1. Isaac Newton and the Binomial Theorem
Callie Edwards and Kristen Johnson

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

3. 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

4. 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

5. 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?

6. 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

7. 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.

8. 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.

9. Newton’s Original Formula

10. 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!!!

11. 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.

12. Negative Exponents

13. Fractional Powers
Your turn!!

14. Can Computers Help?

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

16. 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 …

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

18. 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.

19. 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