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Good Approximations

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

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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?

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Comparing Continued Fractions

Which of these continued fractions is bigger and why?

Not Continued Fractions

Age 14 to 18 Challenge Level:
  1. The key here is that $x$ has to be the integer part of $N$ because the 'continued fraction' part of the expression gives a value less than one.

    As $y$ and $z$ are positive integers (whole numbers), $y + 1/z > 1$ and $1/(y+1/z) < 1$ so we know that this must equal $3/7$ and $x = 1$.

    Hence $y + 1/z = 7/3$. Again $y$ has to be the integer part of $7/3$ so $y = 2$ and $z = 3$.
  2. As in the first part, if $N = 8/5$, then we must have $x = 1$ and $y + 1/z = 5/3$.

    To make $y$ and $z$ positive integers we must have $1/z < 1$ and $y = 1$.

    It then follows that $1/z = 2/3$ so it is impossible to find positive integer values for $x$, $y$ and $z$.