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
14 to 18
Challenge level
Problem
- Find all positive integers $x$, $y$ and $z$ such that: $$x +\cfrac{1}{y + \cfrac{1}{z}} = N = \frac{10}{7}$$
- 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.
Getting Started
What is the integer part of $N$?
Student Solutions
- 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 $z = 3$.
- 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 $z$.