Lecture Notes: Omar Khayyam and His General Cubic Solution

Introduction

• Speaker: Amirul Eduard
• Topic: Omar Khayyam's general cubic solution
• Full Name: Zia-Tal-Din Abdul-Mingh
• Era: 10th century AD
• Origin: Nishankur, Shahirat
• Fields of Contribution:
  • Poetry
  • Mathematics
  • Philosophy
  • Astronomy

General Cubic Solution

• Method: Found through geometry
• Key Question: What challenge did he solve?
• Discovery:
  • Only two objects needed to solve cubic equations
  • Two specific shapes on a graph yield solutions

Method Explanation

• Components:
  • Parabola
  • Circle of radius r
• Key Points:
  • Intersection points at origin and (x1, y1)
  • Intersection point supposed to be the solution

Verification Process

• Step 1: Rewrite parabola and circle equations
• Step 2: Square both sides
• Step 3: Simplification leads to an equation similar to the original cubic equation
• Conclusion: Point (x1, y1) is a valid solution

Real-world Application Example

• Problem: Business profit function = Q^3 - 400Q
• Goal: Find production level for a profit of 100,000 ringgit
• Cubic Equation and Graph:
  • Solutions found at points: 43.547 and 94.8
• Verification:
  • Substitute 43.547 in original equation results in profit close to 100,000 units
  • Conclusion: Method works for given problem

Historical Impact

• Integration: First integration of geometry and algebra
• Influence: Inspired new scholars in mathematics

Conclusion

• Omar Khayyam's contributions in mathematics
• His method effectively provides solutions to cubic equations using geometric shapes

Closing

• Speaker's Note: Thanks for watching!