Understanding Pi: From Ancient Geometry to Newton's Revolutionary Leap

For thousands of years, humanity has been captivated by a simple, yet infinitely complex number: π (pi). This mysterious constant, defined as the ratio of a circle’s circumference to its diameter, has fascinated minds from ancient geometers to modern physicists. But how did we go from slowly carving out polygonal shapes to calculating π with lightning precision using calculus?

Join us on this historical journey—from dusty scrolls to Newtonian brilliance.

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🔢 The Basics of Pi: A Constant Around Every Circle

Before diving into the math, let’s start with what π really is:

• Pi (π) ≈ 3.14159..., a never-ending, non-repeating decimal.

• It appears in formulas for:

  • Circumference: $C = \pi d$

  • Area of a Circle: $A = \pi r^2$

🍕 A Tasty Visualization

Imagine cutting a pizza into strips of crust. Line those pieces up, and they form a little more than three full pizzas—a delicious way to "see" π in action.

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🏛️ Ancient Approaches: Geometry and Hard Work

For more than 2,000 years, early mathematicians worked with what they had: shapes.

➖ Approximating Pi with Polygons

A classic trick was to inscribe and circumscribe regular polygons within and around a circle:

• A hexagon (6 sides) gave:

  • Inner perimeter ≈ 6 → $\pi > 3$

• A square around the circle had:

  • Outer perimeter ≈ 8 → $\pi < 4$

Not bad, but not precise.

🧠 Archimedes' Breakthrough (~250 BC)

The great Greek mathematician Archimedes took it further:

• He used 96-sided polygons to trap π between:

  $$3.1408 < \pi < 3.1429$$

This was astonishingly accurate for his time—and all without a calculator.

🧮 The Long Road to Precision

Over the centuries:

• Francois Viète developed continued fractions for π.

• Ludolph van Ceulen spent 25 years calculating π to 35 decimal places using polygonal methods.

His work was so monumental that the digits were engraved on his tombstone!

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🌪️ Newton’s Revolutionary Leap: A New Era Begins

🦠 In Quarantine, Genius Blooms

In 1666, during the Great Plague of London, a young Isaac Newton retreated to the countryside. There, away from distractions, he quietly redefined mathematics—including how we calculate π.

🔬 Newton’s Method: Infinite Series and Calculus

Newton’s genius lay in leaving geometry behind. Instead, he:

1. Expanded the Binomial Theorem

For any power, including non-integer ones, he generalized:

$$(1 + x)^n = 1 + nx + \frac{n(n-1)}{2}x^2 + \cdots$$

Even for negative or fractional $n$, this expansion worked—a massive leap at the time.

2. Applied to Inverse Functions

He studied:

$$\frac{1}{1 + x} = 1 - x + x^2 - x^3 + \cdots$$

This series converges beautifully for $|x| < 1$, giving an elegant tool for evaluating areas.

3. Integrated to Find Pi

By integrating the above series between 0 and $\frac{1}{2}$, Newton found:

$$\int_0^{1/2} \frac{1}{1 + x^2} \, dx = \tan^{-1}\left(\frac{1}{2}\right)$$

Which leads to an approximation of π:

$$\frac{\pi}{4} = \tan^{-1}(1) = 1 - \frac{1}{3} + \frac{1}{5} - \cdots$$

By choosing better values and smarter expansions, Newton improved convergence drastically.If  your thinking why newton used 1/1+x^2 then it because we was known the result of that integral

🚀 A Few Terms, Big Results

Using just 5 terms, Newton reached an approximation of π accurate to 0.00002—a feat that previously required thousands of polygon sides.

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🧠 Why Newton’s Method Changed Everything

Before Newton:

• Approximation = Hard labor with geometry

• Slower convergence

• Limited by tools

After Newton:

• Approximation = Elegant infinite series

• Fast convergence

• Opened the door to modern analysis

This wasn’t just a better method—it was a paradigm shift.

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🎓 The Legacy of a Formula

Newton’s work didn’t just solve a problem—it transformed how we think about problems.

🗝️ Key Takeaways:

• Ancient methods were ingenious but limited by geometry.

• Archimedes and others pushed those methods to the brink.

• Newton’s calculus ushered in a new era of precision through patterns.

• This story teaches a powerful lesson: The obvious method isn’t always the best.

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💡 Ready to Explore More?

If this journey through the history of π sparked your curiosity, consider exploring more on platforms like Brilliant.org or Khan Academy, where interactive math courses make even complex ideas feel intuitive.

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Thank you for reading! Whether you're slicing pizza or solving integrals, remember: π is everywhere—and so is the beauty of mathematics.