History of Mathematics Lecture Notes

Introduction

• Instructor: Norman Walburger
• Course: History of Mathematics
• University: University of New South Wales
• Textbook: Mathematics and Its History by John Stilwell
  • Available in bookshop and library (open reserve)
• Course structure: 12 lectures covering math history from early times to the 19th century

Course Outline

1. Pythagoras' Theorem
2. Greek Geometry
3. Greek Number Theory
4. Infinity in Greek Mathematics
5. Number Theory in Asia
6. Polynomial Equations
7. Analytic Geometry
8. Projective Geometry
9. Calculus
10. Infinite Series
11. Mechanics
12. Non-Euclidean Geometry

Focus of the Course

• Emphasis on mathematical ideas rather than historical facts
• Understanding the flow of mathematical concepts over time
• Learning fundamental mathematical ideas and theorems
  • Minimal mathematical background required
  • Focus on understanding rather than formal proofs

Pythagoras' Theorem

• One of the most significant theorems in mathematics
• Historical context: Pythagoras lived around 600 BC
• Importance of Greek mathematics

Historical Overview

• Ancient civilizations (2000 BC):
  • Agrarian societies in Mesopotamia and Egypt
  • Other notable cultures: Indus Valley, early Chinese empires
• Greek period (c. 600 BC - 300 AD):
  • Key figures: Pythagoras, Euclid, Archimedes
• Transition to Arabic mathematics, then to European mathematics (1400-1500)
  • Important figures: Newton, Euler, Gauss

Understanding Pythagoras' Theorem

• Known to the Babylonians and Egyptians
  • 3-4-5 triangle example
• Modern formulation:
  • For a right triangle with sides d1, d2, d3:
    • d1² + d2² = d3²
• Ancient Greeks focused on areas rather than lengths
  • Euclid's formulation based on areas

Example of Area Calculation

• Graphical representation of the theorem using areas
• Differences in modern versus Greek approaches to mathematics
• Importance of understanding the geometric representation

Proof of Pythagoras' Theorem

• A geometric proof involving rearranging triangles
• Highlights the innovative Greek thought process

Pythagorean Theorem in Modern Context

• Used in Cartesian coordinates to define distance:
  • Distance (d) between points (x1,y1) and (x2,y2):
    • d = √((x2 - x1)² + (y2 - y1)²)

Irrational Numbers

• Discussion of the irrationality of √2
• Proof by contradiction method:
  • Assume √2 = a/b where a and b have no common factors
  • Leads to conclusion that both a and b must be even, contradicting the initial assumption

Impact of Irrationality on Greek Mathematics

• Pythagorean belief in numbers:
  • Preference for natural numbers and simple fractions
• Discovery of irrational numbers created turmoil among Pythagoreans
  • Secretive about the existence of irrational numbers due to their philosophical implications

Conclusion

• Transition to next topic: Pythagorean triples and description of points on a circle
• Importance of understanding how historical developments relate to modern mathematical concepts