moat onto a perfectly square landmass within, while for some reason only having 19ft 8in logs available.
Realistic? No. So what's the point? Maybe nothing. Could be the authors are weird or a little crazy. But then Ed Burger is a professor and consultant, and Mike Starbird is a distinguished professor at UT and both are professional mathematicians, so maybe they're ivory-tower types giving students flip little problems meant to "teach" something condescending. It could be that's too harsh; maybe the problems were just meant to be readable: math-y enough to communicate something, but easy enough that people don't get turned off to the book.
There is another purpose though. Just like skills in writing and communication, mathematical skills have a point, which they are trying to bring in. Yes the problems are silly, but they do resemble things in the real-world. Of course the real world does not come with its own interpretations or its own structures: it's there, doing its own thing, and we humans have to deal with it. Reality doesn't have problems, humans have problems, and most of us want to deal effectively with them, and of the many ways of dealing with what's out there, the process of formal reasoning, when used properly, is extremely effective. It works like this. Something in reality is encountered and a mental model is formed to reflect it, at least parts which can be understood as a formal structure. Formal or mathematical reasoning can be used to solve the problem in the mental model, and one attempts to translate this into a real-world solution.
The moat-crossing knight is an example. He finds a moat, which he perceives to be a problem (for whatever reason he wants to be on the moat's interior). He ascertains the situation, and creates a mental picture---in this case, a useful interpretation is as a geometry problem. The items around him are also given geometric interpretations: nearby logs can be seen as line segments for instance. His task is to use these line segments, and a few concrete rules, to draw a path across the geometrically-abstracted moat. If this strictly formal problem can be solved, he then attempts to translate the mental solution into the real world.
This is quite a philosophical approach to what is ultimately a collection of very simple story problems, and possibly my observations are pedestrian or ridiculously obvious. Maybe you see some value in this way of thinking. But I'm going through this because I want to encourage you to do likewise: don't just ask how to solve a particular problem from the book (yes, actual problem solving is important too), but why you are solving it. Don't be led by the nose, doing just what the book asks, trying to get through. Be active. Try to see why you're being asked to do it, and ask what of value is in it. You're not in this class to solve ridiculous little problems after all!
Realistic? No. So what's the point? Maybe nothing. Could be the authors are weird or a little crazy. But then Ed Burger is a professor and consultant, and Mike Starbird is a distinguished professor at UT and both are professional mathematicians, so maybe they're ivory-tower types giving students flip little problems meant to "teach" something condescending. It could be that's too harsh; maybe the problems were just meant to be readable: math-y enough to communicate something, but easy enough that people don't get turned off to the book.
There is another purpose though. Just like skills in writing and communication, mathematical skills have a point, which they are trying to bring in. Yes the problems are silly, but they do resemble things in the real-world. Of course the real world does not come with its own interpretations or its own structures: it's there, doing its own thing, and we humans have to deal with it. Reality doesn't have problems, humans have problems, and most of us want to deal effectively with them, and of the many ways of dealing with what's out there, the process of formal reasoning, when used properly, is extremely effective. It works like this. Something in reality is encountered and a mental model is formed to reflect it, at least parts which can be understood as a formal structure. Formal or mathematical reasoning can be used to solve the problem in the mental model, and one attempts to translate this into a real-world solution.
The moat-crossing knight is an example. He finds a moat, which he perceives to be a problem (for whatever reason he wants to be on the moat's interior). He ascertains the situation, and creates a mental picture---in this case, a useful interpretation is as a geometry problem. The items around him are also given geometric interpretations: nearby logs can be seen as line segments for instance. His task is to use these line segments, and a few concrete rules, to draw a path across the geometrically-abstracted moat. If this strictly formal problem can be solved, he then attempts to translate the mental solution into the real world.
This is quite a philosophical approach to what is ultimately a collection of very simple story problems, and possibly my observations are pedestrian or ridiculously obvious. Maybe you see some value in this way of thinking. But I'm going through this because I want to encourage you to do likewise: don't just ask how to solve a particular problem from the book (yes, actual problem solving is important too), but why you are solving it. Don't be led by the nose, doing just what the book asks, trying to get through. Be active. Try to see why you're being asked to do it, and ask what of value is in it. You're not in this class to solve ridiculous little problems after all!
Wednesday, August 28, 2013
Welcome to Math 170
Welcome to Math 170 at upenn! First let's start with some questions, or perhaps meta-questions, about the class. Why take it? What should I (as a student) get out of it? What does Professor Weber want; what are his aims? What does the University want?
The first question has a pretty obvious answer for a lot of people: because I have to! It's a graduation requirement; I need to fill in a circle in my quantitative reasoning section. Others are interested for reasons more closely aligned to the aims of the class. Some want to go further in math, but haven't taken any college mathematics before, and feel they require an introduction before getting to more serious subjects. Others found themselves interested in some of our subject material (and a lot of it I think will be very interesting!).
These are all perfectly valid reasons, but I'd like us to look deeper into the rationale behind the class. In fact, as some students have pointed out in the past, the rationale and aims for Math 170 can seem a little unclear. For example, in a calculus course, the aim is to become proficient in calculus; in an algebra course, the aim is to become proficient in algebra; is an east Asian history course, the aim is to become knowledgeable about east Asian history. In various English courses, one learns grammar, paragraphing, story-telling, critical reading, and effective written communication. What is the aim of Math 170?
Our aim is, in a way, broader than in many other classes you'll take here. This class' students are frequently interested in realms of human knowledge and practice that are less mathematical: the departments I see on my class list circle around Psyc, Biol, Hist, Engl, and so forth, with a decent smattering of UNDC. The purpose of this class is to help develop proficiency in another form of human activity: formal reasoning (which includes, but is far from limited to, reasoning with numbers), and to help develop knowledge in an immensely rich and layered world that is rarely glimpsed: the world of mathematical form and structure.
So that's it: from the point of view of your teacher, me, the class' purpose is for everyone to become more proficient in formal reasoning skills (precision formulation of problems, pattern recognition, shape, formal structures, and, yes, numerical facility), and to gain some knowledge of what mathematics is today and what is being studied and developed in that world. If by semester's end everyone has a greater ability to approach both abstract and real-world problems in a precise, formal way, and everyone has a bit of appreciation for the amazing mathematics that exists in our society today, then I will consider the course a success.
And the University's aims? Of course I can't speak for the university, but I can give my view on the matter. The University wants to graduate students who are knowledgeable and capable, not just in their fields, but in a wide array of human realms---it wants its graduates to possess not just a skill, but what it considers to be an educated mind. That is why engineers take English and history, and why pre-meds and comm. majors take some kind of math. In an even broader sense, the purpose is two-fold: first, it gives you, the student (fundamentally you are the client), value for your money and your trust. The second purpose is that it is a form of advertising for Penn: if it graduates impressive and successful human beings, then Penn's status rises, and the demand for its services rises.
Anyway, I hope this overview of our course goals is valuable to you. As always, your comments and feedback are appreciated!
The first question has a pretty obvious answer for a lot of people: because I have to! It's a graduation requirement; I need to fill in a circle in my quantitative reasoning section. Others are interested for reasons more closely aligned to the aims of the class. Some want to go further in math, but haven't taken any college mathematics before, and feel they require an introduction before getting to more serious subjects. Others found themselves interested in some of our subject material (and a lot of it I think will be very interesting!).
These are all perfectly valid reasons, but I'd like us to look deeper into the rationale behind the class. In fact, as some students have pointed out in the past, the rationale and aims for Math 170 can seem a little unclear. For example, in a calculus course, the aim is to become proficient in calculus; in an algebra course, the aim is to become proficient in algebra; is an east Asian history course, the aim is to become knowledgeable about east Asian history. In various English courses, one learns grammar, paragraphing, story-telling, critical reading, and effective written communication. What is the aim of Math 170?
Our aim is, in a way, broader than in many other classes you'll take here. This class' students are frequently interested in realms of human knowledge and practice that are less mathematical: the departments I see on my class list circle around Psyc, Biol, Hist, Engl, and so forth, with a decent smattering of UNDC. The purpose of this class is to help develop proficiency in another form of human activity: formal reasoning (which includes, but is far from limited to, reasoning with numbers), and to help develop knowledge in an immensely rich and layered world that is rarely glimpsed: the world of mathematical form and structure.
So that's it: from the point of view of your teacher, me, the class' purpose is for everyone to become more proficient in formal reasoning skills (precision formulation of problems, pattern recognition, shape, formal structures, and, yes, numerical facility), and to gain some knowledge of what mathematics is today and what is being studied and developed in that world. If by semester's end everyone has a greater ability to approach both abstract and real-world problems in a precise, formal way, and everyone has a bit of appreciation for the amazing mathematics that exists in our society today, then I will consider the course a success.
And the University's aims? Of course I can't speak for the university, but I can give my view on the matter. The University wants to graduate students who are knowledgeable and capable, not just in their fields, but in a wide array of human realms---it wants its graduates to possess not just a skill, but what it considers to be an educated mind. That is why engineers take English and history, and why pre-meds and comm. majors take some kind of math. In an even broader sense, the purpose is two-fold: first, it gives you, the student (fundamentally you are the client), value for your money and your trust. The second purpose is that it is a form of advertising for Penn: if it graduates impressive and successful human beings, then Penn's status rises, and the demand for its services rises.
Anyway, I hope this overview of our course goals is valuable to you. As always, your comments and feedback are appreciated!
Tuesday, August 27, 2013
Introduction to the blog
Hello all, this blog represents an experiment in teaching. It is meant to be a space for greater direct communication between myself, my students, and teaching assistants or others involved in the classes I'm teaching. It should be a source of information and a forum for questions and discussion, and it should supplement, not replace, other forms of interaction like in-class discussion and office hours.
Before getting started . . . the rules! First, this will be an un-moderated forum, so you will be responsible for policing yourselves as far as civility is concerned. I reserve the right to remove posts, but as long as the discussion remains on-topic and respectful, I don't foresee any reason for doing so. Generally, this blog should be used for the following:
Before getting started . . . the rules! First, this will be an un-moderated forum, so you will be responsible for policing yourselves as far as civility is concerned. I reserve the right to remove posts, but as long as the discussion remains on-topic and respectful, I don't foresee any reason for doing so. Generally, this blog should be used for the following:
- Posting lecture-related material, such as any elaboration of material discussed in class,
- Posting useful course-related material, such as pictures, videos, and outside references,
- Two-way discussions of course material,
- Discussion, amongst students, of assignments, and
- Considered, thoughtful criticisms or discussions of any aspect of class.
- Me answering homework questions! Please use class time, office hours, or email to discuss homework questions with me.
- Unconsidered or unthoughtful criticism, or any abusive or unwarranted harsh language.
Subscribe to:
Posts (Atom)