Leibnizs Harmonic Triangle Gottfried Leibniz 1762 Stephanie Moore
Leibniz’s Harmonic Triangle Gottfried Leibniz, 1762 Stephanie Moore & Jennifer Le. Blanc
Background History of Triangles Leonhard Euler Blaise Pascal Yang Hui Gottfried Leibniz
Pascal’s Triangle Leibniz’s Harmonic Triang 1 1 5 1 1 1 3 4 10 2 3 6 1 10 1 4 5 1 1
Relationships Between the Triangles 1 1 1 2 1 1 4 4 3 2 2 1 1 3 6 4 6 3 1 4 1 5 5 1 12 3 20 4 6 2 3 1 4 4 1 1 2 3 1 12 3 30 6 4 1 20 4 1 5
Patterns Formula To Find Any Term R = Row N = nth term 1 Row 2 Row 3 Row 6 Row 4 Row 5
Diagonals of Leibniz’s Triangl Diagonal 4 Diagonal 3 Diagonal 2 Diagonal 1 1 Diverges
Huygen’s challenge: Find the sum of the triangle reciprocals Triangle Numbers: 1, 3, 6, 10, 15. . . Triangle reciprocals
Diagonal Series Patterns: Diagonal Sum 2 3 4
Summary: • Exterior diagonals Harmonic Series • Interior diagonals Telescoping Series • Each Telescoping Sum Harmonic Series Thanks! ANY QUESTIONS?
THE END
References STONES, I. (1983). THE HARMONIC TRIANGLE: OPPORTUNITIES FOR PATTERN IDENTIFICATI GENERALIZATION. The Mathematics Teacher, 76(5), 350 -354. Retrieved from http: //0 www. jstor. org. wncln. org/stable/27963528 Blaise Pascal [Digital image]. (2012, April 26). Retrieved November 22, 2016, fr https: //commons. wikimedia. org/wiki/File: Blaise _pascal. jpg "Circa 1680, Gottfried Wilhelm Leibniz (1646 -1716), German rationalist philosoph 1675 he founded integral and differential calculus, publishing in 1684 (befo Archive Photos/Getty Images)" -- Image Date: 1/2/1754 -- Image Date: 1/2/175 Retrieved November 26, 2016, from Imh. Leibniz harmonic triangle [Digital image]. (2011, February 18). Retrieved Novemb http: //images. google. de/imgres? imgurl=https%3 A%2 F%2 Fupload. wikimedia. org%2 Fw commons%2 Fthumb%2 Ff 7%2 FLeibniz _harmonic _triangle. svg%2 F 800 px. Leibniz _harmonic _tri angle. svg. png&imgrefurl=https%3 A%2 F%2 Fcommons. wikimedia. org%2 Fwiki%2 FFile%3 A _h armonic _triangle. svg&h=500&w=800&tbnid=K 0 i. EITz. Gw. ENS 7 M%3 A&vet=1&docid=Yh 7 WR 2 B 8 M&ei=PAg 6 WLz 6 OISpmw. Gw 2 Y 2 ADw&tbm=isch&iact=rc&uact=3&dur=1423&page=0&start=0& p=18&ved=0 ah. UKEwi 8 i. M-Bu. Mf. QAh. WE 1 CYKHb. Bs. A _AQMwgd. KAIw. Ag&bih=638&biw=1366
Mac. Kinnon, D. (2009, January 26). [Euler's Number Triangle]. Retrieved November 2 http: //www. mathrecreation. com/2009/01/triangular-numbers-and-eulers-number. ht Pascal, B. (2007, September 27). Tri ángulo de Pascal en el escrito original de Pascal [Digit Retrieved November 22, 2016, from https: //commons. wikimedia. org/wiki/File: Triangulo. Pa Pascal’s triangle. [Image]. In Encyclop ædia Britannica. Retrieved from http: //media 1. school. eb. com/eb-media/03/59803 -004 -88609 FEC. jpg Portrait of Chan Master Yangqi Fanghui [Digital image]. (2014, August 04). Retrie from https: //commons. wikimedia. org/wiki/File: Yangqi. Fanghui. gif Portrait of Swiss mathematician and physicist Leonhard Euler (1707 - 1783), 1740 s of the greatest mathematicians who ever lived, was the first to use the term an expression involving various arguments and was the first to apply calculus Kean Collection/Getty Images). (1754, January 02). Retrieved November 26, 201 STONES, I. (1983). THE HARMONIC TRIANGLE: OPPORTUNITIES FOR PATTERN IDENTIFICATIO GENERALIZATION. The Mathematics Teacher, 76(5), 350 -354. Retrieved from http: //0 www. jstor. org. wncln. org/stable/27963528
Tent, M. B. (2011). Gottfried Wilhelm Leibniz: The polymath who brought us calculus. [Google Books Version]. Retrieved November 21, 2016, from https: //books. google. com/books? id=5 h. TSBQAAQBAJ&pg=PA 78&lpg=PA 78&dq=leibniz+ha +triangle&source
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