The Discovery That Transformed Pi
Veritasium・16 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.
