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The video explores the groundbreaking discoveries made by Sir Isaac Newton in mathematics, particularly in the calculation of pi. Newton observed a pattern in the coefficients of equations that resembled Pascal's triangle and utilized the binomial theorem to expand the triangle with fractions. This allowed him to calculate the square root of 3 with great accuracy and ultimately discover a new way to calculate pi. Newton's innovative thinking and experimentation with mathematical patterns led to the discovery of the Newton series, which revolutionized the calculation of pi and eliminated the need for tedious polygon divisions.

**00:00:00**In this section of the video, the narrator explains how for 200 years, Pi was calculated using a method that was extremely tedious and slow. This method involved dividing polygons until they had numerous sides. Mathematicians around the world used this method to push the limits to calculate Pi to as much precision as possible. However, that changed as Isaac Newton came up with a formula that made it easier to calculate Pi. Newton's discovery forever changed the way Pi was calculated as it no longer relied on dividing polygons into a ridiculous number of sides.**00:05:00**In this section, the video discusses the discovery of the pattern in the coefficients of equations by Sir Isaac Newton in his experimentation to calculate pi. He found that the coefficients resemble the numbers in Pascal's triangle, which has been around since ancient Greece, India, China, and Persia. Newton saw a pattern that would allow him to skip the tedious arithmetic and go straight to the solution, using this triangular system. The beauty of mathematics transcends cultures and time, and Newton's discovery of the binomial theorem was precisely what was seen as coefficients in Pascal's triangle.**00:10:00**In this section, the video discusses the discoveries of Sir Isaac Newton in mathematics, specifically his exploration of the binomial theorem and the Pascal triangle. Newton applied the binomial theorem to negative values of n, and observed a pattern in the triangle that extended beyond integers. He further expanded the triangle with fractions, establishing a continuity of numbers between 0 and 1, and calculating the square root of 3 with great accuracy. Newton then integrated a curve to calculate the area of a circle, and ultimately discovered a new way to calculate Pi. The video explores how Newton's innovative thinking and approach to mathematics led to numerous groundbreaking discoveries.**00:15:00**In this section, the video discusses the discovery of the Newton series, which revolutionized the calculation of pi. This method involves finding a series of equations, each smaller than the last, that approach the value of pi, allowing for faster and more efficient calculations. By using this method, the need for tedious and time-consuming polygon divisions was eliminated. The video also emphasizes the importance of experimentation with mathematical patterns and pushing them beyond their assumed limits to achieve new and innovative solutions.

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