How do you find the square root of 23?

##sqrt(23) ~~ 1151/240 = 4.7958bar(3)##

##23## is a prime number, so it is not possible to simplify its square root, which is an irrational number a little less than ##5 = sqrt(25)##

As such it is not expressible in the form ##p/q## for integers ##p, q##.

We can find rational approximations as follows:

##23 = 5^2-2##

is in the form ##n^2-2##

The square root of a number of the form ##n^2-2## can be expressed as a continued fraction of standard form:

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

In our example ##n=5## and we find:

##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)## terminate it early, just before one of the ##8##'s. For example:

##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.