How do you find the square root of 23?

Aug 31, 2016

$\sqrt{23} \approx \frac{1151}{240} = 4.7958 \overline{3}$

Explanation:

$23$ 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 $\frac{p}{q}$ 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:

$\sqrt{23} \approx 4.79583152$

So our approximation is not bad.