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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?
Age 16 to 18
Challenge Level
Suppose $0 < a < b$. Which of the following continued fractions is bigger and why?
\[ \frac{1}{2+\frac{1}{3+\frac{1}{a}}} \] 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 $a$.
NRICH is part of the family of activities in the Millennium Mathematics Project.