Module 6 homework - Math 3303

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This is a 100 point assignment.

These problems are designed to take you back through earlier topics…a little review for the final is my goal here. AND a preview of the kind of longitudinal summing up that is a feature of the questions on the final.

Homework rules:

Front side only. Keep the questions and your answers in order.

If you send it pdf, send it in a single scanned file.()

If you turn it in personally, have the receptionist date stamp it and put it in my mailbox.

(651 PGH – 8am to 5pm)

1.10 points

  • Create a finite simple continued fraction that has at least 5 fraction bars long.
  • Make a list of tips for anyone who is trying to do this…consider relative size of the numbers, shared factors … how can you pick the numbers efficiently.

2.10 points

  • Find 6 different and interesting facts about continued fractions on the internet – list them in your own words showing the source.
  • Find one worksheet or project that you would like to teach everyone. Print it or write it out with sources listed.

3.10 points

Let’s talk about List where you found these facts:

Represent pi as a nearby rational number:

Represent pi as a decimal to 10 places:

Represent pi as a continued fraction to 7 places:

Give as many mathematical words as you can for pi:

positive and irrational will get you started….
4.10 points

Pick a prime number.

Represent it as an iterated radical (see page 61). How quickly does the iterated radical “become” the prime number?

Represent its square root as a continued fraction. How quickly does the CF “become” the irrational number?

Write a brief essay on the duality of these procedures.

5.10 points

Turn into a solved Diophantine equation – like the authors did with and

at the beginning of Chapter 10 in the book.

6.10 points

Write a brief essay on the role of phi,  in numbers and society. Include 5 facts with explanations about it and list sources at the end.

7.10 points

Note the properties of Fibonacci numbers listed below. Please know the properties of Fibonacci numbers by heart.

  • Every natural number greater than 3 can be written as the sum of distinct Fibonacci numbers starting with f2.

Demonstrate this with 50, 75, and 100

  • Show that the sum of the first n Fibonacci numbers with odd indices (1 through k) is given by the formula:

Illustrate this for n > 8

  • Show that the sum of the first n Fibonacci numbers with even indices is given by the formula:

Illustrate this for n > 9

  • Show that the alternating sum and difference of Fibonacci numbers is given by the formula:

Illustrate this for n > 10

  • Show that the following product is sometimes square-free and sometimes not. Give several examples and see if you can spot a pattern.

8.10 points

Look up “geometric mean”. Give the definition and list 2 different applications of it…not necessarily Fibonacci numbers, try triangles and algebra.

9.10 points

Write a brief essay on the Mystic Pentagram – leave out all the mystic part, focus on the interesting math facts and properties, focus on the connection to phi and to the Fibonacci numbers. List your sources at the end.

10.10 points

Find the Fibonacci Association online. Read and critique one of the offerings from their website.

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