Newton s Work on Infinite Series. Kelly Regan, Nayana Thimmiah, & Arnold Joseph Math 475: History of Mathematics

Transcription

1 Newton s Work on Infinite Series Kelly Regan, Nayana Thimmiah, & Arnold Joseph Math 475: History of Mathematics

2 What is an infinite series? The sum of terms that follow some rule. The series is the sum of the sequence of numbers, not purely the sequence: ½ +¼ +⅛+...+1/2 n = 1 is a series, ½, ¼, ⅛,, 1/2 n is a sequence, not a series

3 Why were infinite series important to Newton? Newton believed that infinite series and power series were important to further developing the analytical field within mathematics, as he explains in Tractatus de methodis serierum et fluxionum (A Treatise on the Methods of Series and Fluxions).

4 Method of Fluxions and Infinite Series Converts a complex quantity into a power series so that it is easier to solve Two major methods: Reduction by Division and Root Extraction

5 Reduction by Division This will convert a complex fraction into a power series. We use a method that we are very familiar with -- long division!

6 Let s start off simple /3

7 Now incorporating variables... (3+7x+4x 2 )/(1+x)

8 Let s try this one together 1/(2+x)

9 Try these problems using Reduction by Division! 1 / (1+x 2 ) a 2 / (b +x)

10 Newton s Work

11 Root Extraction This is a method to solve for the square root It is not as straightforward as reduction by division, but its method is based off an expansion that we are familiar with -- (a+b) 2 = a 2 + 2ab + b 2

12 Let s start off simple... 2

13 Now more generally... (a 2 +x 2 )

14 Newton s Method

15 Now you try! (x -x 2 )

16 Newton s Method

17

18

19 y 3 + 2y - 5 = 0

20 Newton s original insight ignores exponential terms for small quantities allowing for further refining and precision as the method repeats; Newton originally formulated this method without the formal concept of derivatives Newton s method has its drawbacks: doesn t necessarily converge Newton s method provides a methodical way to solve for the roots of these fluent polynomials; this is valuable to Newton for his later insights into physics and also (more historically important) laid a modern foundation for calculus in modern math, Newton s method is just one of many root-finding algorithms: a class of algorithms designed to find the critical points of various types of functions (Newton was concerned with polynomials). There many extensions of Newton s method to handle more complex scenarios

21 Newton vs. Leibniz: who really invented calculus?

22 So who do you think should be credited with the invention of calculus?

23 The End

24 Sources Bardi, Jason Socrates. The Calculus Wars. Basic Books, Newton, Isaac. A Treatise Of The Method Of Fluxions And Infinite Series. T. Woodman, 1737, pp "The Calculus Controversy". Youtube, 2010, Accessed 12 Mar "Newton's Infinitesimal Calculus (1): Reduction By Division/ Long Division". Youtube, 2016, Accessed 15 Mar 2018.

25 In case we run out of time for the video...

26 Newton vs. Leibniz Similarly to Tartaglia and Cardano, there was debate among who truly invented calculus, Isaac Newton or Gottfried Leibniz. Both used what they knew, and adapted it to create something new.

27 Newton -In 1666, sent home from Cambridge University (plague) and developed what is now calculus to solve physics problems -Called it the Method of Fluxions, fluxions are derivatives -Mainly used geometric proofs to justify the method of fluxions -Didn t publish his findings until 1687 in Philosophiæ Naturalis Principia Mathematica

28 Leibniz -Began work on his theory of Calculus in 1674, finding the area under a graph y=f(x) -In 1684 and 1686, he published his work on differentiation and integration, respectively -This was before Newton published his finding, so he got sole credit

29 Where it gets interesting -Newton set out to prove that Leibniz had plagiarized his work -Many of Newtons associates also worked with Leibniz, and it is believed they could ve shared his work. -Newton and Leibniz also corresponded about math problems many times, including Newton's first theories on fluxions. -Leibniz began to lose control of the argument, dying shortly thereafter, leaving Newton as the credited sole creator of calculus -Even after Leibniz death, Newton tried to discredit his work, even though he was fairly accurate (more efficient than Newton s notation)

30 So who is credited with inventing calculus? -Both Newton and Leibniz are credited with creating calculus -Newton s was inspired by motion and physics, while Leibniz was inspired by geometry -Leibniz way was more efficient, so his method was used more frequently than Newtons in the 1700s -Later, it was determined that although Leibniz had probably seen some of Newton s work, he was already far along in his conclusion