---
id: 5900f3a51000cf542c50feb8
challengeType: 5
title: 'Problem 57: Square root convergents'
forumTopicId: 302168
---

## Description
<section id='description'>

It is possible to show that the square root of two can be expressed as an infinite continued fraction.

<div style='text-align: center;'>$\sqrt 2 =1+ \frac 1 {2+ \frac 1 {2 +\frac 1 {2+ \dots}}}$</div>

By expanding this for the first four iterations, we get:

$1 + \frac 1 2 = \frac  32 = 1.5$

$1 + \frac 1 {2 + \frac 1 2} = \frac 7 5 = 1.4$

$1 + \frac 1 {2 + \frac 1 {2+\frac 1 2}} = \frac {17}{12} = 1.41666 \dots$

$1 + \frac 1 {2 + \frac 1 {2+\frac 1 {2+\frac 1 2}}} = \frac {41}{29} = 1.41379 \dots$

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 denominator?

</section>

## Instructions
<section id='instructions'>

</section>

## Tests
<section id='tests'>

```yml
tests:
  - text: <code>squareRootConvergents()</code> should return a number.
    testString: assert(typeof squareRootConvergents() === 'number');
  - text: <code>squareRootConvergents()</code> should return 153.
    testString: assert.strictEqual(squareRootConvergents(), 153);

```

</section>

## Challenge Seed
<section id='challengeSeed'>

<div id='js-seed'>

```js
function squareRootConvergents() {

  return true;
}

squareRootConvergents();
```

</div>



</section>

## Solution
<section id='solution'>

```js
// solution required
```

</section>