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imagine a world where calculating pi was a tedious, time-consuming task. for over 2000 years, mathematicians struggled to find an efficient method to determine this mysterious number. but then, a brilliant mind emerged: isaac newton. he transformed the game with his revolutionary binomial expansion technique, which we'll explore in this article.
the binomial expansion: a game-changer for pi
before newton, the most successful method for calculating pi involved measuring the circumference of a circle and dividing it by its diameter. this method, while accurate, was incredibly slow and required a lot of patience. newton's binomial expansion, on the other hand, allowed mathematicians to calculate pi with greater speed and precision.
what is binomial expansion?
binomial expansion is a mathematical technique used to expand expressions of the form (a + b)^n, where a and b are any real numbers and n is a non-negative integer. this expansion is based on the binomial theorem, which states that the coefficients of the terms in the expansion can be found using the combination formula.
newton's binomial expansion for pi
newton applied the binomial expansion to the calculation of pi by using the series expansion of the square root function. this allowed him to approximate pi with greater accuracy than ever before. the series expansion of the square root function, also known as the binomial series, is given by:
sqrt(1 + x) = 1 + 1/2x - 1/8x^2 + 1/16x^3 - 5/128x^4 + ...
by substituting x = -1 into this series, newton was able to approximate pi using the following formula:
pi = 4(1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...)
this formula, known as the leibniz formula for pi, is a direct result of newton's binomial expansion technique.
the impact of newton's method
newton's binomial expansion technique not only revolutionized the calculation of pi but also had a significant impact on mathematics as a whole. it allowed mathematicians to solve complex problems more efficiently and paved the way for the development of calculus.
the discovery that transformed pi
newton's method for calculating pi was a game-changer. it allowed mathematicians to determine the value of pi with greater accuracy and speed than ever before. this discovery transformed the way we understand and use pi in mathematics and science.
the legacy of newton's binomial expansion
newton's binomial expansion technique has had a lasting impact on mathematics. it has been used to solve a wide range of problems, from calculating the area of a circle to determining the trajectory of a spacecraft. today, the binomial expansion is a fundamental concept in mathematics and is taught in schools around the world.
conclusion
newton's binomial expansion technique revolutionized the calculation of pi and had a significant impact on mathematics as a whole. by using the series expansion of the square root function, newton was able to approximate pi with greater accuracy and speed than ever before. this discovery transformed the way we understand and use pi in mathematics and science, and its legacy continues to this day.
references
arndt, j., & haenel, c. (2001). pi-unleashed. springer science & business media — https://ve42.co/arndt2001
dunham, w. (1990). journey through genius: the great theorems of mathematics. wiley — https://ve42.co/dunham1990
borwein, j. m. (2014). the life of π: from archimedes to eniac and beyond. in from alexandria, through baghdad (pp. 531-561). springer, berlin, heidelberg — https://ve42.co/borwein2012
special thanks to
alex kontorovich, professor of mathematics at rutgers university, and distinguished visiting professor for the public dissemination of mathematics national museum of mathematics momath for being part of this pi day video.
written by
derek muller and alex kontorovich
animation by
ivy tello
filmed by
derek muller and raquel nuno
edited by
derek muller
music by
jonny hyman and petr lebedev
additional music from
https://epidemicsound.com "particle emission", "into the forest", "stavselet", "face of the earth", "firefly in a fairytale"
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