The Discovery That Transformed Pi
Veritasium・2 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.
Get key ideas from YouTube videos. It’s free
Recent questions
How was Pi calculated before Isaac Newton?
Through slow methods like polygon bisection.
Related videos
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.
