Как считали число пи? [Veritasium]
Vert Dider・14 minutes read
Isaac Newton made significant contributions to the calculation of pi, using innovative methods like the Pascal triangle and integrating from 0 to 1/2 for precise results, achieving remarkable accuracy in approximating pi. Newton's strategic integration approach revolutionized the calculation of pi, reducing the time needed from years to just 50 iterations, showcasing the importance of embracing innovative technologies in mathematical advancements.
Insights
- Isaac Newton revolutionized the calculation of pi by introducing innovative methods like using the Pascal triangle and infinite series, leading to precise approximations and groundbreaking advancements.
- Newton's strategic approach of integrating from 0 to 1/2 instead of 0 to 1 significantly expedited pi calculations, showcasing the importance of embracing new technologies and innovative approaches for efficient and accurate mathematical computations.
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Recent questions
How has the method of calculating pi evolved?
The method of calculating pi has evolved over two thousand years, with significant contributions from Isaac Newton. Initially, mathematicians like Archimedes, Dodik, and Ludolf van Selin made progress in approximating pi. Newton's innovative approach involved using the Pascal triangle to simplify calculations and explore beyond natural numbers. He expanded expressions like 1 + x to fractional powers to calculate values like the square root of 3. Newton's integration theory allowed for precise calculations of pi by considering the area under the curve of a quarter circle. His emphasis on innovative ideas and strategic integration approach led to groundbreaking advancements in calculating pi.
What did Isaac Newton contribute to pi calculations?
Isaac Newton made significant contributions to pi calculations by introducing innovative methods. He used the Pascal triangle to simplify calculations and explore beyond natural numbers. Newton's integration theory allowed for precise calculations of pi by considering the area under the curve of a quarter circle. By integrating from 0 to 1/2 instead of 0 to 1, Newton achieved remarkable accuracy in calculating pi, coming close to the value of 3.1416. His approach of expanding expressions to fractional powers and focusing on quickly decreasing terms revolutionized the calculation of pi.
How did Newton's method expedite pi calculations?
Newton's method expedited pi calculations by integrating from 0 to 1/2 instead of 0 to 1. This strategic approach focused on quickly decreasing terms and achieved remarkable accuracy in calculating pi, coming close to the value of 3.1416. By integrating up to 1/2, Newton simplified the calculations and expedited the process, showcasing the efficiency and effectiveness of his innovative method in determining the value of pi.
What technological advancements aided pi calculations?
Technological advancements have aided pi calculations by revolutionizing traditional practices. Newton's method, which previously took years to calculate pi using polygons with four quintillion sides, can now be achieved in 50 iterations. This advancement highlights the importance of embracing new technologies and innovative approaches in mathematical work. Just as technology revolutionized building construction from low-rise structures to towering skyscrapers like those in New York City, it has transformed the process of calculating pi, making it more efficient and accurate.
Why is Newton's approach to pi calculation groundbreaking?
Newton's approach to pi calculation is groundbreaking due to his emphasis on innovative ideas and strategic integration methods. By using the Pascal triangle, expanding expressions to fractional powers, and integrating from 0 to 1/2, Newton achieved remarkable accuracy in calculating pi. His method revolutionized the traditional approach to pi calculations, showcasing the importance of embracing new technologies and innovative strategies in mathematical work. Newton's contributions have paved the way for more efficient and precise methods of determining the value of pi, marking a significant advancement in mathematical calculations.
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Summary
00:00
Evolution of Pi Calculation Through Newton's Innovations
- The method of calculating pi has evolved over two thousand years, with Isaac Newton's contributions being significant.
- To understand the concept, cut a pizza into thin slices and fold them into a rectangle to relate to the circle's area.
- Archimedes and later mathematicians like Dodik and Ludolf van Selin made progress in approximating pi.
- Newton's innovative approach involved using the Pascal triangle to simplify calculations and explore beyond natural numbers.
- Newton's exploration led to the discovery of infinite series and the application of the binomial theorem to negative numbers.
- Newton's method involved expanding expressions like 1 + x to fractional powers to calculate values like the square root of 3.
- Newton's integration theory allowed for precise calculations of pi by considering the area under the curve of a quarter circle.
- Newton's approach of integrating from 0 to 1/2 instead of 0 to 1 expedited calculations by focusing on quickly decreasing terms.
- By integrating up to 1/2, Newton achieved remarkable accuracy in calculating pi, coming close to the value of 3.1416.
- Newton's emphasis on innovative ideas in mathematical work and his strategic integration approach led to groundbreaking advancements in calculating pi.
16:17
Revolutionizing Pi Calculation with Newton's Method
- Calculating pi using polygons with four quintillion sides can now be achieved in 50 iterations with Newton's method, a task that previously took years. This advancement highlights the importance of embracing new technologies and innovative approaches, as demonstrated by the evolution of building construction from low-rise structures to towering skyscrapers like those in New York City, showcasing how technology revolutionizes traditional practices.
