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, 6th edition, Brooks/Cole, Thomson Learning, Inc.

Instructor: Leo Goldmakher


(due on date listed)
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 17 in Babylonian numerals.  
    Oct. 4 Continued discussing the Babylonian number system.
    Discussed expanding fractions in different bases:
  • 13 in decimal.
  • 17 in decimal.
  • 17 in sexagesimal.
  • (Mentioned some nice properties of the decimal expansion of 17.)
    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 + 14 + 142 + 143 + 144 + ... = 43?
    (`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 00
  • 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