1. Leibniz's Harmonic Triangle
Nikki Icard & Andy Hodges
Mat 5930

2. Gottfried Wilhelm Leibniz
Background
Gottfried Wilhelm Leibniz
Born in Leipzig, Germany in 1646.
Read lots of books from his father's library
Obtained Bachelors at the age of 17.
Obtained Doctorate in Law at the age of 20
Estimated IQ of 180 or higher
Interested early on in classical studies but that soon turned into an interest in logic

3. Early Years Worked as counsel and legal advisor to Kings and Princes
In 1672, at the age of 26, visited Paris as a diplomat
Stayed in Paris for much of the next 4 years
Met Christiaan Huygens, who became his mathematical mentor
Huygen's challenged Leibniz to calculate the infinite sum of the reciprocals of the triangular numbers of the form: 1, (1+2), (1+2+3), … with general form (i(i+1))/2 for i=1,2,...
Christiaan Huygens ( )

4. Huygen's Challenge
Calculate the infinite sum of the reciprocals of triangular numbers
Leibniz attacked the problem by recognizing that each term was:
And the sum was:
Simplifying and cancelling:

5. Leibniz's Achievement
Proud of his findings, discussed them with mathematician, John Pell
Pell quickly let Leibniz know he discovered nothing new
Embarrassed by his ignorance, began a period of serious, intense study of mathematics
Concentrated on Pascal's Triangle, his own Harmonic Triangle, and sums and differences of series
Led Leibniz to develop the ideas behind what we now call the Fundamental Theorem of Calculus

6. Pascal's Triangle

7. Leibniz's Harmonic Triangle

8. Properties of both Triangles
Property 1:
Pascal's Triangle - each entry (not in the first row or column) is the sum of the two terms directly above it.
Leibniz's Triangle - each entry is the sum of the two terms directly below it.
Property 2:
Pascal's Triangle - each entry is the difference of the two terms directly below it.
Leibniz's Triangle - each entry (not in the first row) is the difference of the two terms directly above it.
Property 3:
Pascal's Triangle - each entry (not in the first row or column) is the sum of the number (above and to the right) and all left diagonal terms above.
Leibniz's Triangle - each entry is the sum of the number (below and to the right) and all left diagonal terms below.
Property 4:
Each entry in the nth row of Leibniz's Triangle is the reciprocal of the number generated by multiplying n by the corresponding element in the corresponding nth row of Pascal's Triangle.

9. Generating Pascal's Triangle
rth entry in the nth row is given by the binomial coefficient:
Add two consecutive entries to give the entry between them in the row below, by the rule:

10. Generating Harmonic Triangle
rth entry in the nth row is given by the binomial coefficient:
Add two consecutive entries to give the entry between them in the row above, by the rule:

11. Our Lesson Plan
We found that the best way to use Leibniz’s Harmonic Triangle was to tie it in with rational functions.
This activity could be used after the simplification of rational functions has been taught.

12. Our Lesson Plan Continued…
First start with the warm up, which is a review of simplifying rational functions
Then have the students work on the Extension assignment in either groups of 2 or 3
There is also an additional assignment that deals with Harmonic mean, but should also be done as an extension.

13. A Little Bit of Warm Up Simplify COMPLETELY. x + 2 3x + 6
x2 + 2x x2 – 1 x
x2 – x – 12 ÷ +

14. A Little Bit of Extension
a) Show that in diagonal 1 the kth fraction and its successor can be written as 1/k and 1/(k+1).
Show that the sum of the first n fractions in diagonal 2 can be written as
(1/1 – 1/2) + (1/2 – 1/3) +(1/3 – 1/4)+(1/4 – 1/5)+…+(1/n – 1/(n+1))
c) Find a formula for the sum of the first n fractions in diagonal 2. Write your formula as a single fraction.
d) Use your formula to find the sum of the first four fractions in diagonal 2. Check your result by adding the appropriate fractions.
e) What will happen to the sum of the first n fractions in diagonal 2 as n gets larger and larger? Justify your answer.
f) Prove that the kth fraction in diagonal 2 can be written as 1/(k(k+1)).

15. Harmonic Denominator Number Triangle

16. Harmonic Denominator Number Triangle Continued…
The Harmonic Denominator Triangle can be used to find the derivative of a polynomial (and that polynomial can be found from Pascal’s Triangle.