How do you find the square root of 2?

Question

2247 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-08-30T21:33:04

Use a continued fraction to find rational approximations.

$sqrt(2)$ is an irrational number, not expressible in the form $p/q$ 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:

$t = sqrt(2)+1$

$= 2 + (sqrt(2)-1)$

$= 2 + ((sqrt(2)-1)(sqrt(2)+1))/(sqrt(2)+1)$

$= 2 + (2-1)/(sqrt(2)+1)$

$= 2 + 1/(sqrt(2)+1)$

$= 2 + 1/t$

Given that:

$t = 2 + 1/t$

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

$t = 2 + 1/(2+1/t)$

and again:

$t = 2 + 1/(2+1/(2+1/t))$

In fact:

$t = 2 + 1/(2+1/(2+1/(2+1/(2+1/(2+...)))))$

Now remember $t = sqrt(2) + 1$ , so we have:

$sqrt(2) = 1 + 1/(2+1/(2+1/(2+1/(2+1/(2+...)))))$

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 ( $297"mm" xx 210"mm"$ ).

In fact $sqrt(2)$ is closer to $1.41421356237$