### Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

### There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

### Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

# Comparing Continued Fractions

##### Stage: 5 Challenge Level:
Suppose $0 < a < b$. Which of the following continued fractions is bigger and why?

$\frac{1}{2+\frac{1}{3+\frac{1}{a}}}$ \par or $\frac{1}{2+\frac{1}{3+\frac{1}{b}}}$

Suppose the fractions are continued in the same way, then which is the bigger in the following pair and why?

$\frac{1}{2+\frac{1}{3+\frac{1}{4+\frac{1}{a}}}}$

or the same thing with b in place of a.

Now compare: $${1\over\displaystyle 2 + { 1 \over \displaystyle 3+ { 1\over \displaystyle 4 + \dots + {1\over\displaystyle 99+ {1\over \displaystyle {100 + {1 \over \displaystyle a}} }}}}}$$

and the same thing with $b$ in place of $a$.