E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Magical networks
Objective
In this rst lecture, we look at a mysterious mathematical structure. It will almost certainty
be unknown to you. The goal is to see how mathematics can produc
E-320: Teaching Math with a Historical Perspective
O. Knill, 2010-2014
Lecture 1: Mathematical roots
In the same way as one has distinguished the canons of rhetorics: memory, invention, delivery, style, and arrangement, or combined the trivium: grammar, l
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Problem 6
Lecture 9: Quiz
Which Platonic solid is displayed in the picture?
c) Icosahedron
a) Tetrahedron
d) Octahedron
b) Cube
Name:
Problem 1
Which of the following letters are topolo
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Problem 6
Lecture 5: Quiz
The group of all permutations of 4 elements has how many elements:
a) 4
b) 8
c) 16
d) 24
Name:
Problem 7
Problem 1
Who did use letters as variables of algebrai
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 7: Quiz
Name:
The Continuum Hypothesis is:
a) There is a cardinality between the cardinalities of the
c) There exists an innite set.
reals and integers.
d) There exists a contin
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 2: Irrational numbers
Theorem:
3 is not rational.
Proof: 3 = p/q implies 3 = p2 /q 2 or 3q 2 = p2 . If we make a prime factorization, then on the
left hand side contains an odd
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 13: Experimental Mathematics
We look at problems where computers could play a role for the solution. Usually, in the case a
counter example exists.
2. Goldbachs conjecture
The s
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Problem 6
Lecture 6: Quiz
a) What is the theory deals with discontinuities, especially with disappearances of stable minima?
Answer:
b) We looked at the end at a domino problem: How wid
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 12: Dynamical systems and Chaos
5) What do you see if you push the buttons sin, then type 1/x and repeat this process again and
again?
Chaos
Simple transformations can produce c
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010-2014
Proof.
n1
SDf (n) =
[f (k + 1) f (k)] = f (n) f (0) ,
k=0
Lecture 6: Calculus
n1
DSf (n) = [
n1
f (k + 1)
k=0
Calculus formalizes the process of taking dierences and taking sums.
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 3: Worksheets
C
1
O
Thales theorem
It is a result of Thales of Miletus (625 BC -546 BC) stating that if a triangle inscribed in a xed
circle is deformed by moving one of its poi
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Problem 5
Lecture 10: Quiz
Which mathematician has rst described the middle third Cantor set?
a) Smith
b) Cantor
c) Weierstrass
d) Mandelbrot
Name:
Problem 6
Problem 1
Which fractal is
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 8: Probability
The Monty Hall Problem
We want to understand the famous Monty Hall problem
1. Combinatorics
Here are the
most important
combinatorics
problems:
How many ways are
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 8: Quiz
Name:
Problem 1
Who did the st investigations in probability?
a) Betrand.
b) Cardano.
c) Euler.
d) Kolmogorov
Problem 2
We have seen the movie clips in 21: what was the
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 7: Set Theory and Logic
We will mostly focus on the work of two mathematicians: Georg Cantor and Kurt Gdel.
o
Their mathematics changed our way we think about mathematics. In bo
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 12: Quiz
Which of the following dynamical systems have a discrete time?
a) The game of life
c) The Kepler problem
b) Three body system
d) Ulam-Collatz system.
Problem 6
Name:
Wh
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010-2014
Lecture 9: Topology
Topology studies properties of geometric objects which do not change under continuous reversible
deformations. For a topologist, a coee cup with 1 handle is the
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010-2014
Lecture 13: Computing
Computing deals with algorithms and the praxis of programming. While the subject intersects
with computer science, information technology, the theory is by na
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 9: Topology of Alphabet
2. The digits 0-9
1) The numbers 0, 4, 6, 9 are topologically equivalent.
0 4 6 9
1 2 3 5 7
2) The numbers 1, 2, 3, 5, 7 are topologically equivalent.
3)
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010-2014
Lecture 2: Arithmetic
The oldest mathematical discipline is Arithmetic, the theory of manipulating numbers. The rst
steps were done by Babylonian, Egyptian, Chinese, Indian and Gre
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 2: Quiz
Name:
Problem 1
This this important mathematical writing is
Moskow papyrus
Ishango bone
Rynd papyrus
Plympton 322
Bakshali Manuscript
YBC 7289
Problem 2
What was the mea
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Lecture 7: Worksheets
4) How can we write the set dierence A \ B using addition and multiplication. (Note that this
is not the dierence A B.
1. Objective
There are two algebraic operati
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Example
We aim to decrypt the following text:
Lecture 11: Cryptology
xf uif qfpqmf pg uif vojufe tubuft,
Cesar Cypher
jo psefs up gpsn b npsf qfsgfdu vojpo,
In this worksheet we crack t
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010-2014
Lecture 12: Dynamical systems
Dynamical systems theory is the science of time evolution. If time is continuous the evolution is dened by a dierential equation x = f (x). If time is
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Match the birth places:
Euclid of
B Miletus
Hippocrates of
C Chios
Pythagoras of
Name:
A Samos
Thales of
Lecture 3: Quiz
D Alexandria
Problem 1
Problem 6
Who is considered the rst known
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2010
Lecture 11: Cryptography
Cryptography is the theory of codes. Two important aspects of the eld are the encryption
rsp. decryption of information and error correction. Both are crucial i
E-320: Teaching Math with a Historical Perspective
Oliver Knill, 2014
Problem 6
Lecture 4: Quiz
Which theorem assures that 211 2 is divisible by 11?
Name:
a) Wilsons theorem
b) Fermats little theorem
c) Chinese remainder
d) Unique factorization
Problem 1