Mathematics for Poets, Thinkers and Doers (2007)
Teachers: F.Beukers, J. van Maanen
This course is a mathematics course for non-science majors. There are many
aspects of mathematics that appeal to a much wider audience than just scientists.
Think of puzzles (such as the recent Sudoku rage), mathematics in art and
all philosophical matters about which mathematics has something mind-expanding
to say, such as the infinite, the geometry of our world and the secrets of
numbers. Those are the topics that we shall deal with in this course.
Our approach towards mathematics is that it can be understood, if one
is willing to invest some thought into it. That is why active participation
is important during this course. We shall not go into the technical
parts of mathematics and thus avoid the use of formulas. This course is not
the beginning of a regular training in mathematics. After having followed it
you will still have a hard time reading an average math book. In order
to be able to do that, one has to follow regular math courses. As for
this course, our philosophy
is that there is much to enjoy and learn from mathematics without having
the technical skills. In the same way as one can enjoy a good game of soccer.
without being a professional player.
As our guide we use the book by
E.B. Burger & M. Starbird,
The Heart of Mathematics. An invitation to effective thinking.
Key College Publishing, Emeryville, California, 2005 (2nd edition). ISBN: 1-931914-41-9.
Classes take place on Mondays 16:00-18:00 and Thursdays 9:00-11:00
Homework assignments
must be made and handed in by each student individually.
Assignments will always be graded; these grades will partly
determine the overall grade for the course.
Material covered:
- Monday 22/1
Course introduction. Discussion of a few problems in Section 1.1
Assignment I (please click on it).
Homework for next time: Read Sections 2.1 and 2.3.
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Thursday 25/1
Number sense and Number theory, examples. Discussion of BIG numbers,
summing the integers, summing the odd integers and summing cubes
of integers. An unsolved problem (Collatz' problem), see Problem
2.1.II.15 in the book, see also
Collatz applet
Home work: Read Section 2.3, try to do problems 2.3.I.1)2)3)5).
Also try problems 2.3.II.16 (not entirely easy) and 2.3.II.21.
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Monday 29/1
Prime numbers, Section 2.3. Documentation:
Home work for Feb 1.
- We use the Java applet
Factorization usimg
the elliptic curve method. Choose 3-digit numbera at random
and check whether they are prime or not. Write down how many times
it took before you found the first prime. Do the same exercise,
but now with 6-digit numbers. Finally do the experiment with 9-digit
numbers. If you want you can repeat the exercise so that you will
have two counts. In class we will collect as many of those counts
and check if there is some reasonable avarage.
- Use the Mersenne factorisation
table to see if you recognize regularities in the occurrence of
prime factors in the factorsiation of Mersenne numbers. For example,
for which exponents does a given prime factor p occur, or look
for a connection between p-1 and the exponent, etcetera. You
do not need to prove things, just try to discover them.
Assignment II, due February 5, 4 PM.
Please hand in your work in the folder Assignment2 of the section
Work on the AMICO-workspace or hand in the printed version (do not
send me your work by email, this costs me too much time and disk-space.
Ordinary emailmessages are of course OK).
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Thursday 1/2
Section 2.4: modular arithmetic.
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Monday 5/2
Homework for today:
-
Determine a6 modulo 7 for a = 2,3,4,5,6.
- Read section 2.4.
In class we shall discuss:
Modular arithmetic (continued) and Fermat's little theorem.
We make a start with cryptography using the
Mathematica file on RSA.
Assignment 3, deadline Monday, Feb 12.
-
Thursday 8/2
Homework for today:
- determine an integer e, larger than 1, such that
ae = a modulo n for n = 51.
- Same problem , but now with n = 21.
- Read Section 2.5.
In class we shall discuss the idea of public key cryptography and
the RSA-system.
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Monday 12/2
Today will be the day of change: we shall start with a new chapter and a new teacher.
Their names: Chapter 3: Infinity, and Jan van Maanen (maanen@fi.uu.nl) respectively.
In order to prepare for the Monday lecture, do the following
Homework
- Return to Chapter 1, and study Story 5 on pages 8-9, 16 and 21-22.
Remember that in class we worked
on this story. At the end of that session some of you remained with
questions, while others were sure that
Player Two could always play a winning strategy.
Write down your memory of that lesson (what was your conclusion then?).
And what is your position now, after reading?
- Write down an argument in favour of Player Two by which you could convince
even the greatest doubter.
Be prepared to read out your argument in class.
Assignment 4, deadline Monday, Feb 19.
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Thursday 15/2
Today there will be amazing things to discover. Be prepared!
You can be prepared by doing the following things.
- Think about what the fraction 0/0 could possibly mean?
- Read about the rational and irrational numbers in Chapter 2, sections 6 and 7.
- Write down a 'new' number --that is: not an example from the book-- which has
an unending decimal expansion and which is rational (i.e. it can be written as a
fraction of two natural numbers). You can be asked to present the example in class.
- Do the same for a number which is irrational. (You can be asked the same.)
- What numbers are in the majority: the rationals or the irrationals?
- What is the value of the concept of "one-to-one-correspondence" (Chapter 3)?
- Review the text in the book entitled "Rationals equal naturals"
(2nd ed. pages 154-155). This once more explains what we have found
out already in the Monday lecture.
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Monday 19/2
Homework for today:
- Give two arguments for the claim that 0.999999... = 1.
(One is discussed in the book, in the other one you could start to
think about 1/3)
- Read Chapter 3, section 3, and try to understand the difference
between how many rational numbers there are and how
many irrational numbers.
- In the Thursday (15/2) lecture the number 0.123456789101112131415...
was said to be irrational. Find out why this number cannot end
in a pattern of repeating decimals; write down your argument
Assignment 5 deadline: Monday 26/2.
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Thursday 22/2
Homework for today: Read Section 3.5 (1st and 2nd edition) and
answer the following questions: II.6,7,9,10 (I.1,2,4,5 in first
edition). Today we shall look at more strange phenomena
when dealing with the infinite. One of them is story 6
from Chapter 1 (both 1st and 2nd edition). Read the story and see
if you can say something about it.
TERM PROJECT
Off and on you should already be considering ideas for your
term project. We have compiled
a list of instructions and suggestions.
If you have an idea for a subject please make sure it is approved
by the teachers. Also, if need any (more) references or help with your
subject please consult us. Submit your suggestion before the midterm break
so that you can start on the subject after the break.
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Monday 26/2
These two weeks (26 February until 8 March) we shall work on Geometry
(Chapter 4 in the book).
Since we shall make some geometrical constructions, it will be handy if
you take a ruler and
a compass with you (for drawing straight lines and circles). There will
be some spare ones in class,
but better have them for yourself.
In the lecture we shall review two things: Assignment 4 (handed in
Monday 19th) and Story 2 from
Chapter 1. Homework is connected to this:
- Write down your solution to the problem in Story 2 (It was about a
square island,
surrounded by a square moat. This moat was 20 feet across, and there
were two beams available of
19 feet 8 inches long. Is it possible to reach the island?)
- Read Chapter 4, section 1. The text (2nd ed. page 212) displays
some geometrical constructions, executed with ruler and
compass. Revive your memory: have you ever done such constructions?
Could you construct a right angle
by drawing circles and straight lines (not being allowed to measure
angles)?
Assignment 6 deadline: Monday 5/3.
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Thursday 1/3
Homework for today:
- Draw a line segment AB, say 6 cm long.
Draw at least 5 points C such that angle ACB is a right angle.
What sort of shape do these points make? Can you explain?
-
For doing geometry at a certain moment it is practical to do some algebra.
That is why there are some algebraic
questions here, so-called quadratic equations. Solve these equations:
a) x2 = 4
b) (x-1)2 = 4
c) x2- 2x + 1 = 4
d) x2- 2x - 3 = 0
e) x2- 4x + 4 = 9
f) x2- 4x - 5 = 0
g) x2+ 10x - 24 = 0
h) x2 = 1 - x
Here is yet another proof of the Pythagorean Theorem (
to see it click either the Powerpoint
or the
Java Applet).
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Monday 5/3
Some further exploration into constructions: the pentagon and the 30-gon.
Mathematical properties of the golden ratio.
This time there is no assignment, please review the assignments you have made
so far. You will be questioned about them in the oral tests that take
place next week.
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Thursday 8/3
Today we start a new subject, to be continued after Spring break.
We begin with another proof of Pythagoras's theorem using bathroom floor
tilings. Then we continue with the actual subject: regular patterns in the
plane. As a preparation, please read sections 1 and 2 on
David Joyce's webpage
on wallpaper patterns.
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Monday 12/3
Individual interviews on the last 6 assignments. Here is the
schedule. The
interviews today and next Thursday will take place during class time.
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Thursday 15/3
Interviews on the last 6 assignments.
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Monday 19/3
Springbreak
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Wednesday 21/3
For those of you who stay in Utrecht and want to experience something
special in academic life:
Jan van Maanen will deliver his inaugural lecture this Wednesday.
It is a public lecture, open for everyone, followed by a reception.
Mathematical Poets are cordially invited to attend.
At 4 PM, Aula of the university, Domplein
Inaugural lecture by Jan van Maanen:
"De koeiennon. Hoe rekenen en wiskunde te leren, en van wie"
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Thursday 22/3
Springbreak
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Monday 26/3
Class cancelled due to illness of teacher.
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Thursday 29/3
Today we continue with tilings of the plane (wall paper patterns).
To be prepared read as much as you can from
David Joyce's webpage.
You already covered sections 1,2 but may want to reread it.
Bring the patterns that were handed out last time and convince
yourself of the correctness of the symmetry lattices you found in
the tilings. Also bring some colored pencils and rulers, since we
shall be discovering many more symmetries.
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Monday 2/4
Home work for today: Create 3 wall paper patterns using the
Kali program. To get the program, please follow these
Kali-instructions. The designs received are placed in the
Wall paper gallery.
In class we shall start with Section 4.5: Platonic Solids.
For (animated) pictures go to my 4d-cell page
or to the
platonic and archimedean solid page. A complete list
of all uniform solids (convex and non-convex) can be found at
this site.
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Thursday 5/4
Today we look at the 4th dimension, read Section 4.7 of the book.
For a preview see an animation of the
Tesseract.
Do not forget to submit your wallpaper design, they are placed
in the Wall paper gallery. Failure to
do so results in a disfunctioning link.
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Monday 9/4
Second Easter day
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Thursday 12/4
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Monday 16/4
Deadline for handing in the proposals for term project with
a one-page description of what you intend to do.
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Thursday 19/4
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Monday 23/4
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Thursday 26/4
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Monday 30/4
No class (Koninginnedag)
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Thursday 3/5
Presentations. Here is the schedule for today and the next
three times:
Name | Day | Coach | Subject |
A.Beyer | May 7 | JvM | Kovaleskaja |
.Bonneau | May 3 | FB | Pi |
G vd Brand | May 7 | JvM | Music and math |
G.Brenner | May 10 | JvM | Babylonian math |
C. de Bruin | May 10 | FB | Fractal(?) art |
C.Grassnick | May 10 | JvM | Calculating savants |
N.Guastavino | May 10 | FB | Decision math |
F. v Hasselt | May 7 | JvM | Escher tilings |
N. van 't Klooster | May 7 | FB | Mandalas |
N. Makel | May 7 | FB | Cryptography |
M.Martens | May 3 | JvM | Zeno's paradoxes |
A. Poorta | May 3 | FB | Knots |
E.Pronker | May 10 | JvM | 4D geometry |
M. vd Putten | May 7 | JvM | Lunar solar calenders |
C.Reimann | May 7 | FB | Riemann hypothesis |
R. Rochadiat | May 3 | JvM | Tesselation/Islamic art |
M.Tideman | May 10 | ??? | Code breaking |
M. Tselms | May 7 | JvM | Pythagoras |
E.Visser | May 10 | FB | Turing and Enigma |
D. de Vos | May 3 | FB | Magic Squares |
O. Westra | May 3 | JvM | Descartes' innovation |
K. Zeegers | May 3 | FB | Cryptography |
Note that the time of the presentation is 10 minutes at max plus 2 minutes for
questions. Good luck!
The submission deadline for the term paper is May 11. If possible, hand it in earlier!
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Monday 7/5
Final exam week
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Thursday 10/5
Final exam week