How Isaac Newton Revolutionized the Calculation of Pi – From Polygons to Infinite Series by Veritasium info

The Fascinating History of Pi: From Archimedes’ Polygon Method to Newton’s Infinite Series and the Binomial Theorem

Discover how Isaac Newton transformed Pi calculations using infinite series, replacing 2,000 years of polygon methods. A breakthrough in mathematical history.......................

The Discovery That Revolutionized the Way We Calculate Pi

What is the most accurate way to calculate Pi? For over 2,000 years, the process of calculating Pi (π) was painfully slow and complex. The most effective method known was to use polygons—tedious, repetitive, and limited in precision. But that changed dramatically when Sir Isaac Newton entered the scene and transformed the history of Pi.

In this article, we’ll explore how Newton revolutionized the calculation of Pi, why the old methods were so inefficient, and how modern math owes a lot to his genius.

Understanding Pi with Pizza – A Visual Approach

What is Pi? In simple terms, Pi (π ≈ 3.14159) is the ratio of a circle’s circumference to its diameter. But it also defines a circle’s area:

Area = π × r²

To visualize this, imagine cutting the crust off a pizza and stretching it across identical pizzas. You’ll notice the crust spans just over three pizzas, which is how we intuitively understand that π is slightly more than 3.

Now, cut that pizza into ultra-thin slices. Rearranged, these slices form a rough rectangle. The length of this rectangle is half the pizza’s circumference (πr), and the width is the radius (r). Therefore,

Area = πr × r = πr²

This easy pizza method to understand Pi is fun, visual, and educational.

How Did We Calculate Pi Before Newton?

For centuries, mathematicians used polygon approximation to estimate Pi:

• Archimedes' Method (circa 250 BC): He inscribed and circumscribed polygons around a circle and calculated perimeters to trap Pi between two bounds. Starting with a hexagon, he increased the sides to a 96-gon, achieving Pi between 3.1408 and 3.1429.

• Later, Francois Viète and Ludolph van Ceulen pushed this to extremes:

  • Viète calculated Pi using a polygon with 393,216 sides.

  • Van Ceulen went even further, calculating Pi to 35 decimal places using polygons with over 4.6 quintillion sides. These digits were inscribed on his tombstone!

Despite their brilliance, this polygon-based method was painfully slow and required massive computation, often involving square roots of square roots and complex fractions. It was mathematically impressive but not practical.

The Turning Point: Newton’s Infinite Series and the Modern Era of Pi

When did the modern method of calculating Pi begin? In 1666, a 23-year-old Isaac Newton, quarantined during the bubonic plague, made one of the most revolutionary mathematical discoveries. Instead of continuing the polygon approach, he used what we now call Newton’s Binomial Series to calculate Pi with incredible accuracy.

Newton played with expansions of expressions like:

(1 + x)² = 1 + 2x + x²

(1 + x)³ = 1 + 3x + 3x² + x³

These patterns came from Pascal’s Triangle, a concept already known across ancient Chinese, Indian, Persian, and Greek cultures.

But Newton did something radical:

He broke the rules of the binomial theorem by applying it to negative powers.

Traditionally, the binomial expansion worked for positive integers only, but Newton tried:

(1 + x)^-1 = 1 - x + x² - x³ + x⁴ - x⁵ + ...

This infinite alternating series could be used to calculate complex functions like 1 / (1 + x) and eventually led to powerful tools in calculus. Using this method, Newton effectively "speed-ran" Pi, enabling much faster and more accurate approximations.

Does Newton’s Infinite Series Really Work?

This was the big question: Was it mathematically sound?

Yes! Newton demonstrated that multiplying his infinite series for 1 / (1 + x) by (1 + x) simplifies back to 1, confirming the series' validity. This ushered in a new era of infinite series calculations, which could be used to:

• Estimate values of Pi to many decimal places

• Derive other mathematical constants

• Lay the foundation for calculus and mathematical analysis

Today, these series are known as Taylor and Maclaurin Series, and they're core to modern mathematics, physics, and engineering.

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Conclusion: From Polygons to Power Series – Pi's Epic Journey

The story of Pi is a journey from geometry to genius-level algebra. Ancient methods showed how persistent human curiosity is, but it took a young Newton to redefine how we calculate Pi forever. Thanks to him, Pi can now be calculated to trillions of digits, and his infinite series approach is used across mathematics today.

The next time you enjoy a pizza, remember: You're eating a slice of mathematical history.