In 1610, Ludolph Van Ceulen approximated to 30 decimal places using polygons with sides! There had to be a better way! In the 17th century, an entirely new method of approximating was found: series approximation. how Newton approximated Here’s … Find the equation of the semicircle in the picture. Observe that the would satisfy the equation . “full circle” Solve for and show that , in other words, . Use generalized binomial theorem to show Newton’s Find the area of the shaded region with calculus. Now that you have an equation for the semicircle, evaluate a definite integral to show that the area is Find the area of the shaded region with geometry. Find the length of . Find the area of . Notice that is a 30-60-90 triangle. Use this fact to find the area of sector . Notice that (area of shaded region) = (area of sector) (area of triangle). – Conclude that the area of the shaded region is Set the two expressions for area equal to each other to conclude that Use a calculator to approximate using terms in the series and then using terms in the five six series. Compare how the accuracy improves. There are a couple good reasons Newton chose the point to equal . When calculating area by why is a nice number to use? calculus, When calculating area by why is a nice number to use? geometry, use approximating method with the semicircle shaded. entire Let’s Newton’s Use calculus to show that the semicircle has area . Use geometry to show that the semicircle has area . Conclude that The above expression is much simpler than the expression we found in step 4. Why is original series approximation preferable to this one? Newton’s Newton correctly approximated to 15 digits, although he later confessed: "I am Postscript: ashamed to tell you to how many figures I carried these computations, having no other business at the time." Mathematicians have often looked at with curiosity, and sought to find better and better approximations. The motivation is not because better approximation of are more useful, but rather, the motivation comes from the challenge of creating the best possible mathematical method or algorithm for the approximation. Some highlights: 1706: John Machin was the first to compute 100 decimals of π. 1949: the first computer, ENIAC, computed 2037 digits of in 70 hours.
In 1610, Ludolph Van Ceulen approximated to 30 decimal places using polygons with sides! There had to be a better way! In the 17th century, an entirely new method of approximating was found: series approximation. Here’s how Newton approximated … Find the equation of the semicircle in the picture. Observe that the “full circle” would satisfy the equation . Solve for and show that , in other words, . Use Newton’s generalized binomial theorem to show Find the area of the shaded region with calculus. Now that you have an equation for the semicircle, evaluate a definite integral to show that the area is Find the area of the shaded region with geometry. Find the length of . Find the area of . Notice that is a 30-60-90 triangle. Use this fact to find the area of sector . Notice that (area of shaded region) = (area of sector) – (area of triangle). Conclude that the area of the shaded region is Set the two expressions for area equal to each other to conclude that Use a calculator to approximate using five terms in the series and then using six terms in the series. Compare how the accuracy improves. There are a couple good reasons Newton chose the point to equal . When calculating area by calculus, why is a nice number to use? When calculating area by geometry, why is a nice number to use? Let’s use Newton’s approximating method with the entire semicircle shaded. Use calculus to show that the semicircle has area . Use geometry to show that the semicircle has area . Conclude that The above expression is much simpler than the expression we found in step 4. Why is Newton’s original series approximation preferable to this one? Postscript: Newton correctly approximated to 15 digits, although he later confessed: "I am ashamed to tell you to how many figures I carried these computations, having no other business at the time." Mathematicians have often looked at with curiosity, and sought to find better and better approximations. The motivation is not because better approximation of are more useful, but rather, the motivation comes from the challenge of creating the best possible mathematical method or algorithm for the approximation. Some highlights: 1706: John Machin was the first to compute 100 decimals of π. 1949: the first computer, ENIAC, computed 2037 digits of in 70 hours.
Historically_Based_Group_Calc_Projects.docx