The Discovery That Transformed Pi

Veritasium16 minutes read

Isaac Newton revolutionized Pi calculation methods, using the binomial theorem and integration under a curve to increase precision and efficiency significantly. By exploring unconventional approaches like integrating to a half instead of one, Newton achieved a Pi approximation accurate to two parts in 100,000 with just five terms, showcasing the importance of innovation in mathematics.

Insights

  • Newton's innovative use of the binomial theorem and calculus allowed for precise calculations of Pi, showcasing the power of integrating mathematical concepts for increased accuracy.
  • Newton's shift in perspective, integrating from zero to a half instead of zero to one, significantly improved the efficiency of Pi calculations, emphasizing the value of exploring unconventional methods in mathematics to achieve breakthroughs.

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Recent questions

  • How was Pi calculated before Isaac Newton?

    Through slow methods like polygon bisection.

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Summary

00:00

"Revolutionizing Pi Calculation Through History"

  • For 2000 years, calculating Pi was slow until Isaac Newton revolutionized the method.
  • Pi can be visualized using pizzas, with the circumference of a circle being 3.14 times its diameter.
  • Pi's relation to a circle's area is expressed as Pi R squared, demonstrated by cutting a pizza into thin slices.
  • Archimedes improved Pi calculation by bisecting polygons, reaching a range of 3.1408 to 3.1429.
  • Over centuries, mathematicians pushed Pi's precision, with Ludolph van Ceulen achieving 35 decimal places.
  • Newton's breakthrough with the binomial theorem transformed Pi calculations, using Pascal's triangle.
  • Newton extended the binomial theorem to negative values, creating an infinite series for 1 over 1 plus X.
  • Newton explored fractional powers, like square roots, using the binomial theorem for precise calculations.
  • Newton's interest in N equals a half led to efficient methods for finding square roots, like the square root of three.
  • Newton's integration under a curve, a concept from his theory of Fluxions (calculus), allowed for calculating Pi based on the area under the curve.

13:57

Newton's Innovative Approach Revolutionizes Pi Calculation

  • The area of a quarter circle is Pi over four, derived from the area of a unit circle being Pi, with Newton using this knowledge to calculate Pi to high precision by integrating X to some power.
  • Newton introduces a new idea by integrating from zero to a half instead of zero to one, aiming to decrease the size of terms in an infinite series rapidly for more efficient calculations.
  • By integrating to a half, Newton computes the area under the curve as a 30-degree sector of a circle plus a right triangle, leading to a Pi approximation accurate to two parts in 100,000 with just five terms.
  • Newton's method revolutionizes Pi calculations, reducing the time needed significantly compared to polygon bisection methods, highlighting the importance of exploring unconventional approaches in mathematics for enhanced efficiency and insight.
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