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Welcome Art of Numbers CCC-801 Instructor : Neha Gupta Email id : [email protected] Office : B-122 (c) Art of Numbers : Overview • This course deals with two aspects of numbers. In the first part of the course we take up some unexplored patterns that exist in ‘nature’, study them and see some of their applications. • The second part looks at numbers as carriers of information about our lives. Here we learn how to analyze and present data in ways that make sense from the raw data. We use the spreadsheet program in Open Office to present and analyze the data in depth. Detailed Syllabus Part A: Fun with Numbers (~ 12 lectures) • • • • • • Moessner’s Magic Golden Ratio Pascal Triangle, Binomial Theorem Fibonacci Sequence Introductory Number Theory Concepts Continued Fractions and some applications Detailed Syllabus Part B: Handling Data (≤ 3 lectures) • Descriptive Statistics like mean, median, mode, range, standard deviation, percentiles, quartiles • Charts – Bar Charts, column charts, Line Charts, Pie Charts, scatter plots References • The Book of Numbers by John Horton Conway, Richard K. Guy. 2nd edition, Copernicus. • The Heart of Mathematics: An Invitation to Effective Thinking by Edward B. Burger and Michael Starbird. 3rd edition, Wiley. • The Visual Display Of Quantitative Information by Edward Tufte. 2nd edition, Graphics Press. • Excel 2007 for Starters: The Missing Manual by Matthew MacDonald. Shroff/O'Reilly. • Analyzing Business Data with Excel by Gerald Knight. Shroff/O'Reilly. Assessment Scheme Components Attendance Assignments Quizzes Final Term Report Total % 10 10 x 2 15 x 3 25 100 Assignments : 2 ( 10 each ) Quizzes : 3 (15 each) Total marks = 10 + 20 + 45 + 25 = 100 Plan for the Course Part A: Fun with numbers : mostly lectures on Slides/Board Part B: Handling Data (2/3 classes for data Handling using Excel) Quizzes : during Lecture timings Assignments : Submit hard copy on the due date. Final Exam Report: (you can work in groups of ≤ 4 members) • A pool of topics will be uploaded on BB near to the date of submission. • Submission of report(hard copy) by 17th September (Monday), 2018. • Group Members name & topic to be reported by 10th September, 2018. Lets get started Yayyyyy!!! Here, step size = 2 Here, step size = 3 Here, step size = 4 Moessner’s theorem says that the final sequence is 1n , 2n , 3 n, … where n = step size. • This construction is an interesting combinatorial curiosity that has attracted much attention over the years. Moessner’s theorem was never proved by its eponymous discoverer. • 1951 A.Moessner conjures it: (Works for all n ϵ N) • 1952 O.Perron proves it • 1952 I.Paasche and H.Salie generalize it • 1966 Long presents an alternative proof ( and generalizes it) • 2010 Hinze, Rutten & Niqui present new proofs of the thm The initial step size n is constant. What happens if we increase it in each step? Let us repeat the construction starting with a step size of one and increasing the step size by one each time. …, K+1C2 , … First generalization : increasing the step size by one each time Step Size 1 2 4 3 Step Size increasing by 1 5 ( Paasche’s Theorem ) • The proof of the Moessner’s and Paasche’s theorem can be found on a recent paper titled “On Moessner’s Theorem “ by Dexter Kozen and Alexandra Silva • Link : https://www.cs.cornell.edu/~kozen/papers/Moessner.pdf Second generalization : Let us now increment the increment by one in each step. Thus incrementing the step size by 1, 2, 3, 4, . . . in successive steps crossing out 1, 4, 10, 20, . . . , k+2C3 , … 3 Step Size 1 4 1 +2 = 3 6 15 3 +3 = 6 Step size increasing by 1, 2, 3, … ... 20 6 + 4 = 10 • Let us now increment the increment by one in each step, thus incrementing the step size by 1, 2, 3, 4, . . . in successive steps, crossing out 1, 4, 10, 20, . . . , k+2C3 , .... 90 105 362 • The final sequence consists of the superfactorials • 1, 2, 12, 288, . . . = 1!, 2!1!, 3!2!1!, 4!3!2!1!, . . . = 1!!, 2!!, 3!!, 4!!, . . . . • The generalization of Moessner’s theorem that handles these cases is one of the particular cases of Paasche’s theorem. Summary : • Moessner’s theorem describes a procedure for generating a sequence of n integer sequences that lead unexpectedly to the sequence of nth powers 1n , 2n , 3n , . . . . • Paasche’s theorem is a generalization of Moessner’s; by varying the parameters of the procedure, one can obtain the sequence of factorials 1!, 2!, 3!, . . . or the sequence of super factorials 1!!, 2!!, 3!!, • Long’s theorem generalizes Moessner’s in another direction, providing a procedure to generate the sequence a · 1 n-1 ,(a + d) · 2n-1 , (a + 2d) · 3n-1 , . . . • Proofs of these results in the literature are typically based on combinatorics of binomial coefficients or calculational scans. Finish. Yayyyyyy