# README
Square root convergents
It is possible to show that the square root of two can be expressed as an infinite continued fraction.
$$ \sqrt{2} = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \dots}}} $$
By expanding this for the first four iterations, we get:
$1 + \frac{1}{2} = \frac{3}{2} = 1.5$
$1 + \frac{1}{2 + \frac{1}{2}} = \frac{7}{5} = 1.4$
$1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2}}} = \frac{17}{12} = 1.41666\dots$
$1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2}}}} = \frac{41}{29} = 1.41379\dots$
The next three expansions are $\frac{99}{70}$, $\frac{239}{169}$, and $\frac{577}{408}$, but the eighth expansion, $\frac{1393}{985}$, is the first example where the number of digits in the numerator exceeds the number of digits in the denominator.
In the first one-thousand expansions, how many fractions contain a numerator with more digits than the denominator?