# How do you find the square root of 2?

Aug 30, 2016

Use a continued fraction to find rational approximations.

#### Explanation:

$\sqrt{2}$ is an irrational number, not expressible in the form $\frac{p}{q}$ for integers $p , q$.

We can find rational approximations in several ways. Here I show a method called continued fractions...

Consider the number $t = \sqrt{2} + 1$

Then:

$t = \sqrt{2} + 1$

$= 2 + \left(\sqrt{2} - 1\right)$

$= 2 + \frac{\left(\sqrt{2} - 1\right) \left(\sqrt{2} + 1\right)}{\sqrt{2} + 1}$

$= 2 + \frac{2 - 1}{\sqrt{2} + 1}$

$= 2 + \frac{1}{\sqrt{2} + 1}$

$= 2 + \frac{1}{t}$

Given that:

$t = 2 + \frac{1}{t}$

notice that we can substitute this expression for $t$ on the right hand side to find:

$t = 2 + \frac{1}{2 + \frac{1}{t}}$

and again:

$t = 2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{t}}}$

In fact:

$t = 2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \ldots}}}}}$

Now remember $t = \sqrt{2} + 1$, so we have:

$\sqrt{2} = 1 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2 + \ldots}}}}}$

This is called a continued fraction.

There is a shorter notation for a continued fraction using square brackets. Using this notation we can write:

sqrt(2) = [1;2,2,2,2,2,...] = [1;bar(2)]

To find a rational approximation for $\sqrt{2}$ we can truncate this continued fraction early.

For example:

sqrt(2) ~~ [1;2,2,2] = 1+1/(2+1/(2+1/2)) = 1+1/(2+2/5) = 1+5/12 = 17/12 ~~ 1.41bar(6)

For more accuracy, truncate a little later:

sqrt(2) ~~ [1;2,2,2,2,2] = 1+1/(2+1/(2+1/(2+1/(2+1/2)))) = 99/70 = 1.4bar(142857)

This is actually the same accuracy as an approximation to $\sqrt{2}$ as the ratio of the sides of a sheet of A4 ($297 \text{mm" xx 210"mm}$).

In fact $\sqrt{2}$ is closer to $1.41421356237$