# How do you find the square root of 23?

##### 1 Answer

#### Explanation:

As such it is not expressible in the form

We can find rational *approximations* as follows:

#23 = 5^2-2#

is in the form

The square root of a number of the form

#sqrt(n^2-2) = [(n-1); bar(1, (n-2), 1, (2n-2))]#

In our example

#sqrt(23) = [4; bar(1,3,1,8)] = 4+1/(1+1/(3+1/(1+1/(8+1/(1+1/(3+1/(1+...)))))))#

To use this to derive a good approximation for

#sqrt(23) ~~ [4;1,3,1,8,1,3,1] = 4+1/(1+1/(3+1/(1+1/(8+1/(1+1/(3+1/1)))))) = 1151/240 = 4.7958bar(3)#

With a calculator, we find:

#sqrt(23) ~~ 4.79583152#

So our approximation is not bad.