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

Age 16 to 18Challenge Level

Suppose $0 < a < b$. Which of the following continued fractions is bigger and why?

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