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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?

Comparing Continued Fractions

Which of these continued fractions is bigger and why?

Not Continued Fractions

Age 14 to 18
Challenge Level
  1. Find all positive integers $x$, $y$ and $z$ such that: $$x +\cfrac{1}{y + \cfrac{1}{z}} = N = \frac{10}{7}$$
  2. Show that when $N=10/7$ is replaced by $N=8/5$ it is impossible to find positive integer values of $x$, $y$ and $z$ for which the finite continued fraction on the left hand side is equal to $N$. Find another fraction (rational number) $N$ for which the same is true.