Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+... | Summary and Q&A

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December 24, 2019
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Fermat's Christmas theorem: Visualising the hidden circle in pi/4 = 1-1/3+1/5-1/7+...

TL;DR

The Leibniz formula for pi can be derived from the area formula of a circle and the number of lattice points within it.

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Key Insights

  • 🦕 The Leibniz formula connects the odd numbers used in the formula to a hidden circle.
  • ⭕ The approximation of pi using the number of lattice points in a circle improves as the radius of the circle increases.
  • #️⃣ The "good" and "bad" odd numbers play a crucial role in calculating the number of lattice points and have a connection to Fermat's Christmas theorem.
  • 🦡 The 4(good - bad) formula can be used to calculate the number of lattice points by tallying the "good" and "bad" odd numbers.
  • 🍹 The identity that the product of two sums of two integer squares is also a sum of two integer squares is crucial in extending the results to all integers.
  • 🤩 Quadratic reciprocity and Fermat's theorem are key components in fully understanding the proofs of these theorems.

Transcript

(Thanks to Karl for the 2019 Easter egg idea :) Welcome to the 2019 Mathologer Christmas video. In this video we'll investigate that famous and amazing formula over there PI over 4 is equal to 1 minus 1/3 plus 1/5 minus 1/7 and so on. It's usually called a Leibniz formula after Gottfried Wilhelm Leibniz one of the genius inventors of calculus. Sadl... Read More

Questions & Answers

Q: Who first discovered the Leibniz formula for pi?

The formula was first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century.

Q: How can the number of lattice points within a circle be used to approximate its area?

Each lattice point can be considered the center of a unit square, and the total number of lattice points approximates the area of the circle.

Q: What is Fermat's Christmas theorem?

Fermat's Christmas theorem states that "good" primes can be written as sums of two integer squares, while "bad" primes cannot.

Q: How can the 4(good - bad) theorem be used to calculate the number of lattice points in a circle?

By tallying the "good" and "bad" odd numbers and using the 4(good - bad) formula, the number of lattice points can be calculated.

Summary & Key Takeaways

  • The Leibniz formula, which approximates pi, was first discovered by the Indian mathematician Madhava of Sangamagrama in the 14th century, over 200 years before Leibniz.

  • The number of lattice points within a circle can be used to approximate its area.

  • By applying a theorem known as Fermat's Christmas theorem, the number of lattice points in a circle can be calculated using the idea of "good" and "bad" odd numbers.

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