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

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

3. 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
Sir Isaac Newton works on binomial expansions
Image source:

4. 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:

5. 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

6. 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

7. Let’s Practice - Expanding Binomials

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

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

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

11. References
Binomial Distribution | mathematics | Britannica.com. (n.d.). Retrieved December 3, 2016, from
Binomial Theorem. (n.d.). Retrieved December 3, 2016, from
Coolidge, J. L. (1949). The Story of the Binomial Theorem. The American Mathematical Monthly, 56(3), 147–157.
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
NewtonBinomial - NewtonBinomial.pdf. (n.d.). Retrieved from
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.