| 475 Spring 09 Big Problems Big Problem 7: Which Is Bigger? . Compare the cardinalities of the following sets:
- $\mathbb{N}$
- $\mathbb{R}$
- $\mathbb{R}^2$
- $\mathbb{R}^n$
- set of functions from $\mathbb{N} \to \mathbb{N}$
- set of functions from $\mathbb{R} \to \mathbb{R}$
- algebraic numbers (i.e. the roots of finite degree polynomials with integer coefficients)
- the Cantor Set
- and the open interval $(0, 1)$
- set of all things you can describe in finite English sentences
Big Problem 8: Tulie Numbers.
Recall the Tulie Numbers class activity.
- Show that every rational number is dull, i.e. not endless.
- Show that every dull number is rational.
- Explain why the last two statements imply that a number has an infinite continued fraction representation if and only if it is irrational.
- Convince me that $e$ is irrational by finding a pattern in its continued fraction expansion.
Big Problem 9: Irrational Numbers.
- Use an argument similar to Euclid's method to prove $\sqrt[n]{2}$ is irrational for any natural number $n > 1$.
- Use an argument similar to Euclid's method to prove that $\sqrt{p}$ is irrational for $p$ prime.
- Use the continued fractions method to prove the square root of $n^2+1$ is irrational for any natural number $n > 0$. (You can assume the results of Big Problem 8.)
Big Problem 10: Closed Form Formula for the Fibonacci Sequence.
- We call a sequence $A_n$ Fibonacci-esque if it has the property $A_{n+2} = A_{n+1} + A_n$.
- Find all geometric sequences that are Fibonacci-esque. That is, all numbers $C$ such that the sequence $A_n = kC_n$ is Fibonacci-esque (for constant $k$).
- Prove linear combinations of Fibonacci-esque sequences have to be Fibonacci-esque.
- Find a closed-form formula for the famous Fibonacci sequence ${1, 1, 2, 3, 5, 8, \dots}$ by finding a sum of Fibonacci-esque geometric sequences that has the famous values.
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| Math 475.01 Spring 09 [[475 Spring 09 Homework]] [[475 Spring 09 Big Problems]] Official Title. Math 475: Capstone Course . . . 3K - last updated 2009-08-20 21:44 by Eric Hsu |