Isaac Newton and the Binomial Theorem Callie Edwards and Kristen Johnson

What is the Binomial Theorem? The binomial theorem is a method of finding the coefficients for a binomial expansion. Expansion Easily find coefficient

Binomial Theorem History Pingala 3rd Century BC Euclid 4th Century BC Halayudha and Al-Karaji 10th Century AD Knew of something similar to Pascal’s triangle Came up with general binomial theorem Yang Hui 13th Century AD Pingala did orders higher than 2 Euclid did exponent of 2

Binomial Theorem Blaise Pascal (1623-1662) Pascal’s Triangle – 1653 Numbers in the triangle are related to coefficients For example (a+b)2 = a2 + ab+ b2 The exponent corresponds with the row number. 0th row 1st row 2nd row 1 2 1

Binomial Theorem 1,3,3,1 (a+b) 3= 1a3 + 3a2b + 3ab2 + 1b3 Now you do one! (a+b) 3= _a3 + _a2b + _ab2 + _b3 What are the coefficients? 1,3,3,1 (a+b) 3= 1a3 + 3a2b + 3ab2 + 1b3 These are only positive integers! What about fractions and negative numbers?

Isaac Newton Born on December 25, 1642 or January 4, 1643 (depending on the calendar) Lived in England Well known mathematician, physicist, astronomer, philosopher, and theologian Studied at Trinity College and earned his degree in 1665 Did most of calculus work in 1660’s Became Sir Isaac Newton in 1705 Died March 31, 1727

Isaac Newton Generalized the binomial theorem in 1665 Decided that negative numbers and fractions could be included as exponents Less tedious method than expanding Pascal’s triangle, especially with larger exponents Some have called it Newton’s greatest math discovery Used theorem to integrate and to approximate Pi. Newton’s greatest math discovery, used to integrate and approximate PI.

Newton’s Original Formula is the binomial. Mathematics behind newton’s generalization Postive integers terminate Fractions do not Negative numbers do not WHY? 1/(a+b)3 = (a+b)^-3 and when multiplied together, equal 1 Figure out notation for newton’s older notation Show how modern notation comes from Newton’s is the power of expansion. A, B, C, … represent the preceding terms in the expansion.

Newton’s Original Formula

The Binomial Theorem FINITE!!! What if the exponent is a natural number? Is the series finite or infinite? If we let “n” be a natural number, then: So the answer is….. FINITE!!!

Negative Exponents Consider the function, This expansion is equal to, Uh-oh! This series never terminates!! We need to use the properties of negative exponents to verify this result.

Negative Exponents

Fractional Powers Your turn!!

Can Computers Help?

What about convergence? Let’s try the ratio test! Oops! The test failed….

Convergence So to determine where this series converges, we will have to find the radius of convergence! Therefore the Radius of Convergence is 1, and the series converges when …

What do we need to expand Newton’s formula to cover all real numbers?

Binomials in the Classroom Algebra 1 Mainly refers to Pascal’s Triangle for binomial expansion Algebra 2 Pascal’s Triangle for binomial expansion Some books include the Binomial Theorem Discrete Math and Advanced Functions and Modeling Apply the Binomial Theorem for theoretical and experimental probability Pre-Calculus Many NC textbooks use Pascal’s Triangle and the binomial theorem for expansion Calculus Mainly focuses on the theorem for expansion Pascal’s triangle in algebra Binomial theorem in statistics Binomial theorem in Pre-Calc Look up in standards for NC and NCTM Worksheet for binomial theorem for ½ exponent and 1/3 exponent. From book.

Binomials in the Classroom -n is the power -k is the specific term This form looks different from Newton’s binomial theorem, but it works in a similar way. This is found in several high school textbooks Easier for the students to understand the process in this format Easily find specific coefficients Related to Statistics: Combinations nCk