MATC90: Beginnings of Mathematics (Fall 2010)

Lectures: Mondays, 2:10--5:00, in HW 408.

Textbook: Howard Eves, An Introduction to the History of Mathematics, 6^th edition, Brooks/Cole, Thomson Learning, Inc.

Instructor: Leo Goldmakher

Office:

Telephone: 416-207-2783

Email: leo.goldmakher at utoronto dot ca

Office hours: Mondays 12:00--1:15 pm and Thursdays 4:30--6:00; additional office hours by appointment.

Announcements:

Hint has been added to Problem 5.1.

DATE LECTURE SUMMARY ASSIGNMENT
(due on date listed) PROBLEM OF THE DAY DOCUMENTS

Sept. 13 Introductions and policies.
What is history?
What is mathematics?

Introduction to Egyptian mathematics:
counting and the lack of place holders;

the Rosetta stone;

the Rhind papyrus;

Egyptian fractions, and the Erdos-Straus conjecture.
Course syllabus

Sept. 20 Further discussion of Egyptian fractions:
The story of Horus' eye, and other family fun

Greedy algorithm for Egyptian fraction decomposition

More from the Rhind papyrus:
2/n table

Some areas and volumes (including approximation to area of a circle)

Multiplication algorithm

A bit about the Moscow papyrus:
Linear equations (find the aha)

Surface area of hemisphere, volume of frustrum, etc.
Read Chapters 1 and 2 of Eves. Does Egyptian multiplication always work?
In particular, can one write any positive
integer as the sum of distinct powers of 2?

Sept. 27 Brief history of Mesopotamia;
The exploits of Darius, including Pheidippides and the battle of Marathon
The exploits of Xerxes, including the battle of Thermopylae.
The history of cuneiform and its interpretation(s)
The Babylonian number system, pre- and post-300 BC. Assignment 1 Write ¹⁄₇ in Babylonian numerals.

Oct. 4 Continued discussing the Babylonian number system.
Discussed expanding fractions in different bases:
¹⁄₃ in decimal.

¹⁄₇ in decimal.

¹⁄₇ in sexagesimal.
(Mentioned some nice properties of the decimal expansion of ¹⁄₇.)
An example of a Babylonian algebra problem (solved using geometry).
The difference between a theory and a theorem.
The Pythagorean theorem (two statments, not 1!).
"Proved" both statements by drawing appropriate pictures.
Proved the irrationality of √2.
Pythagorean triples (examples, non-examples, primitive vs imprimitive).
Plimpton 322, and three interpretations. Read Chapter 3 of Eves Prove the irrationality of √2

Oct. 11 NO CLASS -- THANKSGIVING Not overeating.

Oct. 18 Discussed some Greek history and legacy.
Discussion of mathematics and the concept of "proof".
Examples showed the important distinction between meaningful pattern and coincidence.
Compass and straightedge contructions.
Plato, Socrates, and a problem from the Meno.
Polyhedra and the Platonic solids.

Sketch of a proof that the only regular polyhedra are the five Platonic solids.
Assignment 2
Figure 2.1 Prove that the only regular polyhedra are the five Platonic solids.

Oct. 25 Corrected definition of regular polyhedron
Intellectual Athens.
Thales and the notion of proof.
Pythagoras and the Pythagoreans.
Three famous (and unsolvable!) Greek problems.
Introduction to Euclid's Elements.
Discussion of postulates vs. common notions.
Determine the area of a 30-60-90 triangle with unit hypotenuse. (No trigonometry allowed!)

Nov. 1 Euclid's Elements.

Congruency and similarity of triangles, other shapes.

Area scales like square of length.

Visualizations of the Pythagorean theorem.

Expressing algebra through geometry.

Unique factorization of integers.

Infinitude of primes.

Rules for divisibility Assignment 3
Figure 3.4 A number is divisible by 3 if and only if the sum of its digits is. Why?

Nov. 8 Twin primes, Sophie Germain primes; are there infinitely many?
Diophantine equations:
Problem of the day

Fermat's Last Theorem
Finding common factors of two numbers: the Euclidean algorithm
Wrapping strings around the equator, and our poor geometric intuition
The difficulty of defining π.
Approximating circles by polygons.
The formula for the area of a circle. Read Chapters 4 and 5 of Eves. A farmer buys exactly 100 animals, using exactly $100. The animals
he buys are chickens, cows, or pigs, and he buys at least one of each.
If chickens cost $0.50 each, cows cost $10 each, and pigs cost $3
each, how many of each did he buy?

Nov. 15 Some more discussion of π:

Curious formulas (by Wallis, Madhava-Gregory-Leibniz, and Euler)

The Landau-Bieberbach controversy

More about Archimedes:

Life, death and the siege of Syracuse

Finding the area of a piece of a parabola (using infinite geometric series)

Heron's formula for the area of a triangle.

The cylinder-sphere theorem (and the story of Cicero)

Eureka! and the law of hydrostatics

Law of the lever

The Archimedes Palimpsest and `The Method'

Early Chinese history, and Pinyin. Assignment 4 Why does 1 + ¹⁄₄ + ¹⁄_4² + ¹⁄_4³ + ¹⁄_4⁴ + ... = ⁴⁄₃?
(`The formula' is not an acceptable answer.)

Nov. 22 Chinese mathematics:
Chinese numerals: written system vs. counting rods

The Imperial Academy and The Ten Classics

Brief discussion of the Zhoubi suanjing (particularly the Gougu rule)

Discussion of the Sunzi suanjing, particularly the Chinese Remainder Theorem.

The Jiuzhang suanshu (Nine Chapters on Mathematical Art): age, contents, commentary, legacy.

Qin Jiushao: a curious character

Indian Mathematics:
The evolution of Hindu-Arabic numerals. (And as a side-note, Durer's contribution to math.)

The work of Brahmagupta (particularly his work on 0, equations, trigonometry, and geometry)

Bhaskara and ⁰⁄₀

Astronomy: phases of the moon, trigonometry, and the relative distances between the Earth, Sun, and Moon.

Al-Khwarizmi Read Chapter 7 of Eves A person has a number of eggs.
When the eggs are counted in threes, there are two eggs left over.

When the eggs are counted in fives, there are three eggs left over.

When the eggs are counted in sevens, there are two eggs left over.
How many eggs are there altogether? Find two possible answers.

Nov. 29 Student Presentations. Final project

Dec. 6 Student Presentations. Assignment 5

DATE	LECTURE SUMMARY	ASSIGNMENT (due on date listed)	PROBLEM OF THE DAY	DOCUMENTS

Sept. 13	Introductions and policies. What is history? What is mathematics? Introduction to Egyptian mathematics: counting and the lack of place holders; the Rosetta stone; the Rhind papyrus; Egyptian fractions, and the Erdos-Straus conjecture.			Course syllabus
Sept. 20	Further discussion of Egyptian fractions: The story of Horus' eye, and other family fun Greedy algorithm for Egyptian fraction decomposition More from the Rhind papyrus: 2/n table Some areas and volumes (including approximation to area of a circle) Multiplication algorithm A bit about the Moscow papyrus: Linear equations (find the aha) Surface area of hemisphere, volume of frustrum, etc.	Read Chapters 1 and 2 of Eves.	Does Egyptian multiplication always work? In particular, can one write any positive integer as the sum of distinct powers of 2?
Sept. 27	Brief history of Mesopotamia; The exploits of Darius, including Pheidippides and the battle of Marathon The exploits of Xerxes, including the battle of Thermopylae. The history of cuneiform and its interpretation(s) The Babylonian number system, pre- and post-300 BC.	Assignment 1	Write ¹⁄₇ in Babylonian numerals.
Oct. 4	Continued discussing the Babylonian number system. Discussed expanding fractions in different bases: ¹⁄₃ in decimal. ¹⁄₇ in decimal. ¹⁄₇ in sexagesimal. (Mentioned some nice properties of the decimal expansion of ¹⁄₇.) An example of a Babylonian algebra problem (solved using geometry). The difference between a theory and a theorem. The Pythagorean theorem (two statments, not 1!). "Proved" both statements by drawing appropriate pictures. Proved the irrationality of √2. Pythagorean triples (examples, non-examples, primitive vs imprimitive). Plimpton 322, and three interpretations.	Read Chapter 3 of Eves	Prove the irrationality of √2
Oct. 11	NO CLASS -- THANKSGIVING		Not overeating.
Oct. 18	Discussed some Greek history and legacy. Discussion of mathematics and the concept of "proof". Examples showed the important distinction between meaningful pattern and coincidence. Compass and straightedge contructions. Plato, Socrates, and a problem from the Meno. Polyhedra and the Platonic solids. Sketch of a proof that the only regular polyhedra are the five Platonic solids.	Assignment 2 Figure 2.1	Prove that the only regular polyhedra are the five Platonic solids.
Oct. 25	Corrected definition of regular polyhedron Intellectual Athens. Thales and the notion of proof. Pythagoras and the Pythagoreans. Three famous (and unsolvable!) Greek problems. Introduction to Euclid's Elements. Discussion of postulates vs. common notions.		Determine the area of a 30-60-90 triangle with unit hypotenuse. (No trigonometry allowed!)
Nov. 1	Euclid's Elements. Congruency and similarity of triangles, other shapes. Area scales like square of length. Visualizations of the Pythagorean theorem. Expressing algebra through geometry. Unique factorization of integers. Infinitude of primes. Rules for divisibility	Assignment 3 Figure 3.4	A number is divisible by 3 if and only if the sum of its digits is. Why?
Nov. 8	Twin primes, Sophie Germain primes; are there infinitely many? Diophantine equations: Problem of the day Fermat's Last Theorem Finding common factors of two numbers: the Euclidean algorithm Wrapping strings around the equator, and our poor geometric intuition The difficulty of defining π. Approximating circles by polygons. The formula for the area of a circle.	Read Chapters 4 and 5 of Eves.	A farmer buys exactly 100 animals, using exactly $100. The animals he buys are chickens, cows, or pigs, and he buys at least one of each. If chickens cost $0.50 each, cows cost $10 each, and pigs cost $3 each, how many of each did he buy?
Nov. 15	Some more discussion of π: Curious formulas (by Wallis, Madhava-Gregory-Leibniz, and Euler) The Landau-Bieberbach controversy More about Archimedes: Life, death and the siege of Syracuse Finding the area of a piece of a parabola (using infinite geometric series) Heron's formula for the area of a triangle. The cylinder-sphere theorem (and the story of Cicero) Eureka! and the law of hydrostatics Law of the lever The Archimedes Palimpsest and `The Method' Early Chinese history, and Pinyin.	Assignment 4	Why does 1 + ¹⁄₄ + ¹⁄_4² + ¹⁄_4³ + ¹⁄_4⁴ + ... = ⁴⁄₃? (`The formula' is not an acceptable answer.)
Nov. 22	Chinese mathematics: Chinese numerals: written system vs. counting rods The Imperial Academy and The Ten Classics Brief discussion of the Zhoubi suanjing (particularly the Gougu rule) Discussion of the Sunzi suanjing, particularly the Chinese Remainder Theorem. The Jiuzhang suanshu (Nine Chapters on Mathematical Art): age, contents, commentary, legacy. Qin Jiushao: a curious character Indian Mathematics: The evolution of Hindu-Arabic numerals. (And as a side-note, Durer's contribution to math.) The work of Brahmagupta (particularly his work on 0, equations, trigonometry, and geometry) Bhaskara and ⁰⁄₀ Astronomy: phases of the moon, trigonometry, and the relative distances between the Earth, Sun, and Moon. Al-Khwarizmi	Read Chapter 7 of Eves	A person has a number of eggs. When the eggs are counted in threes, there are two eggs left over. When the eggs are counted in fives, there are three eggs left over. When the eggs are counted in sevens, there are two eggs left over. How many eggs are there altogether? Find two possible answers.
Nov. 29	Student Presentations.	Final project
Dec. 6	Student Presentations.	Assignment 5