Newton & the Binomial Theorem

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Presentation transcript:

Newton & the Binomial Theorem K. Bowman, F. Miller, & T. Spain

The Binomial Theorem What is seen in the classroom: What Newton actually proposed:

Historical Progression Binomial expansions known since Euclid The idea for squares and cubes can be traced from Euclid to Aryahbata and then to Omar Khayyam. 1300s - mathematician Chu-Shih-Chien established triangular diagram for coefficients. Pascal’s Triangle 1664-65 Sir Isaac Newton works on binomial expansions Image source: http://www.dbriers.com/tutorials/wp-content/uploads/2010/02/pascal.gif

Newton’s Work Studied Wallis’ Arithmetica Infinitorum Worked methods backwards to include negative integers. He hypothesized that he could expand Pascal’s Triangle to include rational exponents. He was studying the area under various curves Beginning with circles and then hyperbolas Didn’t publish anything about this until 1676 when communicating with Leibniz Image source: http://www.biography.com/people/isaac-newton-9422656

Proof Jakob Bernoulli proved theorem for non-negative integers Published in 1713, after his death Leonhard Euler provided flawed proof for negative and rational powers Niels Abel completed proof for theorem in early 1800s Bernoulli Abel

How is it useful? How can we use it in our classrooms? Expanding binomials Pascal’s triangle expansion Finding specific terms of binomials Binomial probability

Let’s Practice - Expanding Binomials

Let’s Practice - Pascal’s Triangle Expansion (3v-4w)5

Let’s Practice - Finding Specific Terms Find the sixth term of (2x+5)8

Let’s Practice - Binomial Probability The probability of a car driving by with tinted windows is 0.15. What is the probability that at most 2 of the next 10 cars driving by have tinted windows?

References Binomial Distribution | mathematics | Britannica.com. (n.d.). Retrieved December 3, 2016, from https://www.britannica.com/topic/binomial-distribution Binomial Theorem. (n.d.). Retrieved December 3, 2016, from https://trans4mind.com/personal_development/mathematics/series/binomialTheorem.htm Coolidge, J. L. (1949). The Story of the Binomial Theorem. The American Mathematical Monthly, 56(3), 147–157. https://doi.org/10.2307/2305028 Edwards, A.W.F. (2002) Pascal’s arithmetic triangle: The story of a mathematical idea. Baltimore MD: Johns Hopkins University Press. Microsoft Word - newtonl1-student.doc - newton1-student.pdf. (n.d.). Retrieved from http://www.wwu.edu/teachingmathhistory/docs/psfile/newton1-student.pdf NewtonBinomial - NewtonBinomial.pdf. (n.d.). Retrieved from http://quadrivium.info/MathInt/Notes/NewtonBinomial.pdf STEWART, L. (1960). The binomial theorem. The Mathematics Teacher, 53(5), 344–348. Whiteside, D. T. (1961). Newton’s Discovery of the General Binomial Theorem. The Mathematical Gazette, 45(353), 175–180. https://doi.org/10.2307/3612767