The Evolution of Pi in Ancient India

Introduction

• The circle and sphere are common geometric shapes found everywhere.
• The ratio of circumference to diameter is known as Pi (π), a crucial element in understanding the properties of circles.
• Common values of Pi: 22/7 or 3.14.
• This lecture focuses on the historical evolution of Pi specifically through contributions from ancient India.

Purpose of Pi

• Pi has been a subject of mathematical obsession for thousands of years.
• Its significance lies in measurement, particularly in geometry and astronomy.

Measurement Concepts

• Measurement of length has evolved from using hand spans to more complex geometrical constructs.
• Area measurement introduced a new dimension:
  • Example: Area of a square = side × side (a × a = a²).
• Introduction of circular measurements led to the question of finding a constant value (Pi) to relate diameter and circumference.

Ancient Indian Contributions to Pi

Milestones of Pi

1. Vedic Period (circa 3700 BCE)

   • Purpose: Geometric principles for fire altars in rituals.
   • Proof: Found in the Kalpa Sutras, specifically in the Sulba Sutras.
   • Precession: Approximation of Pi was about 3, derived from a relationship between circumference and diameter.

2. Aryabhata (6th Century CE)

   • Purpose: Precise calculations for astronomy and geometry.
   • Proof: Found in Aryabhatiya.
   • Precession: Provided an approximation of Pi as 3.1416.

3. Madhava (16th Century CE)

   • Purpose: Developed the infinite series for calculating Pi, which revolutionized mathematical calculations.
   • Proof: Described in "Mahajjana Prakarana."
   • Precession: Introduced infinite series for Pi:
     • Formula: π = 4 × (1 - 1/3 + 1/5 - 1/7 + ...)

4. Srinivasa Ramanujan (20th Century)

   • Purpose: Fast calculations of Pi values for advanced mathematics.
   • Proof: Formulas in his notebooks yielded extremely quick results for Pi.
   • Precession: Provided a method for quickly converging on the value of Pi.

Conclusion

• The evolution of Pi in ancient India demonstrates significant mathematical milestones, from early approximations to advanced series.
• Contributions from various Indian mathematicians have greatly enriched the understanding and calculation of Pi, surpassing many other civilizations in some aspects.
• Understanding Pi's history is essential beyond just its numerical value (22/7).

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• The lecture emphasizes the rich and deep history of Pi in ancient India, showcasing its purpose, proof, and precision throughout different periods.